[mlir][sparse] scalarize reductions in for-loops during sparse codegen

[aartbik]

Summary

Reductions in innermost loops become harder for the backend to disambiguate
after bufferization into memrefs, resulting in less efficient load-update-store
cycles. By scalarizing innermost reductions, the backend is more likely to assign
a register to perform the reduction (also prepares vectorization). Even though
we could scalarize reductions for more outer loops and while-loops as well,
currently scalarization is only done for chains of innermost for-loops, where
it matters most, to avoid complicating codegen unnecessary (viz. adding lots
of yield instructions).

This CL also refactors condition simplification into the merger class,
where it belongs, so that conditions are simplified only once per loop
nest and not repeatedly as was currently done. This CL also fixes a few
minor bugs, some layout issues, and comments.

[aartbik]

rebase

[aartbik]

trigger test

[aartbik]

rebase

[aartbik]

rebase

[penpornk]

I'm so sorry for the delay!

mlir/lib/Dialect/Linalg/Transforms/Sparsification.cpp

55

It's great that we are switching to enum. In the format extension paper there are 6 types [1]. :)

[1] http://tensor-compiler.org/chou-oopsla18-taco-formats.pdf -- page 123:7

103

Should "last tensor" be "2nd-to-last tensor" or "last non-artificial tensor"? (This sounds verbose but it technically isn't the last tensor anymore so we should be clear.)

104

Nit: Maybe add "at the end" after "non-existing tensor"?

157

What does /\_op mean? Do you mean to draw disjunction \/?

216

Should the quick check p0 != p1 be in latGT instead? Or do you want to avoid the function call? (It's small enough and is likely inlined though.)

[aartbik]

Thanks for the review Penporn!

mlir/lib/Dialect/Linalg/Transforms/Sparsification.cpp

55

Yes, the initial goal was to make "not-used" a bit more apparent (rather than sweeping them under the dense carpet ;-), but you are right this also prepares future extension!

103

Yes, last was relative to the given #, but I agree this could be phrased better. Reworded.

157

No, it really is conjunction operator. This may be confusing at first, but check Fred's paper, section 5.1

216

No, because I really want LT, and not LE in this case. I could add a latEQ if you think that is better, and then combine the two into a latLE?

[aartbik]

addressed comments

[penpornk]

mlir/lib/Dialect/Linalg/Transforms/Sparsification.cpp

216

I think this is fine. Thank you for the clarification! :)

mlir/lib/Dialect/Linalg/Transforms/Sparsification.cpp

 #include "mlir/Dialect/SCF/SCF.h"
 #include "mlir/Dialect/StandardOps/IR/Ops.h"
 
 using namespace mlir;
 
 namespace {
 
 enum class Kind { kTensor, kInvariant, kMulF, kMulI, kAddF, kAddI };
+enum class Dim { kSparse, kDense, kUndef };
 
 /// Tensor expression. Represents a MLIR expression in tensor index notation.
 /// For tensors, e0 denotes the tensor index. For invariants, the IR value is
 /// stored directly. For binary operations, e0 and e1 denote the index of the
 /// children tensor expressions.
 struct TensorExp {
   TensorExp(Kind k, unsigned x, unsigned y, Value v)
       : kind(k), e0(x), e1(y), val(v) {

 /// Lattice point. Each lattice point consists of a conjunction of tensor
 /// loop indices (encoded in a bitvector) and the index of the corresponding
 /// tensor expression.
 struct LatPoint {
   LatPoint(unsigned n, unsigned e, unsigned b) : bits(n, false), exp(e) {
     bits.set(b);
   }
   LatPoint(const llvm::BitVector &b, unsigned e) : bits(b), exp(e) {}
-  /// Conjunction of tensor loop indices as bitvector.
+  /// Conjunction of tensor loop indices as bitvector. This represents
+  /// all indices involved in the tensor expression
   llvm::BitVector bits;
+  /// Simplified conjunction of tensor loop indices as bitvector. This
+  /// represents a simplified condition under which this tensor expression
+  /// must execute. Pre-computed during codegen to avoid repeated eval.
+  llvm::BitVector simple;
   /// Index of the tensor expresssion.
   unsigned exp;
 };
 
 /// A class to handle all iteration lattice operations. This class abstracts
 /// away from some implementation details of storing iteration lattices and
 /// tensor expressions. This allows for fine-tuning performance characteristics
 /// independently from the basic algorithm if bottlenecks are identified.
 class Merger {
 public:
+  /// Constructs a merger for the given number of tensors and loops. The
+  /// user supplies the number of tensors involved in the kernel, with the
+  /// last tensor in this set denoting the output tensor. The merger adds an
+  /// additional synthetic tensor at the end of this set to represent all
+  /// invariant expressions in the kernel.
   Merger(unsigned t, unsigned l)
-      : numTensors(t), numLoops(l), isSparse(t, std::vector<bool>(l, false)) {}
+      : outTensor(t - 1), numTensors(t + 1), numLoops(l),
+        dims(t + 1, std::vector<Dim>(l, Dim::kUndef)) {}
 
   /// Adds a tensor expression. Returns its index.
   unsigned addExp(Kind k, unsigned e0, unsigned e1 = -1u, Value v = Value()) {
     unsigned e = tensorExps.size();
     tensorExps.push_back(TensorExp(k, e0, e1, v));
     return e;
   }
   unsigned addExp(Kind k, Value v) { return addExp(k, -1u, -1u, v); }
unsigned conjLatPoint(Kind kind, unsigned p0, unsigned p1) {
     unsigned p = latPoints.size();
     llvm::BitVector nb = llvm::BitVector(latPoints[p0].bits);
     nb |= latPoints[p1].bits;
     unsigned e = addExp(kind, latPoints[p0].exp, latPoints[p1].exp);
     latPoints.push_back(LatPoint(nb, e));
     return p;
   }
 
-  /// Conjunctive merge of L1 and L2 is conjunction of cartesian product.
-  /// Returns the index of the new set.
+  /// Conjunctive merge of two lattice sets L0 and L1 is conjunction of
+  /// cartesian product. Returns the index of the new set.
   unsigned takeConj(Kind kind, unsigned s0, unsigned s1) {
     unsigned s = addSet();
     for (unsigned p0 : latSets[s0])
       for (unsigned p1 : latSets[s1])
         latSets[s].push_back(conjLatPoint(kind, p0, p1));
     return s;
   }
 
-  /// Disjunctive merge of L0 and L1 is (L0 /\_op L1, L0, L1).
+  /// Disjunctive merge of two lattice sets L0 and L1 is (L0 /\_op L1, L0, L1).
   /// Returns the index of the new set.
   unsigned takeDisj(Kind kind, unsigned s0, unsigned s1) {
     unsigned s = takeConj(kind, s0, s1);
     for (unsigned p : latSets[s0])
       latSets[s].push_back(p);
     for (unsigned p : latSets[s1])
       latSets[s].push_back(p);
     return s;
   }
 
   /// Optimizes the iteration lattice points in the given set. This
   /// method should be called right before code generation to avoid
   /// generating redundant loops and conditions.
-  unsigned optimize(unsigned s0) {
+  unsigned optimizeSet(unsigned s0) {
     unsigned s = addSet();
     assert(latSets[s0].size() != 0);
     unsigned p0 = latSets[s0][0];
     for (unsigned p1 : latSets[s0]) {
       bool add = true;
+      llvm::BitVector simple = simplifyCond(s0, p1);
       if (p0 != p1) {
         // Is this a straightforward copy?
         unsigned e = latPoints[p1].exp;
-        if (exp(e).kind == Kind::kTensor && exp(e).e0 == numTensors - 1)
+        if (exp(e).kind == Kind::kTensor && exp(e).e0 == outTensor)
           continue;
-        // Is any dense index exhausted?
+        // Only dense exhausted?
         llvm::BitVector tmp = latPoints[p1].bits;
         tmp ^= latPoints[p0].bits;
-        if (hasAnyOf(tmp, false))
+        if (!hasAnyDimOf(tmp, Dim::kSparse))
           continue;
-        // Is this a direct duplication of an earlier conjunction?
+        // Duplication of an earlier conjunction?
         for (unsigned p2 : latSets[s]) {
-          tmp = latPoints[p1].bits;
-          tmp ^= latPoints[p2].bits;
+          tmp = simple;
+          tmp ^= latPoints[p2].simple;
           if (tmp.count() == 0) {
             add = false;
             break;
           }
         }
         assert(!add || latGT(p0, p1));
       }
-      if (add)
+      if (add) {
         latSets[s].push_back(p1);
+        latPoints[latSets[s].back()].simple = simple;
+      }
     }
     return s;
   }
 
-  // Returns true if Li > Lj.
+  /// Simplifies the conditions in a conjunction of a given lattice point
+  /// within the given set using just two basic rules:
+  /// (1) multiple dense conditions are reduced to single dense, and
+  /// (2) a *singleton* sparse/dense is reduced to sparse/random access.
+  llvm::BitVector simplifyCond(unsigned s, unsigned p0) {
+    // First determine if this lattice point is a *singleton*, i.e.,
+    // the last point in a lattice, no other is less than this one.
+    bool isSingleton = true;
+    for (unsigned p1 : latSets[s]) {
+      if (p0 != p1 && latGT(p0, p1)) {
+        unsigned e = latPoints[p1].exp;
+        if (exp(e).kind == Kind::kTensor && exp(e).e0 == outTensor)
+          continue;
+        llvm::BitVector tmp = latPoints[p1].bits;
+        tmp ^= latPoints[p0].bits;
+        if (hasAnyDimOf(tmp, Dim::kSparse)) {
+          isSingleton = false;
+          break;
+        }
+      }
+    }
+    // Now apply the two basic rules.
+    llvm::BitVector simple = latPoints[p0].bits;
+    bool reset = isSingleton && hasAnyDimOf(simple, Dim::kSparse);
+    for (unsigned b = 0, be = simple.size(); b < be; b++) {
+      if (simple[b] && !isDim(b, Dim::kSparse)) {
+        if (reset)
+          simple.reset(b);
+        reset = true;
+      }
+    }
+    return simple;
+  }
+
+  /// Returns true if Li > Lj.
   bool latGT(unsigned i, unsigned j) const {
     const llvm::BitVector &bitsi = latPoints[i].bits;
     const llvm::BitVector &bitsj = latPoints[j].bits;
     assert(bitsi.size() == bitsj.size());
     if (bitsi.count() > bitsj.count()) {
       for (unsigned b = 0, be = bitsj.size(); b < be; b++)
         if (bitsj[b] && !bitsi[b])
           return false;
       return true;
     }
     return false;
   }
 
-  // Bit translation.
+  /// Bit translation.
   unsigned tensor(unsigned b) const { return b % numTensors; }
   unsigned index(unsigned b) const { return b / numTensors; }
 
-  // Returns true if bit corresponds to sparse access.
-  bool isSparseBit(unsigned b) const {
-    return isSparseAccess(tensor(b), index(b));
-  }
+  /// Returns true if bit corresponds to queried dim.
+  bool isDim(unsigned b, Dim d) const { return isDim(tensor(b), index(b), d); }
 
-  // Returns true if tensor access at given index is sparse.
-  bool isSparseAccess(unsigned t, unsigned i) const {
+  /// Returns true if tensor access at given index has queried dim.
+  bool isDim(unsigned t, unsigned i, Dim d) const {
     assert(t < numTensors && i < numLoops);
-    return isSparse[t][i];
+    return dims[t][i] == d;
   }
 
-  // Returns true if any set bit corresponds to sparse/dense access.
-  bool hasAnyOf(const llvm::BitVector &bits, bool sparse) const {
+  /// Returns true if any set bit corresponds to queried dim.
+  bool hasAnyDimOf(const llvm::BitVector &bits, Dim d) const {
     for (unsigned b = 0, be = bits.size(); b < be; b++)
-      if (bits[b] && isSparseBit(b) == sparse)
+      if (bits[b] && isDim(b, d))
         return true;
     return false;
   }
 
-  // Getters.
-  std::vector<std::vector<bool>> &sparse() { return isSparse; }
+  // Setter
+  void setDim(unsigned t, unsigned i, Dim d) { dims[t][i] = d; }
+
+  /// Getters.
   TensorExp &exp(unsigned e) { return tensorExps[e]; }
   LatPoint &lat(unsigned l) { return latPoints[l]; }
   SmallVector<unsigned, 16> &set(unsigned s) { return latSets[s]; }
 
 private:
+  const unsigned outTensor;
   const unsigned numTensors;
   const unsigned numLoops;
 
-  std::vector<std::vector<bool>> isSparse;
+  std::vector<std::vector<Dim>> dims;
   llvm::SmallVector<TensorExp, 32> tensorExps;
   llvm::SmallVector<LatPoint, 16> latPoints;
   llvm::SmallVector<SmallVector<unsigned, 16>, 8> latSets;
 };
 
 // Code generation.
 struct CodeGen {
   CodeGen(linalg::SparsificationOptions o, unsigned numTensors,
           unsigned numLoops)
       : options(o), loops(numLoops), sizes(numLoops), buffers(numTensors),
         pointers(numTensors, std::vector<Value>(numLoops)),
         indices(numTensors, std::vector<Value>(numLoops)),
         highs(numTensors, std::vector<Value>(numLoops)),
         pidxs(numTensors, std::vector<Value>(numLoops)),
-        idxs(numTensors, std::vector<Value>(numLoops)) {}
-  // Sparsification options.
+        idxs(numTensors, std::vector<Value>(numLoops)), redExp(-1u), redVal() {}
+  /// Sparsification options.
   linalg::SparsificationOptions options;
-  // Universal dense indices and upper bounds (by index). The loops array
-  // is updated with the value of the universal dense index in the current
-  // loop. The sizes array is set once with the inferred dimension sizes.
+  /// Universal dense indices and upper bounds (by index). The loops array
+  /// is updated with the value of the universal dense index in the current
+  /// loop. The sizes array is set once with the inferred dimension sizes.
   std::vector<Value> loops;
   std::vector<Value> sizes;
-  // Buffers for storing dense and sparse numerical values (by tensor).
-  // This array is set once during bufferization of all tensors.
+  /// Buffers for storing dense and sparse numerical values (by tensor).
+  /// This array is set once during bufferization of all tensors.
   std::vector<Value> buffers;
-  // Sparse storage schemes (1-D): pointers and indices (by tensor and index).
-  // This array is set once during bufferization of all sparse tensors.
+  /// Sparse storage schemes (1-D): pointers and indices (by tensor and index).
+  /// This array is set once during bufferization of all sparse tensors.
   std::vector<std::vector<Value>> pointers;
   std::vector<std::vector<Value>> indices;
-  // Sparse iteration information (by tensor and index). These arrays
-  // are updated to remain current within the current loop.
+  /// Sparse iteration information (by tensor and index). These arrays
+  /// are updated to remain current within the current loop.
   std::vector<std::vector<Value>> highs;
   std::vector<std::vector<Value>> pidxs;
   std::vector<std::vector<Value>> idxs;
+  /// Current reduction, updated during code generation. When indices of a
+  /// reduction are exhausted,  all inner loops can "scalarize" the reduction.
+  // TODO: currently only done for (a chain of) innermost for-loops, where it
+  // is most effective; we could generalize to more outer and while-loops.
+  unsigned redExp;
+  Value redVal;
 };
 
 } // namespace
 
 /// Helper method to inspect sparse annotations in the linalg operation.
 /// Fills the per-dimension sparsity information for all tensors.
-static void findSparseAnnotations(linalg::GenericOp op,
-                                  std::vector<std::vector<bool>> &isSparse) {
+static void findSparseAnnotations(Merger &merger, linalg::GenericOp op) {
   unsigned numTensors = op.getNumInputsAndOutputs();
   ArrayAttr sparseAttr = op.sparseAttr();
   for (unsigned t = 0; t < numTensors; t++) {
     auto map = op.getIndexingMap(t);
     auto dimAttr = sparseAttr[t].cast<ArrayAttr>();
     // For each tensor, we accept a per-dimension Sparse or Dense annotation.
     // This is translated to the loop index that indexes that dimension.
     unsigned rank = op.getShapedType(t).getRank();
-    for (unsigned d = 0; d < rank; d++)
-      if (isSparseDim(dimAttr[d])) {
-        unsigned idx = map.getDimPosition(d);
-        isSparse[t][idx] = true;
+    for (unsigned d = 0; d < rank; d++) {
+      if (isSparseDim(dimAttr[d])) {
+        merger.setDim(t, idx, Dim::kSparse);
       } else {
         assert(isDenseDim(dimAttr[d]));
-      }
+        merger.setDim(t, idx, Dim::kDense);
+      }
   }
 }
 
 /// A DFS helper to compute a topological sort. Note that recursion is
 /// bounded by the number of implicit loops, which is always small.
 /// Returns false when a cycle is detected.
 static bool topSortDFS(unsigned i, std::vector<unsigned> &visit,
                        std::vector<unsigned> &topSort,

 
 /// Builds the iteration lattices in a bottom-up traversal given the remaining
 /// tensor (sub)expression and the next loop index in the iteration graph.
 static unsigned buildLattices(Merger &merger, linalg::GenericOp op,
                               unsigned exp, unsigned idx) {
   Kind kind = merger.exp(exp).kind;
   if (kind == Kind::kTensor || kind == Kind::kInvariant) {
     // Either the index is really used in the tensor expression, or it is
-    // set to the "non-existing dense index" in that dimension. Invariant
-    // expressions borrow the output tensor indices.
+    // set to the undefined index in that dimension. An invariant expression
+    // is set to a synthetic tensor with undefined indices only.
     unsigned s = merger.addSet();
     unsigned t = kind == Kind::kTensor ? merger.exp(exp).e0
-                                       : op.getNumInputsAndOutputs() - 1;
+                                       : op.getNumInputsAndOutputs();
     merger.set(s).push_back(merger.addLat(t, idx, exp));
     return s;
   }
   unsigned s0 = buildLattices(merger, op, merger.exp(exp).e0, idx);
   unsigned s1 = buildLattices(merger, op, merger.exp(exp).e1, idx);
   switch (kind) {
   case Kind::kTensor:
   case Kind::kInvariant:
for (unsigned t = 0; t < numTensors; t++) {
     auto shape = tensorType.getShape();
     auto map = op.getIndexingMap(t);
     // Scan all dimensions of current tensor.
     bool allDense = true;
     args.clear();
     for (unsigned d = 0, rank = shape.size(); d < rank; d++) {
       unsigned i = map.getDimPosition(d);
       // Handle sparse storage schemes.
-      if (merger.isSparseAccess(t, i)) {
+      if (merger.isDim(t, i, Dim::kSparse)) {
         allDense = false;
         auto dynShape = {ShapedType::kDynamicSize};
         auto ptrTp = MemRefType::get(
             dynShape, genIntType(rewriter, codegen.options.ptrType));
         auto indTp = MemRefType::get(
             dynShape, genIntType(rewriter, codegen.options.indType));
         codegen.pointers[t][i] = rewriter.create<AllocaOp>(loc, ptrTp, unknown);
         codegen.indices[t][i] = rewriter.create<AllocaOp>(loc, indTp, unknown);

 }
 
 /// Generates a load on a dense or sparse tensor.
 static Value genTensorLoad(Merger &merger, CodeGen &codegen,
                            PatternRewriter &rewriter, linalg::GenericOp op,
                            unsigned exp) {
   // Test if the load was hoisted to a higher loop nest.
   Value val = merger.exp(exp).val;
-  if (val) {
-    merger.exp(exp).val = Value(); // reset
+  if (val)
     return val;
-  }
   // Actual load.
   SmallVector<Value, 4> args;
   unsigned tensor = merger.exp(exp).e0;
   auto map = op.getIndexingMap(tensor);
   bool sparse = false;
   for (unsigned i = 0, m = map.getNumResults(); i < m; ++i) {
     unsigned idx = map.getDimPosition(i);
     args.push_back(codegen.loops[idx]); // universal dense index
-    if (sparse || merger.isSparseAccess(tensor, idx)) {
+    if (sparse || merger.isDim(tensor, idx, Dim::kSparse)) {
       sparse = true;
       args.clear();
       args.push_back(codegen.pidxs[tensor][idx]); // position index
     }
   }
   Location loc = op.getLoc();
   Value ptr = codegen.buffers[tensor];
   return rewriter.create<LoadOp>(loc, ptr, args);
 }
 
 /// Generates a store on a dense tensor.
 static void genTensorStore(Merger &merger, CodeGen &codegen,
                            PatternRewriter &rewriter, linalg::GenericOp op,
                            unsigned tensor, Value rhs) {
+  // Test if this is a scalarized reduction.
+  unsigned lhs = op.getNumInputsAndOutputs() - 1;
+  if (lhs == tensor && codegen.redVal) {
+    codegen.redVal = rhs;
+    return;
+  }
+  // Actual load.
   SmallVector<Value, 4> args;
   auto map = op.getIndexingMap(tensor);
   for (unsigned i = 0, m = map.getNumResults(); i < m; ++i) {
     unsigned idx = map.getDimPosition(i);
     args.push_back(codegen.loops[idx]); // universal dense index
   }
   Location loc = op.getLoc();
   Value ptr = codegen.buffers[tensor];
static Value genExp(Merger &merger, CodeGen &codegen, PatternRewriter &rewriter,
   case Kind::kAddI:
     return rewriter.create<AddIOp>(op.getLoc(), v0, v1);
   }
 }
 
 /// Hoists loop invariant tensor loads for which indices have been exhausted.
 static void genInvariants(Merger &merger, CodeGen &codegen,
                           PatternRewriter &rewriter, linalg::GenericOp op,
-                          unsigned exp) {
+                          unsigned exp, unsigned ldx, bool hoist) {
   if (merger.exp(exp).kind == Kind::kTensor) {
-    unsigned lhs = op.getNumInputsAndOutputs() - 1;
+    // Inspect tensor indices.
+    bool atLevel = ldx == -1u;
     unsigned tensor = merger.exp(exp).e0;
-    if (tensor == lhs)
-      return; // TODO: scalarize reduction as well (using scf.yield)
     auto map = op.getIndexingMap(tensor);
     for (unsigned i = 0, m = map.getNumResults(); i < m; ++i) {
       unsigned idx = map.getDimPosition(i);
       if (!codegen.loops[idx])
         return; // still in play
+      else if (idx == ldx)
+        atLevel = true;
+    }
+    // All exhausted at this level (atLevel denotes exactly at this level).
+    unsigned lhs = op.getNumInputsAndOutputs() - 1;
+    if (lhs == tensor) {
+      codegen.redExp = hoist ? exp : -1u;
+    } else if (atLevel) {
+      merger.exp(exp).val =
+          hoist ? genTensorLoad(merger, codegen, rewriter, op, exp) : Value();
     }
-    // All exhausted at this level.
-    merger.exp(exp).val = genTensorLoad(merger, codegen, rewriter, op, exp);
-
   } else if (merger.exp(exp).kind != Kind::kInvariant) {
     // Traverse into the binary operations. Note that we only hoist
     // tensor loads, since subsequent MLIR/LLVM passes know how to
     // deal with all other kinds of derived loop invariants.
-    genInvariants(merger, codegen, rewriter, op, merger.exp(exp).e0);
-    genInvariants(merger, codegen, rewriter, op, merger.exp(exp).e1);
+    unsigned e0 = merger.exp(exp).e0;
+    unsigned e1 = merger.exp(exp).e1;
+    genInvariants(merger, codegen, rewriter, op, e0, ldx, hoist);
+    genInvariants(merger, codegen, rewriter, op, e1, ldx, hoist);
   }
 }
 
 /// Generates initialization code for the subsequent loop sequence at
 /// current index level. Returns true if the loop sequence needs to
 /// maintain the universal index.
 static bool genInit(Merger &merger, CodeGen &codegen, PatternRewriter &rewriter,
                     linalg::GenericOp op, std::vector<unsigned> &topSort,
                     unsigned at, llvm::BitVector &inits) {
   bool needsUniv = false;
   Location loc = op.getLoc();
   unsigned idx = topSort[at];
 
   // Initialize sparse positions.
   for (unsigned b = 0, be = inits.size(); b < be; b++) {
     if (inits[b]) {
       unsigned tensor = merger.tensor(b);
       assert(idx == merger.index(b));
-      if (merger.isSparseBit(b)) {
+      if (merger.isDim(b, Dim::kSparse)) {
         // Initialize sparse index.
         unsigned pat = at;
         for (; pat != 0; pat--) {
           if (codegen.pidxs[tensor][topSort[pat - 1]])
             break;
         }
         Value ptr = codegen.pointers[tensor][idx];
         Value one = rewriter.create<ConstantIndexOp>(loc, 1);
static Operation *genFor(Merger &merger, CodeGen &codegen,
   unsigned fb = indices.find_first();
   unsigned tensor = merger.tensor(fb);
   assert(idx == merger.index(fb));
 
   // Parallelization strategy. Any implicit loop in the Linalg operation that
   // is marked "parallel" is a candidate. Whether it is actually converted to
   // a parallel operation depends on the requested strategy.
   auto iteratorTypes = op.iterator_types().getValue();
-  bool isSparse = merger.isSparseBit(fb);
+  bool isSparse = merger.isDim(fb, Dim::kSparse);
   bool isParallel = linalg::isParallelIteratorType(iteratorTypes[idx]);
   switch (codegen.options.parallelizationStrategy) {
   case linalg::SparseParallelizationStrategy::kNone:
     isParallel = false;
     break;
   case linalg::SparseParallelizationStrategy::kDenseOuterLoop:
     isParallel &= isOuter && !isSparse;
     break;
if (isParallel) {
     if (isSparse)
       codegen.pidxs[tensor][idx] = parOp.getInductionVars()[0];
     else
       codegen.loops[idx] = parOp.getInductionVars()[0];
     rewriter.setInsertionPointToStart(parOp.getBody());
     return parOp;
   }
 
-  // Emit a sequential loop.
-  scf::ForOp forOp = rewriter.create<scf::ForOp>(loc, lo, hi, step);
+  // Emit a sequential loop, potentially with a scalarized reduction.
+  bool scalarRed = isInner && codegen.redExp != -1u;
+  SmallVector<Value, 4> operands;
+  if (scalarRed) {
+    Value load =
+        codegen.redVal
+            ? codegen.redVal // chained with previous for-loop
+            : genTensorLoad(merger, codegen, rewriter, op, codegen.redExp);
+    operands.push_back(load);
+  }
+  scf::ForOp forOp = rewriter.create<scf::ForOp>(loc, lo, hi, step, operands);
+  if (scalarRed) {
+    codegen.redVal = merger.exp(codegen.redExp).val =
+        forOp.getRegionIterArgs().front();
+  }
+  // Assign induction variable to sparse or dense index.
   if (isSparse)
     codegen.pidxs[tensor][idx] = forOp.getInductionVar();
   else
     codegen.loops[idx] = forOp.getInductionVar();
   rewriter.setInsertionPointToStart(forOp.getBody());
   return forOp;
 }
 
 /// Emit a while-loop for co-iteration over multiple indices.
 static Operation *genWhile(Merger &merger, CodeGen &codegen,
                            PatternRewriter &rewriter, linalg::GenericOp op,
                            unsigned idx, bool needsUniv,
                            llvm::BitVector &indices) {
   SmallVector<Type, 4> types;
   SmallVector<Value, 4> operands;
   // Construct the while-loop with a parameter for each index.
   Type indexType = rewriter.getIndexType();
   for (unsigned b = 0, be = indices.size(); b < be; b++) {
-    if (indices[b] && merger.isSparseBit(b)) {
+    if (indices[b] && merger.isDim(b, Dim::kSparse)) {
       unsigned tensor = merger.tensor(b);
       assert(idx == merger.index(b));
       types.push_back(indexType);
       operands.push_back(codegen.pidxs[tensor][idx]);
     }
   }
   if (needsUniv) {
     types.push_back(indexType);
     operands.push_back(codegen.loops[idx]);
   }
   Location loc = op.getLoc();
   scf::WhileOp whileOp = rewriter.create<scf::WhileOp>(loc, types, operands);
   Block *before = rewriter.createBlock(&whileOp.before(), {}, types);
   Block *after = rewriter.createBlock(&whileOp.after(), {}, types);
 
   // Build the "before" region, which effectively consists
   // of a conjunction of "i < upper" tests on all induction.
   rewriter.setInsertionPointToStart(&whileOp.before().front());
   Value cond;
   unsigned o = 0;
   for (unsigned b = 0, be = indices.size(); b < be; b++) {
-    if (indices[b] && merger.isSparseBit(b)) {
+    if (indices[b] && merger.isDim(b, Dim::kSparse)) {
       unsigned tensor = merger.tensor(b);
       assert(idx == merger.index(b));
       Value op1 = before->getArgument(o);
       Value op2 = codegen.highs[tensor][idx];
       Value opc = rewriter.create<CmpIOp>(loc, CmpIPredicate::ult, op1, op2);
       cond = cond ? rewriter.create<AndOp>(loc, cond, opc) : opc;
       codegen.pidxs[tensor][idx] = after->getArgument(o++);
     }
static void genLocals(Merger &merger, CodeGen &codegen,
                       std::vector<unsigned> &topSort, unsigned at,
                       bool needsUniv, llvm::BitVector &locals) {
   Location loc = op.getLoc();
   unsigned idx = topSort[at];
 
   // Initialize sparse indices.
   Value min;
   for (unsigned b = 0, be = locals.size(); b < be; b++) {
-    if (locals[b] && merger.isSparseBit(b)) {
+    if (locals[b] && merger.isDim(b, Dim::kSparse)) {
       unsigned tensor = merger.tensor(b);
       assert(idx == merger.index(b));
       Value ptr = codegen.indices[tensor][idx];
       Value s = codegen.pidxs[tensor][idx];
       Value load = genLoad(rewriter, loc, ptr, s);
       codegen.idxs[tensor][idx] = load;
       if (!needsUniv) {
         if (min) {
static void genLocals(Merger &merger, CodeGen &codegen,
   // Merge dense universal index over minimum.
   if (min) {
     assert(!needsUniv);
     codegen.loops[idx] = min;
   }
 
   // Initialize dense positions.
   for (unsigned b = 0, be = locals.size(); b < be; b++) {
-    if (locals[b] && !merger.isSparseBit(b)) {
+    if (locals[b] && merger.isDim(b, Dim::kDense)) {
       unsigned tensor = merger.tensor(b);
       assert(idx == merger.index(b));
-      if (!codegen.highs[tensor][idx])
-        continue; // unused dimension
       unsigned pat = at;
       for (; pat != 0; pat--)
         if (codegen.pidxs[tensor][topSort[pat - 1]])
           break;
       Value p = (pat == 0) ? rewriter.create<ConstantIndexOp>(loc, 0)
                            : codegen.pidxs[tensor][topSort[pat - 1]];
       Value m = rewriter.create<MulIOp>(loc, codegen.sizes[idx], p);
       codegen.pidxs[tensor][idx] =
           rewriter.create<AddIOp>(loc, m, codegen.loops[idx]);
     }
   }
 }
 
 /// Generates the induction structure for a while-loop.
 static void genWhileInduction(Merger &merger, CodeGen &codegen,
                               PatternRewriter &rewriter, linalg::GenericOp op,
                               unsigned idx, bool needsUniv,
                               llvm::BitVector &induction, ResultRange results) {
   Location loc = op.getLoc();
   unsigned o = 0;
   SmallVector<Value, 4> operands;
   Value one = rewriter.create<ConstantIndexOp>(loc, 1);
-  for (unsigned b = 0, be = induction.size(); b < be; b++)
-    if (induction[b] && merger.isSparseBit(b)) {
+  for (unsigned b = 0, be = induction.size(); b < be; b++) {
+    if (induction[b] && merger.isDim(b, Dim::kSparse)) {
       unsigned tensor = merger.tensor(b);
       assert(idx == merger.index(b));
       Value op1 = codegen.idxs[tensor][idx];
       Value op2 = codegen.loops[idx];
       Value op3 = codegen.pidxs[tensor][idx];
       Value cmp = rewriter.create<CmpIOp>(loc, CmpIPredicate::eq, op1, op2);
       Value add = rewriter.create<AddIOp>(loc, op3, one);
       operands.push_back(rewriter.create<SelectOp>(loc, cmp, add, op3));
       codegen.pidxs[tensor][idx] = results[o++];
     }
+  }
   if (needsUniv) {
     operands.push_back(rewriter.create<AddIOp>(loc, codegen.loops[idx], one));
     codegen.loops[idx] = results[o++];
   }
   assert(o == operands.size());
   rewriter.create<scf::YieldOp>(loc, operands);
 }
 
 /// Generates a single if-statement within a while-loop.
-static void genIf(Merger &merger, CodeGen &codegen, PatternRewriter &rewriter,
-                  linalg::GenericOp op, unsigned idx,
-                  llvm::BitVector &conditions, scf::IfOp &ifOp) {
+static scf::IfOp genIf(Merger &merger, CodeGen &codegen,
+                       PatternRewriter &rewriter, linalg::GenericOp op,
+                       unsigned idx, llvm::BitVector &conditions) {
   Location loc = op.getLoc();
-  if (ifOp)
-    rewriter.setInsertionPointToStart(&ifOp.elseRegion().front());
   Value cond;
   for (unsigned b = 0, be = conditions.size(); b < be; b++) {
     if (conditions[b]) {
       unsigned tensor = merger.tensor(b);
       assert(idx == merger.index(b));
       Value clause;
-      if (merger.isSparseBit(b)) {
+      if (merger.isDim(b, Dim::kSparse)) {
         Value op1 = codegen.idxs[tensor][idx];
         Value op2 = codegen.loops[idx];
         clause = rewriter.create<CmpIOp>(loc, CmpIPredicate::eq, op1, op2);
       } else {
         clause = rewriter.create<ConstantIntOp>(loc, 1, 1); // true
       }
       cond = cond ? rewriter.create<AndOp>(loc, cond, clause) : clause;
     }
   }
-  ifOp = rewriter.create<scf::IfOp>(loc, cond, /*else*/ true);
+  scf::IfOp ifOp = rewriter.create<scf::IfOp>(loc, cond, /*else*/ true);
   rewriter.setInsertionPointToStart(&ifOp.thenRegion().front());
-}
-
-/// Optimize the loop indices of Li with two rules rules:
-/// (1) convert multiple dense to single dense, and
-/// (2) convert singleton sparse/dense to sparse/random access.
-static void optimizeIndices(Merger merger, unsigned lsize,
-                            llvm::BitVector &indices) {
-  if (merger.hasAnyOf(indices, false)) {
-    bool reset = lsize == 1 && merger.hasAnyOf(indices, true);
-    for (unsigned b = 0, be = indices.size(); b < be; b++) {
-      if (indices[b] && !merger.isSparseBit(b)) {
-        if (reset)
-          indices.reset(b);
-        reset = true;
-      }
-    }
-  }
+  return ifOp;
 }
 
 /// Recursively generates code while computing iteration lattices in order
 /// to manage the complexity of implementing co-iteration over unions
 /// and intersections of sparse iterations spaces.
 static void genStmt(Merger &merger, CodeGen &codegen, PatternRewriter &rewriter,
                     linalg::GenericOp op, std::vector<unsigned> &topSort,
                     unsigned exp, unsigned at) {
   // At each leaf, assign remaining tensor (sub)expression to output tensor.
   if (at == topSort.size()) {
     unsigned lhs = op.getNumInputsAndOutputs() - 1;
     Value rhs = genExp(merger, codegen, rewriter, op, exp);
     genTensorStore(merger, codegen, rewriter, op, lhs, rhs);
     return;
   }
 
   // Construct iteration lattices for current loop index, with L0 at top.
   // Then emit initialization code for the loop sequence at this level.
   // We maintain the universal dense index if dense indices are still
   // in play for a non-singleton loop sequence.
+  // Location loc = op.getLoc();
   unsigned idx = topSort[at];
-  unsigned lts = merger.optimize(buildLattices(merger, op, exp, idx));
+  unsigned lts = merger.optimizeSet(buildLattices(merger, op, exp, idx));
   unsigned lsize = merger.set(lts).size();
   assert(lsize != 0);
   unsigned l0 = merger.set(lts)[0];
-  LatPoint lat0 = merger.lat(l0);
-  genInvariants(merger, codegen, rewriter, op, exp);
-  bool needsUniv =
-      genInit(merger, codegen, rewriter, op, topSort, at, lat0.bits) &&
+  unsigned ldx = at == 0 ? -1u : topSort[at - 1];
+  genInvariants(merger, codegen, rewriter, op, exp, ldx, /*hoist=*/true);
+  bool needsUniv = genInit(merger, codegen, rewriter, op, topSort, at,
+                           merger.lat(l0).bits) &&
                    lsize > 1;
 
   // Emit a loop for every lattice point L0 >= Li.
-  for (unsigned li : merger.set(lts)) {
-    LatPoint lati = merger.lat(li);
+  for (unsigned i = 0; i < lsize; i++) {
+    unsigned li = merger.set(lts)[i];
 
     // Emit loop.
-    llvm::BitVector indices = lati.bits;
-    optimizeIndices(merger, lsize, indices);
+    llvm::BitVector indices = merger.lat(li).simple;
     Operation *loop =
         genLoop(merger, codegen, rewriter, op, topSort, at, needsUniv, indices);
-    genLocals(merger, codegen, rewriter, op, topSort, at, needsUniv, lati.bits);
+    genLocals(merger, codegen, rewriter, op, topSort, at, needsUniv,
+              merger.lat(li).bits);
 
     // Visit all lattices points with Li >= Lj to generate the
     // loop-body, possibly with if statements for coiteration.
     bool isWhile = dyn_cast<scf::WhileOp>(loop) != nullptr;
-    scf::IfOp ifOp;
-    for (unsigned lj : merger.set(lts)) {
+    for (unsigned j = 0; j < lsize; j++) {
+      unsigned lj = merger.set(lts)[j];
+      unsigned ej = merger.lat(lj).exp;
       if (li == lj || merger.latGT(li, lj)) {
-        LatPoint latj = merger.lat(lj);
-        llvm::BitVector tmp = latj.bits;
-        tmp ^= lati.bits;
-        if (merger.hasAnyOf(tmp, false))
-          continue; // dense exhausted within if/else
+        if (li != lj) {
+          llvm::BitVector tmp = merger.lat(lj).bits;
+          tmp ^= merger.lat(li).bits;
+          if (!merger.hasAnyDimOf(tmp, Dim::kSparse))
+            continue; // only dense exhausted within if/else
+        }
         // Recurse into body of each branch.
-        if (isWhile)
-          genIf(merger, codegen, rewriter, op, idx, latj.bits, ifOp);
-        genStmt(merger, codegen, rewriter, op, topSort, latj.exp, at + 1);
+        if (isWhile) {
+          scf::IfOp ifOp =
+              genIf(merger, codegen, rewriter, op, idx, merger.lat(lj).simple);
+          genStmt(merger, codegen, rewriter, op, topSort, ej, at + 1);
+          rewriter.setInsertionPointToStart(&ifOp.elseRegion().front());
+        } else {
+          genStmt(merger, codegen, rewriter, op, topSort, ej, at + 1);
+        }
       }
     }
 
     // Wrap-up induction and restore insertion point.
     if (isWhile) {
       scf::WhileOp whileOp = cast<scf::WhileOp>(loop);
       rewriter.setInsertionPointToEnd(&whileOp.after().front());
       genWhileInduction(merger, codegen, rewriter, op, idx, needsUniv,
-                        lati.bits, whileOp.results());
+                        merger.lat(li).bits, whileOp.results());
     } else {
       needsUniv = false;
+      if (codegen.redVal) {
+        rewriter.create<scf::YieldOp>(op.getLoc(), codegen.redVal);
+        codegen.redVal = loop->getResult(0);
+      }
     }
     rewriter.setInsertionPointAfter(loop);
   }
+
+  // Wrap-up loop sequence.
+  Value red = codegen.redVal;
+  if (red) {
+    codegen.redVal = merger.exp(codegen.redExp).val = Value(); // end chain
+    unsigned lhs = op.getNumInputsAndOutputs() - 1;
+    genTensorStore(merger, codegen, rewriter, op, lhs, red);
+  }
   codegen.loops[idx] = Value();
+  genInvariants(merger, codegen, rewriter, op, exp, ldx, /*hoist=*/false);
 }
 
 namespace {
 
 /// Sparse rewriting rule for generic Lingalg operation.
 struct GenericOpSparsifier : public OpRewritePattern<linalg::GenericOp> {
 public:
   GenericOpSparsifier(MLIRContext *context, linalg::SparsificationOptions o)
       : OpRewritePattern<linalg::GenericOp>(context), options(o) {}
 
   LogicalResult matchAndRewrite(linalg::GenericOp op,
                                 PatternRewriter &rewriter) const override {
     // Detects sparse annotations and translate the per-dimension sparsity
     // information for all tensors to loop indices in the kernel.
     if (!op.hasSparseSemantics())
       return failure();
     assert(op.getNumOutputs() == 1);
     unsigned numTensors = op.getNumInputsAndOutputs();
     unsigned numLoops = op.iterator_types().getValue().size();
     Merger merger(numTensors, numLoops);
-    findSparseAnnotations(op, merger.sparse());
+    findSparseAnnotations(merger, op);
 
     // Computes a topologically sorted iteration graph to ensure
     // tensors are visited in natural index order. Fails on cycles.
     // This assumes that higher-level passes have already put the
     // tensors in each tensor expression in a feasible order.
     // TODO: try again without *dense* constraints on failure or
     //       even try to insert sparse reorderings to resolve cycles
     std::vector<unsigned> topSort;

mlir/test/Dialect/Linalg/sparse_1d.mlir

%0 = linalg.generic #trait_ss
     ins(%arga, %argb: tensor<32xf32>, tensor<32xf32>) {
       ^bb(%a: f32, %b: f32):
         %0 = mulf %a, %b  : f32
         linalg.yield %0 : f32
   } -> tensor<32xf32>
   return %0 : tensor<32xf32>
 }
 
+#trait_two_way_inv = {
+  indexing_maps = [
+    affine_map<(i) -> (i)>, // a
+    affine_map<(i) -> (i)>, // b
+    affine_map<(i) -> (i)>  // x (out)
+  ],
+  sparse = [
+    [ "S" ], // a
+    [ "S" ], // b
+    [ "D" ]  // x
+  ],
+  iterator_types = ["parallel"],
+  doc = "x(i) = a(i) * c + b(i) * c"
+}
+
+// CHECK-LABEL:   func @two_way_inv(
+// CHECK-SAME:                      %[[VAL_0:.*0]]: tensor<16xf32>,
+// CHECK-SAME:                      %[[VAL_1:.*1]]: tensor<16xf32>,
+// CHECK-SAME:                      %[[VAL_2:.*2]]: f32) -> tensor<16xf32> {
+// CHECK:           %[[VAL_3:.*]] = constant 999 : index
+// CHECK:           %[[VAL_4:.*]] = constant 0 : index
+// CHECK:           %[[VAL_5:.*]] = constant 1 : index
+// CHECK:           %[[VAL_6:.*]] = alloca(%[[VAL_3]]) : memref<?xindex>
+// CHECK:           %[[VAL_7:.*]] = alloca(%[[VAL_3]]) : memref<?xindex>
+// CHECK:           %[[VAL_8:.*]] = alloca(%[[VAL_3]]) : memref<?xf32>
+// CHECK:           %[[VAL_9:.*]] = alloca(%[[VAL_3]]) : memref<?xindex>
+// CHECK:           %[[VAL_10:.*]] = alloca(%[[VAL_3]]) : memref<?xindex>
+// CHECK:           %[[VAL_11:.*]] = alloca(%[[VAL_3]]) : memref<?xf32>
+// CHECK:           %[[VAL_12:.*]] = alloca() : memref<16xf32>
+// CHECK:           %[[VAL_13:.*]] = load %[[VAL_6]]{{\[}}%[[VAL_4]]] : memref<?xindex>
+// CHECK:           %[[VAL_14:.*]] = load %[[VAL_6]]{{\[}}%[[VAL_5]]] : memref<?xindex>
+// CHECK:           %[[VAL_15:.*]] = load %[[VAL_9]]{{\[}}%[[VAL_4]]] : memref<?xindex>
+// CHECK:           %[[VAL_16:.*]] = load %[[VAL_9]]{{\[}}%[[VAL_5]]] : memref<?xindex>
+// CHECK:           %[[VAL_17:.*]]:3 = scf.while (%[[VAL_18:.*]] = %[[VAL_13]], %[[VAL_19:.*]] = %[[VAL_15]], %[[VAL_20:.*]] = %[[VAL_4]]) : (index, index, index) -> (index, index, index) {
+// CHECK:             %[[VAL_21:.*]] = cmpi "ult", %[[VAL_18]], %[[VAL_14]] : index
+// CHECK:             %[[VAL_22:.*]] = cmpi "ult", %[[VAL_19]], %[[VAL_16]] : index
+// CHECK:             %[[VAL_23:.*]] = and %[[VAL_21]], %[[VAL_22]] : i1
+// CHECK:             scf.condition(%[[VAL_23]]) %[[VAL_18]], %[[VAL_19]], %[[VAL_20]] : index, index, index
+// CHECK:           } do {
+// CHECK:           ^bb0(%[[VAL_24:.*]]: index, %[[VAL_25:.*]]: index, %[[VAL_26:.*]]: index):
+// CHECK:             %[[VAL_27:.*]] = load %[[VAL_7]]{{\[}}%[[VAL_24]]] : memref<?xindex>
+// CHECK:             %[[VAL_28:.*]] = load %[[VAL_10]]{{\[}}%[[VAL_25]]] : memref<?xindex>
+// CHECK:             %[[VAL_29:.*]] = cmpi "eq", %[[VAL_27]], %[[VAL_26]] : index
+// CHECK:             %[[VAL_30:.*]] = cmpi "eq", %[[VAL_28]], %[[VAL_26]] : index
+// CHECK:             %[[VAL_31:.*]] = and %[[VAL_29]], %[[VAL_30]] : i1
+// CHECK:             scf.if %[[VAL_31]] {
+// CHECK:               %[[VAL_32:.*]] = load %[[VAL_8]]{{\[}}%[[VAL_24]]] : memref<?xf32>
+// CHECK:               %[[VAL_33:.*]] = mulf %[[VAL_32]], %[[VAL_2]] : f32
+// CHECK:               %[[VAL_34:.*]] = load %[[VAL_11]]{{\[}}%[[VAL_25]]] : memref<?xf32>
+// CHECK:               %[[VAL_35:.*]] = mulf %[[VAL_34]], %[[VAL_2]] : f32
+// CHECK:               %[[VAL_36:.*]] = addf %[[VAL_33]], %[[VAL_35]] : f32
+// CHECK:               store %[[VAL_36]], %[[VAL_12]]{{\[}}%[[VAL_26]]] : memref<16xf32>
+// CHECK:             } else {
+// CHECK:               %[[VAL_37:.*]] = cmpi "eq", %[[VAL_27]], %[[VAL_26]] : index
+// CHECK:               scf.if %[[VAL_37]] {
+// CHECK:                 %[[VAL_38:.*]] = load %[[VAL_8]]{{\[}}%[[VAL_24]]] : memref<?xf32>
+// CHECK:                 %[[VAL_39:.*]] = mulf %[[VAL_38]], %[[VAL_2]] : f32
+// CHECK:                 store %[[VAL_39]], %[[VAL_12]]{{\[}}%[[VAL_26]]] : memref<16xf32>
+// CHECK:               } else {
+// CHECK:                 %[[VAL_40:.*]] = cmpi "eq", %[[VAL_28]], %[[VAL_26]] : index
+// CHECK:                 scf.if %[[VAL_40]] {
+// CHECK:                   %[[VAL_41:.*]] = load %[[VAL_11]]{{\[}}%[[VAL_25]]] : memref<?xf32>
+// CHECK:                   %[[VAL_42:.*]] = mulf %[[VAL_41]], %[[VAL_2]] : f32
+// CHECK:                   store %[[VAL_42]], %[[VAL_12]]{{\[}}%[[VAL_26]]] : memref<16xf32>
+// CHECK:                 } else {
+// CHECK:                 }
+// CHECK:               }
+// CHECK:             }
+// CHECK:             %[[VAL_43:.*]] = cmpi "eq", %[[VAL_27]], %[[VAL_26]] : index
+// CHECK:             %[[VAL_44:.*]] = addi %[[VAL_24]], %[[VAL_5]] : index
+// CHECK:             %[[VAL_45:.*]] = select %[[VAL_43]], %[[VAL_44]], %[[VAL_24]] : index
+// CHECK:             %[[VAL_46:.*]] = cmpi "eq", %[[VAL_28]], %[[VAL_26]] : index
+// CHECK:             %[[VAL_47:.*]] = addi %[[VAL_25]], %[[VAL_5]] : index
+// CHECK:             %[[VAL_48:.*]] = select %[[VAL_46]], %[[VAL_47]], %[[VAL_25]] : index
+// CHECK:             %[[VAL_49:.*]] = addi %[[VAL_26]], %[[VAL_5]] : index
+// CHECK:             scf.yield %[[VAL_45]], %[[VAL_48]], %[[VAL_49]] : index, index, index
+// CHECK:           }
+// CHECK:           scf.for %[[VAL_50:.*]] = %[[VAL_51:.*]]#0 to %[[VAL_14]] step %[[VAL_5]] {
+// CHECK:             %[[VAL_52:.*]] = load %[[VAL_8]]{{\[}}%[[VAL_50]]] : memref<?xf32>
+// CHECK:             %[[VAL_53:.*]] = mulf %[[VAL_52]], %[[VAL_2]] : f32
+// CHECK:             store %[[VAL_53]], %[[VAL_12]]{{\[}}%[[VAL_51]]#2] : memref<16xf32>
+// CHECK:           }
+// CHECK:           scf.for %[[VAL_54:.*]] = %[[VAL_55:.*]]#1 to %[[VAL_16]] step %[[VAL_5]] {
+// CHECK:             %[[VAL_56:.*]] = load %[[VAL_10]]{{\[}}%[[VAL_54]]] : memref<?xindex>
+// CHECK:             %[[VAL_57:.*]] = load %[[VAL_11]]{{\[}}%[[VAL_54]]] : memref<?xf32>
+// CHECK:             %[[VAL_58:.*]] = mulf %[[VAL_57]], %[[VAL_2]] : f32
+// CHECK:             store %[[VAL_58]], %[[VAL_12]]{{\[}}%[[VAL_56]]] : memref<16xf32>
+// CHECK:           }
+// CHECK:           %[[VAL_59:.*]] = tensor_load %[[VAL_12]] : memref<16xf32>
+// CHECK:           return %[[VAL_59]] : tensor<16xf32>
+// CHECK:         }
+func @two_way_inv(%arga: tensor<16xf32>,
+                  %argb: tensor<16xf32>, %argc: f32) -> tensor<16xf32> {
+  %0 = linalg.generic #trait_two_way_inv
+    ins(%arga, %argb : tensor<16xf32>, tensor<16xf32>) {
+      ^bb(%a : f32, %b : f32):
+        %0 = mulf %a, %argc : f32
+        %1 = mulf %b, %argc : f32
+        %2 = addf %0, %1 : f32
+        linalg.yield %2: f32
+  } -> tensor<16xf32>
+  return %0 : tensor<16xf32>
+}
+
+// CHECK-LABEL:   func @two_way_inv_alt(
+// CHECK-SAME:                          %[[VAL_0:.*0]]: tensor<16xf32>,
+// CHECK-SAME:                          %[[VAL_1:.*1]]: tensor<16xf32>,
+// CHECK-SAME:                          %[[VAL_2:.*2]]: f32) -> tensor<16xf32> {
+// CHECK:           %[[VAL_3:.*]] = constant 999 : index
+// CHECK:           %[[VAL_4:.*]] = constant 0 : index
+// CHECK:           %[[VAL_5:.*]] = constant 1 : index
+// CHECK:           %[[VAL_6:.*]] = alloca(%[[VAL_3]]) : memref<?xindex>
+// CHECK:           %[[VAL_7:.*]] = alloca(%[[VAL_3]]) : memref<?xindex>
+// CHECK:           %[[VAL_8:.*]] = alloca(%[[VAL_3]]) : memref<?xf32>
+// CHECK:           %[[VAL_9:.*]] = alloca(%[[VAL_3]]) : memref<?xindex>
+// CHECK:           %[[VAL_10:.*]] = alloca(%[[VAL_3]]) : memref<?xindex>
+// CHECK:           %[[VAL_11:.*]] = alloca(%[[VAL_3]]) : memref<?xf32>
+// CHECK:           %[[VAL_12:.*]] = alloca() : memref<16xf32>
+// CHECK:           %[[VAL_13:.*]] = load %[[VAL_6]]{{\[}}%[[VAL_4]]] : memref<?xindex>
+// CHECK:           %[[VAL_14:.*]] = load %[[VAL_6]]{{\[}}%[[VAL_5]]] : memref<?xindex>
+// CHECK:           %[[VAL_15:.*]] = load %[[VAL_9]]{{\[}}%[[VAL_4]]] : memref<?xindex>
+// CHECK:           %[[VAL_16:.*]] = load %[[VAL_9]]{{\[}}%[[VAL_5]]] : memref<?xindex>
+// CHECK:           %[[VAL_17:.*]]:3 = scf.while (%[[VAL_18:.*]] = %[[VAL_13]], %[[VAL_19:.*]] = %[[VAL_15]], %[[VAL_20:.*]] = %[[VAL_4]]) : (index, index, index) -> (index, index, index) {
+// CHECK:             %[[VAL_21:.*]] = cmpi "ult", %[[VAL_18]], %[[VAL_14]] : index
+// CHECK:             %[[VAL_22:.*]] = cmpi "ult", %[[VAL_19]], %[[VAL_16]] : index
+// CHECK:             %[[VAL_23:.*]] = and %[[VAL_21]], %[[VAL_22]] : i1
+// CHECK:             scf.condition(%[[VAL_23]]) %[[VAL_18]], %[[VAL_19]], %[[VAL_20]] : index, index, index
+// CHECK:           } do {
+// CHECK:           ^bb0(%[[VAL_24:.*]]: index, %[[VAL_25:.*]]: index, %[[VAL_26:.*]]: index):
+// CHECK:             %[[VAL_27:.*]] = load %[[VAL_7]]{{\[}}%[[VAL_24]]] : memref<?xindex>
+// CHECK:             %[[VAL_28:.*]] = load %[[VAL_10]]{{\[}}%[[VAL_25]]] : memref<?xindex>
+// CHECK:             %[[VAL_29:.*]] = cmpi "eq", %[[VAL_27]], %[[VAL_26]] : index
+// CHECK:             %[[VAL_30:.*]] = cmpi "eq", %[[VAL_28]], %[[VAL_26]] : index
+// CHECK:             %[[VAL_31:.*]] = and %[[VAL_29]], %[[VAL_30]] : i1
+// CHECK:             scf.if %[[VAL_31]] {
+// CHECK:               %[[VAL_32:.*]] = load %[[VAL_8]]{{\[}}%[[VAL_24]]] : memref<?xf32>
+// CHECK:               %[[VAL_33:.*]] = load %[[VAL_11]]{{\[}}%[[VAL_25]]] : memref<?xf32>
+// CHECK:               %[[VAL_34:.*]] = addf %[[VAL_32]], %[[VAL_33]] : f32
+// CHECK:               %[[VAL_35:.*]] = mulf %[[VAL_34]], %[[VAL_2]] : f32
+// CHECK:               store %[[VAL_35]], %[[VAL_12]]{{\[}}%[[VAL_26]]] : memref<16xf32>
+// CHECK:             } else {
+// CHECK:               %[[VAL_36:.*]] = cmpi "eq", %[[VAL_27]], %[[VAL_26]] : index
+// CHECK:               scf.if %[[VAL_36]] {
+// CHECK:                 %[[VAL_37:.*]] = load %[[VAL_8]]{{\[}}%[[VAL_24]]] : memref<?xf32>
+// CHECK:                 %[[VAL_38:.*]] = mulf %[[VAL_37]], %[[VAL_2]] : f32
+// CHECK:                 store %[[VAL_38]], %[[VAL_12]]{{\[}}%[[VAL_26]]] : memref<16xf32>
+// CHECK:               } else {
+// CHECK:                 %[[VAL_39:.*]] = cmpi "eq", %[[VAL_28]], %[[VAL_26]] : index
+// CHECK:                 scf.if %[[VAL_39]] {
+// CHECK:                   %[[VAL_40:.*]] = load %[[VAL_11]]{{\[}}%[[VAL_25]]] : memref<?xf32>
+// CHECK:                   %[[VAL_41:.*]] = mulf %[[VAL_40]], %[[VAL_2]] : f32
+// CHECK:                   store %[[VAL_41]], %[[VAL_12]]{{\[}}%[[VAL_26]]] : memref<16xf32>
+// CHECK:                 } else {
+// CHECK:                 }
+// CHECK:               }
+// CHECK:             }
+// CHECK:             %[[VAL_42:.*]] = cmpi "eq", %[[VAL_27]], %[[VAL_26]] : index
+// CHECK:             %[[VAL_43:.*]] = addi %[[VAL_24]], %[[VAL_5]] : index
+// CHECK:             %[[VAL_44:.*]] = select %[[VAL_42]], %[[VAL_43]], %[[VAL_24]] : index
+// CHECK:             %[[VAL_45:.*]] = cmpi "eq", %[[VAL_28]], %[[VAL_26]] : index
+// CHECK:             %[[VAL_46:.*]] = addi %[[VAL_25]], %[[VAL_5]] : index
+// CHECK:             %[[VAL_47:.*]] = select %[[VAL_45]], %[[VAL_46]], %[[VAL_25]] : index
+// CHECK:             %[[VAL_48:.*]] = addi %[[VAL_26]], %[[VAL_5]] : index
+// CHECK:             scf.yield %[[VAL_44]], %[[VAL_47]], %[[VAL_48]] : index, index, index
+// CHECK:           }
+// CHECK:           scf.for %[[VAL_49:.*]] = %[[VAL_50:.*]]#0 to %[[VAL_14]] step %[[VAL_5]] {
+// CHECK:             %[[VAL_51:.*]] = load %[[VAL_8]]{{\[}}%[[VAL_49]]] : memref<?xf32>
+// CHECK:             %[[VAL_52:.*]] = mulf %[[VAL_51]], %[[VAL_2]] : f32
+// CHECK:             store %[[VAL_52]], %[[VAL_12]]{{\[}}%[[VAL_50]]#2] : memref<16xf32>
+// CHECK:           }
+// CHECK:           scf.for %[[VAL_53:.*]] = %[[VAL_54:.*]]#1 to %[[VAL_16]] step %[[VAL_5]] {
+// CHECK:             %[[VAL_55:.*]] = load %[[VAL_10]]{{\[}}%[[VAL_53]]] : memref<?xindex>
+// CHECK:             %[[VAL_56:.*]] = load %[[VAL_11]]{{\[}}%[[VAL_53]]] : memref<?xf32>
+// CHECK:             %[[VAL_57:.*]] = mulf %[[VAL_56]], %[[VAL_2]] : f32
+// CHECK:             store %[[VAL_57]], %[[VAL_12]]{{\[}}%[[VAL_55]]] : memref<16xf32>
+// CHECK:           }
+// CHECK:           %[[VAL_58:.*]] = tensor_load %[[VAL_12]] : memref<16xf32>
+// CHECK:           return %[[VAL_58]] : tensor<16xf32>
+// CHECK:         }
+func @two_way_inv_alt(%arga: tensor<16xf32>,
+                      %argb: tensor<16xf32>, %argc: f32) -> tensor<16xf32> {
+  // Same kernel, but now expressed as "x(i) = (a(i) + b(i)) * c".
+  %0 = linalg.generic #trait_two_way_inv
+    ins(%arga, %argb : tensor<16xf32>, tensor<16xf32>) {
+      ^bb(%a : f32, %b : f32):
+        %0 = addf %a, %b : f32
+        %1 = mulf %0, %argc : f32
+        linalg.yield %1: f32
+  } -> tensor<16xf32>
+  return %0 : tensor<16xf32>
+}
+
 #trait_sum_reduction = {
   indexing_maps = [
     affine_map<(i) -> (i)>,  // a
     affine_map<(i) -> ()>    // x (scalar out)
   ],
   sparse = [
     [ "S" ],  // a
     [  ]      // x
   ],
   iterator_types = ["reduction"],
-  doc = "x = SUM_i a(i)"
+  doc = "x += SUM_i a(i)"
 }
 
 // CHECK-LABEL:   func @sum_reduction(
 // CHECK-SAME:                        %[[VAL_0:.*]]: tensor<?xf32>,
 // CHECK-SAME:                        %[[VAL_1:.*]]: tensor<f32>) -> tensor<f32> {
 // CHECK:           %[[VAL_2:.*]] = constant 999 : index
 // CHECK:           %[[VAL_3:.*]] = constant 0 : index
 // CHECK:           %[[VAL_4:.*]] = constant 1 : index
 // CHECK:           %[[VAL_5:.*]] = alloca(%[[VAL_2]]) : memref<?xindex>
 // CHECK:           %[[VAL_6:.*]] = alloca(%[[VAL_2]]) : memref<?xindex>
 // CHECK:           %[[VAL_7:.*]] = alloca(%[[VAL_2]]) : memref<?xf32>
 // CHECK:           %[[VAL_8:.*]] = alloca() : memref<f32>
 // CHECK:           %[[VAL_9:.*]] = load %[[VAL_5]]{{\[}}%[[VAL_3]]] : memref<?xindex>
 // CHECK:           %[[VAL_10:.*]] = load %[[VAL_5]]{{\[}}%[[VAL_4]]] : memref<?xindex>
-// CHECK:           scf.for %[[VAL_11:.*]] = %[[VAL_9]] to %[[VAL_10]] step %[[VAL_4]] {
-// CHECK:             %[[VAL_12:.*]] = load %[[VAL_8]][] : memref<f32>
-// CHECK:             %[[VAL_13:.*]] = load %[[VAL_7]]{{\[}}%[[VAL_11]]] : memref<?xf32>
-// CHECK:             %[[VAL_14:.*]] = addf %[[VAL_12]], %[[VAL_13]] : f32
-// CHECK:             store %[[VAL_14]], %[[VAL_8]][] : memref<f32>
-// CHECK:           }
-// CHECK:           %[[VAL_15:.*]] = tensor_load %[[VAL_8]] : memref<f32>
-// CHECK:           return %[[VAL_15]] : tensor<f32>
+// CHECK:           %[[VAL_11:.*]] = load %[[VAL_8]][] : memref<f32>
+// CHECK:           %[[VAL_12:.*]] = scf.for %[[VAL_13:.*]] = %[[VAL_9]] to %[[VAL_10]] step %[[VAL_4]] iter_args(%[[VAL_14:.*]] = %[[VAL_11]]) -> (f32) {
+// CHECK:             %[[VAL_15:.*]] = load %[[VAL_7]]{{\[}}%[[VAL_13]]] : memref<?xf32>
+// CHECK:             %[[VAL_16:.*]] = addf %[[VAL_14]], %[[VAL_15]] : f32
+// CHECK:             scf.yield %[[VAL_16]] : f32
+// CHECK:           }
+// CHECK:           store %[[VAL_17:.*]], %[[VAL_8]][] : memref<f32>
+// CHECK:           %[[VAL_18:.*]] = tensor_load %[[VAL_8]] : memref<f32>
+// CHECK:           return %[[VAL_18]] : tensor<f32>
 // CHECK:         }
 func @sum_reduction(%arga: tensor<?xf32>, %argx: tensor<f32>) -> tensor<f32> {
   %0 = linalg.generic #trait_sum_reduction
     ins(%arga : tensor<?xf32>)
     init(%argx : tensor<f32>) {
       ^bb(%a : f32, %x : f32):
         %0 = addf %x, %a  : f32
         linalg.yield %0: f32
   } -> tensor<f32>
   return %0 : tensor<f32>
 }
+
+#trait_sum_reduction_ss = {
+  indexing_maps = [
+    affine_map<(i) -> (i)>, // a
+    affine_map<(i) -> (i)>, // b
+    affine_map<(i)-> ()>    // x (scalar out)
+  ],
+  sparse = [
+    [ "S" ],  // a
+    [ "S" ],  // b
+    [     ]   // x
+  ],
+  iterator_types = ["reduction"],
+  doc = "x += SUM_i a(i) + b(i)"
+}
+
+// CHECK-LABEL:   func @sum_reduction_ss(
+// CHECK-SAME:                           %[[VAL_0:.*0]]: tensor<16xf32>,
+// CHECK-SAME:                           %[[VAL_1:.*1]]: tensor<16xf32>,
+// CHECK-SAME:                           %[[VAL_2:.*2]]: tensor<f32>) -> tensor<f32> {
+// CHECK:           %[[VAL_3:.*]] = constant 999 : index
+// CHECK:           %[[VAL_4:.*]] = constant 0 : index
+// CHECK:           %[[VAL_5:.*]] = constant 1 : index
+// CHECK:           %[[VAL_6:.*]] = alloca(%[[VAL_3]]) : memref<?xindex>
+// CHECK:           %[[VAL_7:.*]] = alloca(%[[VAL_3]]) : memref<?xindex>
+// CHECK:           %[[VAL_8:.*]] = alloca(%[[VAL_3]]) : memref<?xf32>
+// CHECK:           %[[VAL_9:.*]] = alloca(%[[VAL_3]]) : memref<?xindex>
+// CHECK:           %[[VAL_10:.*]] = alloca(%[[VAL_3]]) : memref<?xindex>
+// CHECK:           %[[VAL_11:.*]] = alloca(%[[VAL_3]]) : memref<?xf32>
+// CHECK:           %[[VAL_12:.*]] = alloca() : memref<f32>
+// CHECK:           %[[VAL_13:.*]] = load %[[VAL_6]]{{\[}}%[[VAL_4]]] : memref<?xindex>
+// CHECK:           %[[VAL_14:.*]] = load %[[VAL_6]]{{\[}}%[[VAL_5]]] : memref<?xindex>
+// CHECK:           %[[VAL_15:.*]] = load %[[VAL_9]]{{\[}}%[[VAL_4]]] : memref<?xindex>
+// CHECK:           %[[VAL_16:.*]] = load %[[VAL_9]]{{\[}}%[[VAL_5]]] : memref<?xindex>
+// CHECK:           %[[VAL_17:.*]]:3 = scf.while (%[[VAL_18:.*]] = %[[VAL_13]], %[[VAL_19:.*]] = %[[VAL_15]], %[[VAL_20:.*]] = %[[VAL_4]]) : (index, index, index) -> (index, index, index) {
+// CHECK:             %[[VAL_21:.*]] = cmpi "ult", %[[VAL_18]], %[[VAL_14]] : index
+// CHECK:             %[[VAL_22:.*]] = cmpi "ult", %[[VAL_19]], %[[VAL_16]] : index
+// CHECK:             %[[VAL_23:.*]] = and %[[VAL_21]], %[[VAL_22]] : i1
+// CHECK:             scf.condition(%[[VAL_23]]) %[[VAL_18]], %[[VAL_19]], %[[VAL_20]] : index, index, index
+// CHECK:           } do {
+// CHECK:           ^bb0(%[[VAL_24:.*]]: index, %[[VAL_25:.*]]: index, %[[VAL_26:.*]]: index):
+// CHECK:             %[[VAL_27:.*]] = load %[[VAL_7]]{{\[}}%[[VAL_24]]] : memref<?xindex>
+// CHECK:             %[[VAL_28:.*]] = load %[[VAL_10]]{{\[}}%[[VAL_25]]] : memref<?xindex>
+// CHECK:             %[[VAL_29:.*]] = cmpi "eq", %[[VAL_27]], %[[VAL_26]] : index
+// CHECK:             %[[VAL_30:.*]] = cmpi "eq", %[[VAL_28]], %[[VAL_26]] : index
+// CHECK:             %[[VAL_31:.*]] = and %[[VAL_29]], %[[VAL_30]] : i1
+// CHECK:             scf.if %[[VAL_31]] {
+// CHECK:               %[[VAL_32:.*]] = load %[[VAL_12]][] : memref<f32>
+// CHECK:               %[[VAL_33:.*]] = load %[[VAL_8]]{{\[}}%[[VAL_24]]] : memref<?xf32>
+// CHECK:               %[[VAL_34:.*]] = load %[[VAL_11]]{{\[}}%[[VAL_25]]] : memref<?xf32>
+// CHECK:               %[[VAL_35:.*]] = addf %[[VAL_33]], %[[VAL_34]] : f32
+// CHECK:               %[[VAL_36:.*]] = addf %[[VAL_32]], %[[VAL_35]] : f32
+// CHECK:               store %[[VAL_36]], %[[VAL_12]][] : memref<f32>
+// CHECK:             } else {
+// CHECK:               %[[VAL_37:.*]] = cmpi "eq", %[[VAL_27]], %[[VAL_26]] : index
+// CHECK:               scf.if %[[VAL_37]] {
+// CHECK:                 %[[VAL_38:.*]] = load %[[VAL_12]][] : memref<f32>
+// CHECK:                 %[[VAL_39:.*]] = load %[[VAL_8]]{{\[}}%[[VAL_24]]] : memref<?xf32>
+// CHECK:                 %[[VAL_40:.*]] = addf %[[VAL_38]], %[[VAL_39]] : f32
+// CHECK:                 store %[[VAL_40]], %[[VAL_12]][] : memref<f32>
+// CHECK:               } else {
+// CHECK:                 %[[VAL_41:.*]] = cmpi "eq", %[[VAL_28]], %[[VAL_26]] : index
+// CHECK:                 scf.if %[[VAL_41]] {
+// CHECK:                   %[[VAL_42:.*]] = load %[[VAL_12]][] : memref<f32>
+// CHECK:                   %[[VAL_43:.*]] = load %[[VAL_11]]{{\[}}%[[VAL_25]]] : memref<?xf32>
+// CHECK:                   %[[VAL_44:.*]] = addf %[[VAL_42]], %[[VAL_43]] : f32
+// CHECK:                   store %[[VAL_44]], %[[VAL_12]][] : memref<f32>
+// CHECK:                 } else {
+// CHECK:                 }
+// CHECK:               }
+// CHECK:             }
+// CHECK:             %[[VAL_45:.*]] = cmpi "eq", %[[VAL_27]], %[[VAL_26]] : index
+// CHECK:             %[[VAL_46:.*]] = addi %[[VAL_24]], %[[VAL_5]] : index
+// CHECK:             %[[VAL_47:.*]] = select %[[VAL_45]], %[[VAL_46]], %[[VAL_24]] : index
+// CHECK:             %[[VAL_48:.*]] = cmpi "eq", %[[VAL_28]], %[[VAL_26]] : index
+// CHECK:             %[[VAL_49:.*]] = addi %[[VAL_25]], %[[VAL_5]] : index
+// CHECK:             %[[VAL_50:.*]] = select %[[VAL_48]], %[[VAL_49]], %[[VAL_25]] : index
+// CHECK:             %[[VAL_51:.*]] = addi %[[VAL_26]], %[[VAL_5]] : index
+// CHECK:             scf.yield %[[VAL_47]], %[[VAL_50]], %[[VAL_51]] : index, index, index
+// CHECK:           }
+// CHECK:           %[[VAL_52:.*]] = load %[[VAL_12]][] : memref<f32>
+// CHECK:           %[[VAL_53:.*]] = scf.for %[[VAL_54:.*]] = %[[VAL_55:.*]]#0 to %[[VAL_14]] step %[[VAL_5]] iter_args(%[[VAL_56:.*]] = %[[VAL_52]]) -> (f32) {
+// CHECK:             %[[VAL_57:.*]] = load %[[VAL_8]]{{\[}}%[[VAL_54]]] : memref<?xf32>
+// CHECK:             %[[VAL_58:.*]] = addf %[[VAL_56]], %[[VAL_57]] : f32
+// CHECK:             scf.yield %[[VAL_58]] : f32
+// CHECK:           }
+// CHECK:           %[[VAL_59:.*]] = scf.for %[[VAL_60:.*]] = %[[VAL_61:.*]]#1 to %[[VAL_16]] step %[[VAL_5]] iter_args(%[[VAL_62:.*]] = %[[VAL_63:.*]]) -> (f32) {
+// CHECK:             %[[VAL_64:.*]] = load %[[VAL_11]]{{\[}}%[[VAL_60]]] : memref<?xf32>
+// CHECK:             %[[VAL_65:.*]] = addf %[[VAL_62]], %[[VAL_64]] : f32
+// CHECK:             scf.yield %[[VAL_65]] : f32
+// CHECK:           }
+// CHECK:           store %[[VAL_66:.*]], %[[VAL_12]][] : memref<f32>
+// CHECK:           %[[VAL_67:.*]] = tensor_load %[[VAL_12]] : memref<f32>
+// CHECK:           return %[[VAL_67]] : tensor<f32>
+// CHECK:         }
+func @sum_reduction_ss(%arga: tensor<16xf32>,
+                       %argb: tensor<16xf32>,
+                       %argx: tensor<f32>) -> tensor<f32> {
+  // Just for testing. This case would be better expressed
+  // as two separate reductions kernels.
+  %0 = linalg.generic #trait_sum_reduction_ss
+    ins(%arga, %argb: tensor<16xf32>, tensor<16xf32>)
+    init(%argx : tensor<f32>) {
+      ^bb(%a : f32, %b : f32, %x : f32):
+        %0 = addf %a, %b  : f32
+        %1 = addf %x, %0  : f32
+        linalg.yield %1: f32
+  } -> tensor<f32>
+  return %0 : tensor<f32>
+}
+
+#trait_sum_reduction_inv_ss = {
+  indexing_maps = [
+    affine_map<(i) -> (i)>, // a
+    affine_map<(i) -> ()>,  // b
+    affine_map<(i) -> (i)>, // c
+    affine_map<(i) -> ()>   // x (out)
+  ],
+  sparse = [
+    [ "S" ], // a
+    [     ], // b
+    [ "S" ], // c
+    [     ]  // x
+  ],
+  iterator_types = ["reduction"],
+  doc = "x += SUM_i a(i) * b + c(i)"
+}
+
+// CHECK-LABEL:   func @sum_reduction_inv(
+// CHECK-SAME:                            %[[VAL_0:.*0]]: tensor<16xf32>,
+// CHECK-SAME:                            %[[VAL_1:.*1]]: tensor<f32>,
+// CHECK-SAME:                            %[[VAL_2:.*2]]: tensor<16xf32>,
+// CHECK-SAME:                            %[[VAL_3:.*3]]: tensor<f32>) -> tensor<f32> {
+// CHECK:           %[[VAL_4:.*]] = constant 999 : index
+// CHECK:           %[[VAL_5:.*]] = constant 0 : index
+// CHECK:           %[[VAL_6:.*]] = constant 1 : index
+// CHECK:           %[[VAL_7:.*]] = alloca(%[[VAL_4]]) : memref<?xindex>
+// CHECK:           %[[VAL_8:.*]] = alloca(%[[VAL_4]]) : memref<?xindex>
+// CHECK:           %[[VAL_9:.*]] = alloca(%[[VAL_4]]) : memref<?xf32>
+// CHECK:           %[[VAL_10:.*]] = alloca() : memref<f32>
+// CHECK:           %[[VAL_11:.*]] = alloca(%[[VAL_4]]) : memref<?xindex>
+// CHECK:           %[[VAL_12:.*]] = alloca(%[[VAL_4]]) : memref<?xindex>
+// CHECK:           %[[VAL_13:.*]] = alloca(%[[VAL_4]]) : memref<?xf32>
+// CHECK:           %[[VAL_14:.*]] = alloca() : memref<f32>
+// CHECK:           %[[VAL_15:.*]] = load %[[VAL_10]][] : memref<f32>
+// CHECK:           %[[VAL_16:.*]] = load %[[VAL_7]]{{\[}}%[[VAL_5]]] : memref<?xindex>
+// CHECK:           %[[VAL_17:.*]] = load %[[VAL_7]]{{\[}}%[[VAL_6]]] : memref<?xindex>
+// CHECK:           %[[VAL_18:.*]] = load %[[VAL_11]]{{\[}}%[[VAL_5]]] : memref<?xindex>
+// CHECK:           %[[VAL_19:.*]] = load %[[VAL_11]]{{\[}}%[[VAL_6]]] : memref<?xindex>
+// CHECK:           %[[VAL_20:.*]]:3 = scf.while (%[[VAL_21:.*]] = %[[VAL_16]], %[[VAL_22:.*]] = %[[VAL_18]], %[[VAL_23:.*]] = %[[VAL_5]]) : (index, index, index) -> (index, index, index) {
+// CHECK:             %[[VAL_24:.*]] = cmpi "ult", %[[VAL_21]], %[[VAL_17]] : index
+// CHECK:             %[[VAL_25:.*]] = cmpi "ult", %[[VAL_22]], %[[VAL_19]] : index
+// CHECK:             %[[VAL_26:.*]] = and %[[VAL_24]], %[[VAL_25]] : i1
+// CHECK:             scf.condition(%[[VAL_26]]) %[[VAL_21]], %[[VAL_22]], %[[VAL_23]] : index, index, index
+// CHECK:           } do {
+// CHECK:           ^bb0(%[[VAL_27:.*]]: index, %[[VAL_28:.*]]: index, %[[VAL_29:.*]]: index):
+// CHECK:             %[[VAL_30:.*]] = load %[[VAL_8]]{{\[}}%[[VAL_27]]] : memref<?xindex>
+// CHECK:             %[[VAL_31:.*]] = load %[[VAL_12]]{{\[}}%[[VAL_28]]] : memref<?xindex>
+// CHECK:             %[[VAL_32:.*]] = cmpi "eq", %[[VAL_30]], %[[VAL_29]] : index
+// CHECK:             %[[VAL_33:.*]] = cmpi "eq", %[[VAL_31]], %[[VAL_29]] : index
+// CHECK:             %[[VAL_34:.*]] = and %[[VAL_32]], %[[VAL_33]] : i1
+// CHECK:             scf.if %[[VAL_34]] {
+// CHECK:               %[[VAL_35:.*]] = load %[[VAL_14]][] : memref<f32>
+// CHECK:               %[[VAL_36:.*]] = load %[[VAL_9]]{{\[}}%[[VAL_27]]] : memref<?xf32>
+// CHECK:               %[[VAL_37:.*]] = mulf %[[VAL_36]], %[[VAL_15]] : f32
+// CHECK:               %[[VAL_38:.*]] = load %[[VAL_13]]{{\[}}%[[VAL_28]]] : memref<?xf32>
+// CHECK:               %[[VAL_39:.*]] = addf %[[VAL_37]], %[[VAL_38]] : f32
+// CHECK:               %[[VAL_40:.*]] = addf %[[VAL_35]], %[[VAL_39]] : f32
+// CHECK:               store %[[VAL_40]], %[[VAL_14]][] : memref<f32>
+// CHECK:             } else {
+// CHECK:               %[[VAL_41:.*]] = cmpi "eq", %[[VAL_30]], %[[VAL_29]] : index
+// CHECK:               scf.if %[[VAL_41]] {
+// CHECK:                 %[[VAL_42:.*]] = load %[[VAL_14]][] : memref<f32>
+// CHECK:                 %[[VAL_43:.*]] = load %[[VAL_9]]{{\[}}%[[VAL_27]]] : memref<?xf32>
+// CHECK:                 %[[VAL_44:.*]] = mulf %[[VAL_43]], %[[VAL_15]] : f32
+// CHECK:                 %[[VAL_45:.*]] = addf %[[VAL_42]], %[[VAL_44]] : f32
+// CHECK:                 store %[[VAL_45]], %[[VAL_14]][] : memref<f32>
+// CHECK:               } else {
+// CHECK:                 %[[VAL_46:.*]] = cmpi "eq", %[[VAL_31]], %[[VAL_29]] : index
+// CHECK:                 scf.if %[[VAL_46]] {
+// CHECK:                   %[[VAL_47:.*]] = load %[[VAL_14]][] : memref<f32>
+// CHECK:                   %[[VAL_48:.*]] = load %[[VAL_13]]{{\[}}%[[VAL_28]]] : memref<?xf32>
+// CHECK:                   %[[VAL_49:.*]] = addf %[[VAL_47]], %[[VAL_48]] : f32
+// CHECK:                   store %[[VAL_49]], %[[VAL_14]][] : memref<f32>
+// CHECK:                 } else {
+// CHECK:                 }
+// CHECK:               }
+// CHECK:             }
+// CHECK:             %[[VAL_50:.*]] = cmpi "eq", %[[VAL_30]], %[[VAL_29]] : index
+// CHECK:             %[[VAL_51:.*]] = addi %[[VAL_27]], %[[VAL_6]] : index
+// CHECK:             %[[VAL_52:.*]] = select %[[VAL_50]], %[[VAL_51]], %[[VAL_27]] : index
+// CHECK:             %[[VAL_53:.*]] = cmpi "eq", %[[VAL_31]], %[[VAL_29]] : index
+// CHECK:             %[[VAL_54:.*]] = addi %[[VAL_28]], %[[VAL_6]] : index
+// CHECK:             %[[VAL_55:.*]] = select %[[VAL_53]], %[[VAL_54]], %[[VAL_28]] : index
+// CHECK:             %[[VAL_56:.*]] = addi %[[VAL_29]], %[[VAL_6]] : index
+// CHECK:             scf.yield %[[VAL_52]], %[[VAL_55]], %[[VAL_56]] : index, index, index
+// CHECK:           }
+// CHECK:           %[[VAL_57:.*]] = load %[[VAL_14]][] : memref<f32>
+// CHECK:           %[[VAL_58:.*]] = scf.for %[[VAL_59:.*]] = %[[VAL_60:.*]]#0 to %[[VAL_17]] step %[[VAL_6]] iter_args(%[[VAL_61:.*]] = %[[VAL_57]]) -> (f32) {
+// CHECK:             %[[VAL_62:.*]] = load %[[VAL_9]]{{\[}}%[[VAL_59]]] : memref<?xf32>
+// CHECK:             %[[VAL_63:.*]] = mulf %[[VAL_62]], %[[VAL_15]] : f32
+// CHECK:             %[[VAL_64:.*]] = addf %[[VAL_61]], %[[VAL_63]] : f32
+// CHECK:             scf.yield %[[VAL_64]] : f32
+// CHECK:           }
+// CHECK:           %[[VAL_65:.*]] = scf.for %[[VAL_66:.*]] = %[[VAL_67:.*]]#1 to %[[VAL_19]] step %[[VAL_6]] iter_args(%[[VAL_68:.*]] = %[[VAL_69:.*]]) -> (f32) {
+// CHECK:             %[[VAL_70:.*]] = load %[[VAL_13]]{{\[}}%[[VAL_66]]] : memref<?xf32>
+// CHECK:             %[[VAL_71:.*]] = addf %[[VAL_68]], %[[VAL_70]] : f32
+// CHECK:             scf.yield %[[VAL_71]] : f32
+// CHECK:           }
+// CHECK:           store %[[VAL_72:.*]], %[[VAL_14]][] : memref<f32>
+// CHECK:           %[[VAL_73:.*]] = tensor_load %[[VAL_14]] : memref<f32>
+// CHECK:           return %[[VAL_73]] : tensor<f32>
+// CHECK:         }
+func @sum_reduction_inv(%arga: tensor<16xf32>,
+                        %argb: tensor<f32>,
+                        %argc: tensor<16xf32>,
+                        %argx: tensor<f32>) -> tensor<f32> {
+  // Just for testing. This case would be better expressed
+  // as two separate reductions kernels.
+  %0 = linalg.generic #trait_sum_reduction_inv_ss
+    ins(%arga, %argb, %argc : tensor<16xf32>, tensor<f32>, tensor<16xf32>)
+    init(%argx : tensor<f32>) {
+      ^bb(%a : f32, %b : f32, %c : f32, %x : f32):
+        %0 = mulf %a, %b  : f32
+        %1 = addf %0, %c  : f32
+        %2 = addf %x, %1  : f32
+        linalg.yield %2: f32
+  } -> tensor<f32>
+  return %0 : tensor<f32>
+}

mlir/test/Dialect/Linalg/sparse_2d.mlir

indexing_maps = [
     affine_map<(i,j) -> (i)>     // x (out)
   ],
   sparse = [
     [ "D", "S" ],  // A
     [ "D" ],       // b
     [ "D" ]        // x
   ],
   iterator_types = ["parallel", "reduction"],
-  doc = "x(i) += A(i,j) * b(j)"
+  doc = "x(i) += SUM_j A(i,j) * b(j)"
 }
 
 // CHECK-LABEL:   func @matvec(
 // CHECK-SAME:                 %[[VAL_0:.*]]: tensor<16x32xf32>,
 // CHECK-SAME:                 %[[VAL_1:.*]]: tensor<32xf32>,
 // CHECK-SAME:                 %[[VAL_2:.*]]: tensor<16xf32>) -> tensor<16xf32> {
 // CHECK:           %[[VAL_3:.*]] = constant 999 : index
 // CHECK:           %[[VAL_4:.*]] = constant 16 : index
 // CHECK:           %[[VAL_5:.*]] = constant 0 : index
 // CHECK:           %[[VAL_6:.*]] = constant 1 : index
 // CHECK:           %[[VAL_7:.*]] = alloca(%[[VAL_3]]) : memref<?xindex>
 // CHECK:           %[[VAL_8:.*]] = alloca(%[[VAL_3]]) : memref<?xindex>
 // CHECK:           %[[VAL_9:.*]] = alloca(%[[VAL_3]]) : memref<?xf32>
 // CHECK:           %[[VAL_10:.*]] = alloca() : memref<32xf32>
 // CHECK:           %[[VAL_11:.*]] = alloca() : memref<16xf32>
 // CHECK:           scf.for %[[VAL_12:.*]] = %[[VAL_5]] to %[[VAL_4]] step %[[VAL_6]] {
 // CHECK:             %[[VAL_13:.*]] = load %[[VAL_7]]{{\[}}%[[VAL_12]]] : memref<?xindex>
 // CHECK:             %[[VAL_14:.*]] = addi %[[VAL_12]], %[[VAL_6]] : index
 // CHECK:             %[[VAL_15:.*]] = load %[[VAL_7]]{{\[}}%[[VAL_14]]] : memref<?xindex>
-// CHECK:             scf.for %[[VAL_16:.*]] = %[[VAL_13]] to %[[VAL_15]] step %[[VAL_6]] {
-// CHECK:               %[[VAL_17:.*]] = load %[[VAL_8]]{{\[}}%[[VAL_16]]] : memref<?xindex>
-// CHECK:               %[[VAL_18:.*]] = load %[[VAL_9]]{{\[}}%[[VAL_16]]] : memref<?xf32>
-// CHECK:               %[[VAL_19:.*]] = load %[[VAL_10]]{{\[}}%[[VAL_17]]] : memref<32xf32>
-// CHECK:               %[[VAL_20:.*]] = mulf %[[VAL_18]], %[[VAL_19]] : f32
-// CHECK:               %[[VAL_21:.*]] = load %[[VAL_11]]{{\[}}%[[VAL_12]]] : memref<16xf32>
-// CHECK:               %[[VAL_22:.*]] = addf %[[VAL_20]], %[[VAL_21]] : f32
-// CHECK:               store %[[VAL_22]], %[[VAL_11]]{{\[}}%[[VAL_12]]] : memref<16xf32>
+// CHECK:             %[[VAL_16:.*]] = load %[[VAL_11]]{{\[}}%[[VAL_12]]] : memref<16xf32>
+// CHECK:             %[[VAL_17:.*]] = scf.for %[[VAL_18:.*]] = %[[VAL_13]] to %[[VAL_15]] step %[[VAL_6]] iter_args(%[[VAL_19:.*]] = %[[VAL_16]]) -> (f32) {
+// CHECK:               %[[VAL_20:.*]] = load %[[VAL_8]]{{\[}}%[[VAL_18]]] : memref<?xindex>
+// CHECK:               %[[VAL_21:.*]] = load %[[VAL_9]]{{\[}}%[[VAL_18]]] : memref<?xf32>
+// CHECK:               %[[VAL_22:.*]] = load %[[VAL_10]]{{\[}}%[[VAL_20]]] : memref<32xf32>
+// CHECK:               %[[VAL_23:.*]] = mulf %[[VAL_21]], %[[VAL_22]] : f32
+// CHECK:               %[[VAL_24:.*]] = addf %[[VAL_23]], %[[VAL_19]] : f32
+// CHECK:               scf.yield %[[VAL_24]] : f32
 // CHECK:             }
+// CHECK:             store %[[VAL_25:.*]], %[[VAL_11]]{{\[}}%[[VAL_12]]] : memref<16xf32>
 // CHECK:           }
-// CHECK:           %[[VAL_23:.*]] = tensor_load %[[VAL_11]] : memref<16xf32>
-// CHECK:           return %[[VAL_23]] : tensor<16xf32>
+// CHECK:           %[[VAL_26:.*]] = tensor_load %[[VAL_11]] : memref<16xf32>
+// CHECK:           return %[[VAL_26]] : tensor<16xf32>
 // CHECK:         }
 func @matvec(%argA: tensor<16x32xf32>, %argb: tensor<32xf32>, %argx: tensor<16xf32>) -> tensor<16xf32> {
   %0 = linalg.generic #trait_matvec
       ins(%argA, %argb : tensor<16x32xf32>, tensor<32xf32>)
       init(%argx : tensor<16xf32>) {
     ^bb(%A: f32, %b: f32, %x: f32):
       %0 = mulf %A, %b : f32
       %1 = addf %0, %x : f32
       linalg.yield %1 : f32
   } -> tensor<16xf32>
   return %0 : tensor<16xf32>
 }
 
 #trait_sum_reduction = {
   indexing_maps = [
-    affine_map<(i,j) -> (i,j)>,  // a
-    affine_map<(i,j) -> ()>      // x (scalar out)
+    affine_map<(i,j) -> (i,j)>, // A
+    affine_map<(i,j) -> ()>     // x (scalar out)
   ],
   sparse = [
-    [ "D","S" ],  // a
+    [ "D", "S" ], // A
     [ ]           // x
   ],
   iterator_types = ["reduction", "reduction"],
-  doc = "x = SUM_ij a(i,j)"
+  doc = "x += SUM_ij A(i,j)"
 }
 
 // CHECK-LABEL:   func @sum_reduction(
-// CHECK-SAME:                        %[[VAL_0:.*0]]: tensor<10x20xf32>,
-// CHECK-SAME:                        %[[VAL_1:.*1]]: tensor<f32>) -> tensor<f32> {
+// CHECK-SAME:                        %[[VAL_0:.*]]: tensor<10x20xf32>,
+// CHECK-SAME:                        %[[VAL_1:.*]]: tensor<f32>) -> tensor<f32> {
 // CHECK:           %[[VAL_2:.*]] = constant 999 : index
 // CHECK:           %[[VAL_3:.*]] = constant 10 : index
 // CHECK:           %[[VAL_4:.*]] = constant 0 : index
 // CHECK:           %[[VAL_5:.*]] = constant 1 : index
 // CHECK:           %[[VAL_6:.*]] = alloca(%[[VAL_2]]) : memref<?xindex>
 // CHECK:           %[[VAL_7:.*]] = alloca(%[[VAL_2]]) : memref<?xindex>
 // CHECK:           %[[VAL_8:.*]] = alloca(%[[VAL_2]]) : memref<?xf32>
 // CHECK:           %[[VAL_9:.*]] = alloca() : memref<f32>
 // CHECK:           scf.for %[[VAL_10:.*]] = %[[VAL_4]] to %[[VAL_3]] step %[[VAL_5]] {
 // CHECK:             %[[VAL_11:.*]] = load %[[VAL_6]]{{\[}}%[[VAL_10]]] : memref<?xindex>
 // CHECK:             %[[VAL_12:.*]] = addi %[[VAL_10]], %[[VAL_5]] : index
 // CHECK:             %[[VAL_13:.*]] = load %[[VAL_6]]{{\[}}%[[VAL_12]]] : memref<?xindex>
-// CHECK:             scf.for %[[VAL_14:.*]] = %[[VAL_11]] to %[[VAL_13]] step %[[VAL_5]] {
-// CHECK:               %[[VAL_15:.*]] = load %[[VAL_9]][] : memref<f32>
-// CHECK:               %[[VAL_16:.*]] = load %[[VAL_8]]{{\[}}%[[VAL_14]]] : memref<?xf32>
-// CHECK:               %[[VAL_17:.*]] = addf %[[VAL_15]], %[[VAL_16]] : f32
-// CHECK:               store %[[VAL_17]], %[[VAL_9]][] : memref<f32>
+// CHECK:             %[[VAL_14:.*]] = load %[[VAL_9]][] : memref<f32>
+// CHECK:             %[[VAL_15:.*]] = scf.for %[[VAL_16:.*]] = %[[VAL_11]] to %[[VAL_13]] step %[[VAL_5]] iter_args(%[[VAL_17:.*]] = %[[VAL_14]]) -> (f32) {
+// CHECK:               %[[VAL_18:.*]] = load %[[VAL_8]]{{\[}}%[[VAL_16]]] : memref<?xf32>
+// CHECK:               %[[VAL_19:.*]] = addf %[[VAL_17]], %[[VAL_18]] : f32
+// CHECK:               scf.yield %[[VAL_19]] : f32
 // CHECK:             }
+// CHECK:             store %[[VAL_20:.*]], %[[VAL_9]][] : memref<f32>
 // CHECK:           }
-// CHECK:           %[[VAL_18:.*]] = tensor_load %[[VAL_9]] : memref<f32>
-// CHECK:           return %[[VAL_18]] : tensor<f32>
+// CHECK:           %[[VAL_21:.*]] = tensor_load %[[VAL_9]] : memref<f32>
+// CHECK:           return %[[VAL_21]] : tensor<f32>
 // CHECK:         }
 func @sum_reduction(%arga: tensor<10x20xf32>, %argx: tensor<f32>) -> tensor<f32> {
   %0 = linalg.generic #trait_sum_reduction
     ins(%arga : tensor<10x20xf32>)
     init(%argx : tensor<f32>) {
       ^bb(%a : f32, %x : f32):
         %0 = addf %x, %a  : f32
         linalg.yield %0: f32
#trait_sampled_dense_dense = {
   ],
   sparse = [
     [ "S", "S" ],  // S
     [ "D", "D" ],  // A
     [ "D", "D" ],  // B
     [ "D", "D" ]   // X
   ],
   iterator_types = ["parallel", "parallel", "reduction"],
-  doc = "X(i,j) = S(i,j) SUM_k A(i,k) B(k,j)"
+  doc = "X(i,j) += S(i,j) SUM_k A(i,k) B(k,j)"
 }
 
 // CHECK-LABEL:   func @sampled_dense_dense(
 // CHECK-SAME:                              %[[VAL_0:.*0]]: tensor<?x?xf32>,
 // CHECK-SAME:                              %[[VAL_1:.*1]]: tensor<?x?xf32>,
 // CHECK-SAME:                              %[[VAL_2:.*2]]: tensor<?x?xf32>,
 // CHECK-SAME:                              %[[VAL_3:.*3]]: tensor<?x?xf32>) -> tensor<?x?xf32> {
 // CHECK:           %[[VAL_4:.*]] = constant 999 : index
init(%argx : tensor<?x?xf32>) {
       ^bb(%s : f32, %a : f32, %b : f32, %x : f32):
         %0 = mulf %a, %b  : f32
         %1 = mulf %s, %0  : f32
         %2 = addf %x, %1  : f32
         linalg.yield %2: f32
   } -> tensor<?x?xf32>
   return %0 : tensor<?x?xf32>
 }
+
+#trait_sum_kernel_with_inv = {
+  indexing_maps = [
+    affine_map<(i,j) -> (i,j)>,  // A
+    affine_map<(i,j) -> (i,j)>,  // B
+    affine_map<(i,j) -> (i,j)>,  // C
+    affine_map<(i,j) -> (i)>,    // d
+    affine_map<(i,j) -> ()>,     // e
+    affine_map<(i,j) -> (i)>     // x (out)
+  ],
+  sparse = [
+    [ "S", "S" ], // A
+    [ "D", "S" ], // B
+    [ "D", "S" ], // C
+    [ "D"  ],     // d
+    [      ],     // e
+    [ "D"  ]      // x
+  ],
+  iterator_types = ["parallel", "reduction"],
+  doc = "x(i) = SUM_j A(i,j) * B(i,j) * d(i) * e + C(i,j)"
+}
+
+// CHECK-LABEL:   func @sum_kernel_with_inv(
+// CHECK-SAME:                              %[[VAL_0:.*0]]: tensor<?x?xf32>,
+// CHECK-SAME:                              %[[VAL_1:.*1]]: tensor<?x?xf32>,
+// CHECK-SAME:                              %[[VAL_2:.*2]]: tensor<?x?xf32>,
+// CHECK-SAME:                              %[[VAL_3:.*3]]: tensor<?xf32>,
+// CHECK-SAME:                              %[[VAL_4:.*4]]: tensor<f32>,
+// CHECK-SAME:                              %[[VAL_5:.*5]]: tensor<?xf32>) -> tensor<?xf32> {
+// CHECK:           %[[VAL_6:.*]] = constant 999 : index
+// CHECK:           %[[VAL_7:.*]] = constant 0 : index
+// CHECK:           %[[VAL_8:.*]] = constant true
+// CHECK:           %[[VAL_9:.*]] = constant 1 : index
+// CHECK:           %[[VAL_10:.*]] = alloca(%[[VAL_6]]) : memref<?xindex>
+// CHECK:           %[[VAL_11:.*]] = alloca(%[[VAL_6]]) : memref<?xindex>
+// CHECK:           %[[VAL_12:.*]] = alloca(%[[VAL_6]]) : memref<?xindex>
+// CHECK:           %[[VAL_13:.*]] = alloca(%[[VAL_6]]) : memref<?xindex>
+// CHECK:           %[[VAL_14:.*]] = alloca(%[[VAL_6]]) : memref<?xf32>
+// CHECK:           %[[VAL_15:.*]] = alloca(%[[VAL_6]]) : memref<?xindex>
+// CHECK:           %[[VAL_16:.*]] = alloca(%[[VAL_6]]) : memref<?xindex>
+// CHECK:           %[[VAL_17:.*]] = alloca(%[[VAL_6]]) : memref<?xf32>
+// CHECK:           %[[VAL_18:.*]] = alloca(%[[VAL_6]]) : memref<?xindex>
+// CHECK:           %[[VAL_19:.*]] = alloca(%[[VAL_6]]) : memref<?xindex>
+// CHECK:           %[[VAL_20:.*]] = alloca(%[[VAL_6]]) : memref<?xf32>
+// CHECK:           %[[VAL_21:.*]] = dim %[[VAL_3]], %[[VAL_7]] : tensor<?xf32>
+// CHECK:           %[[VAL_22:.*]] = alloca(%[[VAL_21]]) : memref<?xf32>
+// CHECK:           %[[VAL_23:.*]] = alloca() : memref<f32>
+// CHECK:           %[[VAL_24:.*]] = dim %[[VAL_5]], %[[VAL_7]] : tensor<?xf32>
+// CHECK:           %[[VAL_25:.*]] = alloca(%[[VAL_24]]) : memref<?xf32>
+// CHECK:           %[[VAL_26:.*]] = load %[[VAL_23]][] : memref<f32>
+// CHECK:           %[[VAL_27:.*]] = load %[[VAL_10]]{{\[}}%[[VAL_7]]] : memref<?xindex>
+// CHECK:           %[[VAL_28:.*]] = load %[[VAL_10]]{{\[}}%[[VAL_9]]] : memref<?xindex>
+// CHECK:           %[[VAL_29:.*]]:2 = scf.while (%[[VAL_30:.*]] = %[[VAL_27]], %[[VAL_31:.*]] = %[[VAL_7]]) : (index, index) -> (index, index) {
+// CHECK:             %[[VAL_32:.*]] = cmpi "ult", %[[VAL_30]], %[[VAL_28]] : index
+// CHECK:             scf.condition(%[[VAL_32]]) %[[VAL_30]], %[[VAL_31]] : index, index
+// CHECK:           } do {
+// CHECK:           ^bb0(%[[VAL_33:.*]]: index, %[[VAL_34:.*]]: index):
+// CHECK:             %[[VAL_35:.*]] = load %[[VAL_11]]{{\[}}%[[VAL_33]]] : memref<?xindex>
+// CHECK:             %[[VAL_36:.*]] = cmpi "eq", %[[VAL_35]], %[[VAL_34]] : index
+// CHECK:             scf.if %[[VAL_36]] {
+// CHECK:               %[[VAL_37:.*]] = load %[[VAL_22]]{{\[}}%[[VAL_34]]] : memref<?xf32>
+// CHECK:               %[[VAL_38:.*]] = load %[[VAL_12]]{{\[}}%[[VAL_33]]] : memref<?xindex>
+// CHECK:               %[[VAL_39:.*]] = addi %[[VAL_33]], %[[VAL_9]] : index
+// CHECK:               %[[VAL_40:.*]] = load %[[VAL_12]]{{\[}}%[[VAL_39]]] : memref<?xindex>
+// CHECK:               %[[VAL_41:.*]] = load %[[VAL_15]]{{\[}}%[[VAL_34]]] : memref<?xindex>
+// CHECK:               %[[VAL_42:.*]] = addi %[[VAL_34]], %[[VAL_9]] : index
+// CHECK:               %[[VAL_43:.*]] = load %[[VAL_15]]{{\[}}%[[VAL_42]]] : memref<?xindex>
+// CHECK:               %[[VAL_44:.*]] = load %[[VAL_18]]{{\[}}%[[VAL_34]]] : memref<?xindex>
+// CHECK:               %[[VAL_45:.*]] = addi %[[VAL_34]], %[[VAL_9]] : index
+// CHECK:               %[[VAL_46:.*]] = load %[[VAL_18]]{{\[}}%[[VAL_45]]] : memref<?xindex>
+// CHECK:               %[[VAL_47:.*]]:4 = scf.while (%[[VAL_48:.*]] = %[[VAL_38]], %[[VAL_49:.*]] = %[[VAL_41]], %[[VAL_50:.*]] = %[[VAL_44]], %[[VAL_51:.*]] = %[[VAL_7]]) : (index, index, index, index) -> (index, index, index, index) {
+// CHECK:                 %[[VAL_52:.*]] = cmpi "ult", %[[VAL_48]], %[[VAL_40]] : index
+// CHECK:                 %[[VAL_53:.*]] = cmpi "ult", %[[VAL_49]], %[[VAL_43]] : index
+// CHECK:                 %[[VAL_54:.*]] = and %[[VAL_52]], %[[VAL_53]] : i1
+// CHECK:                 %[[VAL_55:.*]] = cmpi "ult", %[[VAL_50]], %[[VAL_46]] : index
+// CHECK:                 %[[VAL_56:.*]] = and %[[VAL_54]], %[[VAL_55]] : i1
+// CHECK:                 scf.condition(%[[VAL_56]]) %[[VAL_48]], %[[VAL_49]], %[[VAL_50]], %[[VAL_51]] : index, index, index, index
+// CHECK:               } do {
+// CHECK:               ^bb0(%[[VAL_57:.*]]: index, %[[VAL_58:.*]]: index, %[[VAL_59:.*]]: index, %[[VAL_60:.*]]: index):
+// CHECK:                 %[[VAL_61:.*]] = load %[[VAL_13]]{{\[}}%[[VAL_57]]] : memref<?xindex>
+// CHECK:                 %[[VAL_62:.*]] = load %[[VAL_16]]{{\[}}%[[VAL_58]]] : memref<?xindex>
+// CHECK:                 %[[VAL_63:.*]] = load %[[VAL_19]]{{\[}}%[[VAL_59]]] : memref<?xindex>
+// CHECK:                 %[[VAL_64:.*]] = cmpi "eq", %[[VAL_61]], %[[VAL_60]] : index
+// CHECK:                 %[[VAL_65:.*]] = cmpi "eq", %[[VAL_62]], %[[VAL_60]] : index
+// CHECK:                 %[[VAL_66:.*]] = and %[[VAL_64]], %[[VAL_65]] : i1
+// CHECK:                 %[[VAL_67:.*]] = cmpi "eq", %[[VAL_63]], %[[VAL_60]] : index
+// CHECK:                 %[[VAL_68:.*]] = and %[[VAL_66]], %[[VAL_67]] : i1
+// CHECK:                 scf.if %[[VAL_68]] {
+// CHECK:                   %[[VAL_69:.*]] = load %[[VAL_25]]{{\[}}%[[VAL_34]]] : memref<?xf32>
+// CHECK:                   %[[VAL_70:.*]] = load %[[VAL_14]]{{\[}}%[[VAL_57]]] : memref<?xf32>
+// CHECK:                   %[[VAL_71:.*]] = load %[[VAL_17]]{{\[}}%[[VAL_58]]] : memref<?xf32>
+// CHECK:                   %[[VAL_72:.*]] = mulf %[[VAL_70]], %[[VAL_71]] : f32
+// CHECK:                   %[[VAL_73:.*]] = mulf %[[VAL_72]], %[[VAL_37]] : f32
+// CHECK:                   %[[VAL_74:.*]] = mulf %[[VAL_73]], %[[VAL_26]] : f32
+// CHECK:                   %[[VAL_75:.*]] = load %[[VAL_20]]{{\[}}%[[VAL_59]]] : memref<?xf32>
+// CHECK:                   %[[VAL_76:.*]] = addf %[[VAL_74]], %[[VAL_75]] : f32
+// CHECK:                   %[[VAL_77:.*]] = addf %[[VAL_69]], %[[VAL_76]] : f32
+// CHECK:                   store %[[VAL_77]], %[[VAL_25]]{{\[}}%[[VAL_34]]] : memref<?xf32>
+// CHECK:                 } else {
+// CHECK:                   %[[VAL_78:.*]] = cmpi "eq", %[[VAL_61]], %[[VAL_60]] : index
+// CHECK:                   %[[VAL_79:.*]] = cmpi "eq", %[[VAL_62]], %[[VAL_60]] : index
+// CHECK:                   %[[VAL_80:.*]] = and %[[VAL_78]], %[[VAL_79]] : i1
+// CHECK:                   scf.if %[[VAL_80]] {
+// CHECK:                     %[[VAL_81:.*]] = load %[[VAL_25]]{{\[}}%[[VAL_34]]] : memref<?xf32>
+// CHECK:                     %[[VAL_82:.*]] = load %[[VAL_14]]{{\[}}%[[VAL_57]]] : memref<?xf32>
+// CHECK:                     %[[VAL_83:.*]] = load %[[VAL_17]]{{\[}}%[[VAL_58]]] : memref<?xf32>
+// CHECK:                     %[[VAL_84:.*]] = mulf %[[VAL_82]], %[[VAL_83]] : f32
+// CHECK:                     %[[VAL_85:.*]] = mulf %[[VAL_84]], %[[VAL_37]] : f32
+// CHECK:                     %[[VAL_86:.*]] = mulf %[[VAL_85]], %[[VAL_26]] : f32
+// CHECK:                     %[[VAL_87:.*]] = addf %[[VAL_81]], %[[VAL_86]] : f32
+// CHECK:                     store %[[VAL_87]], %[[VAL_25]]{{\[}}%[[VAL_34]]] : memref<?xf32>
+// CHECK:                   } else {
+// CHECK:                     %[[VAL_88:.*]] = cmpi "eq", %[[VAL_63]], %[[VAL_60]] : index
+// CHECK:                     scf.if %[[VAL_88]] {
+// CHECK:                       %[[VAL_89:.*]] = load %[[VAL_25]]{{\[}}%[[VAL_34]]] : memref<?xf32>
+// CHECK:                       %[[VAL_90:.*]] = load %[[VAL_20]]{{\[}}%[[VAL_59]]] : memref<?xf32>
+// CHECK:                       %[[VAL_91:.*]] = addf %[[VAL_89]], %[[VAL_90]] : f32
+// CHECK:                       store %[[VAL_91]], %[[VAL_25]]{{\[}}%[[VAL_34]]] : memref<?xf32>
+// CHECK:                     } else {
+// CHECK:                     }
+// CHECK:                   }
+// CHECK:                 }
+// CHECK:                 %[[VAL_92:.*]] = cmpi "eq", %[[VAL_61]], %[[VAL_60]] : index
+// CHECK:                 %[[VAL_93:.*]] = addi %[[VAL_57]], %[[VAL_9]] : index
+// CHECK:                 %[[VAL_94:.*]] = select %[[VAL_92]], %[[VAL_93]], %[[VAL_57]] : index
+// CHECK:                 %[[VAL_95:.*]] = cmpi "eq", %[[VAL_62]], %[[VAL_60]] : index
+// CHECK:                 %[[VAL_96:.*]] = addi %[[VAL_58]], %[[VAL_9]] : index
+// CHECK:                 %[[VAL_97:.*]] = select %[[VAL_95]], %[[VAL_96]], %[[VAL_58]] : index
+// CHECK:                 %[[VAL_98:.*]] = cmpi "eq", %[[VAL_63]], %[[VAL_60]] : index
+// CHECK:                 %[[VAL_99:.*]] = addi %[[VAL_59]], %[[VAL_9]] : index
+// CHECK:                 %[[VAL_100:.*]] = select %[[VAL_98]], %[[VAL_99]], %[[VAL_59]] : index
+// CHECK:                 %[[VAL_101:.*]] = addi %[[VAL_60]], %[[VAL_9]] : index
+// CHECK:                 scf.yield %[[VAL_94]], %[[VAL_97]], %[[VAL_100]], %[[VAL_101]] : index, index, index, index
+// CHECK:               }
+// CHECK:               %[[VAL_102:.*]]:3 = scf.while (%[[VAL_103:.*]] = %[[VAL_104:.*]]#0, %[[VAL_105:.*]] = %[[VAL_104]]#1, %[[VAL_106:.*]] = %[[VAL_104]]#3) : (index, index, index) -> (index, index, index) {
+// CHECK:                 %[[VAL_107:.*]] = cmpi "ult", %[[VAL_103]], %[[VAL_40]] : index
+// CHECK:                 %[[VAL_108:.*]] = cmpi "ult", %[[VAL_105]], %[[VAL_43]] : index
+// CHECK:                 %[[VAL_109:.*]] = and %[[VAL_107]], %[[VAL_108]] : i1
+// CHECK:                 scf.condition(%[[VAL_109]]) %[[VAL_103]], %[[VAL_105]], %[[VAL_106]] : index, index, index
+// CHECK:               } do {
+// CHECK:               ^bb0(%[[VAL_110:.*]]: index, %[[VAL_111:.*]]: index, %[[VAL_112:.*]]: index):
+// CHECK:                 %[[VAL_113:.*]] = load %[[VAL_13]]{{\[}}%[[VAL_110]]] : memref<?xindex>
+// CHECK:                 %[[VAL_114:.*]] = load %[[VAL_16]]{{\[}}%[[VAL_111]]] : memref<?xindex>
+// CHECK:                 %[[VAL_115:.*]] = cmpi "eq", %[[VAL_113]], %[[VAL_112]] : index
+// CHECK:                 %[[VAL_116:.*]] = cmpi "eq", %[[VAL_114]], %[[VAL_112]] : index
+// CHECK:                 %[[VAL_117:.*]] = and %[[VAL_115]], %[[VAL_116]] : i1
+// CHECK:                 scf.if %[[VAL_117]] {
+// CHECK:                   %[[VAL_118:.*]] = load %[[VAL_25]]{{\[}}%[[VAL_34]]] : memref<?xf32>
+// CHECK:                   %[[VAL_119:.*]] = load %[[VAL_14]]{{\[}}%[[VAL_110]]] : memref<?xf32>
+// CHECK:                   %[[VAL_120:.*]] = load %[[VAL_17]]{{\[}}%[[VAL_111]]] : memref<?xf32>
+// CHECK:                   %[[VAL_121:.*]] = mulf %[[VAL_119]], %[[VAL_120]] : f32
+// CHECK:                   %[[VAL_122:.*]] = mulf %[[VAL_121]], %[[VAL_37]] : f32
+// CHECK:                   %[[VAL_123:.*]] = mulf %[[VAL_122]], %[[VAL_26]] : f32
+// CHECK:                   %[[VAL_124:.*]] = addf %[[VAL_118]], %[[VAL_123]] : f32
+// CHECK:                   store %[[VAL_124]], %[[VAL_25]]{{\[}}%[[VAL_34]]] : memref<?xf32>
+// CHECK:                 } else {
+// CHECK:                 }
+// CHECK:                 %[[VAL_125:.*]] = cmpi "eq", %[[VAL_113]], %[[VAL_112]] : index
+// CHECK:                 %[[VAL_126:.*]] = addi %[[VAL_110]], %[[VAL_9]] : index
+// CHECK:                 %[[VAL_127:.*]] = select %[[VAL_125]], %[[VAL_126]], %[[VAL_110]] : index
+// CHECK:                 %[[VAL_128:.*]] = cmpi "eq", %[[VAL_114]], %[[VAL_112]] : index
+// CHECK:                 %[[VAL_129:.*]] = addi %[[VAL_111]], %[[VAL_9]] : index
+// CHECK:                 %[[VAL_130:.*]] = select %[[VAL_128]], %[[VAL_129]], %[[VAL_111]] : index
+// CHECK:                 %[[VAL_131:.*]] = addi %[[VAL_112]], %[[VAL_9]] : index
+// CHECK:                 scf.yield %[[VAL_127]], %[[VAL_130]], %[[VAL_131]] : index, index, index
+// CHECK:               }
+// CHECK:               %[[VAL_132:.*]] = load %[[VAL_25]]{{\[}}%[[VAL_34]]] : memref<?xf32>
+// CHECK:               %[[VAL_133:.*]] = scf.for %[[VAL_134:.*]] = %[[VAL_135:.*]]#2 to %[[VAL_46]] step %[[VAL_9]] iter_args(%[[VAL_136:.*]] = %[[VAL_132]]) -> (f32) {
+// CHECK:                 %[[VAL_137:.*]] = load %[[VAL_20]]{{\[}}%[[VAL_134]]] : memref<?xf32>
+// CHECK:                 %[[VAL_138:.*]] = addf %[[VAL_136]], %[[VAL_137]] : f32
+// CHECK:                 scf.yield %[[VAL_138]] : f32
+// CHECK:               }
+// CHECK:               store %[[VAL_139:.*]], %[[VAL_25]]{{\[}}%[[VAL_34]]] : memref<?xf32>
+// CHECK:             } else {
+// CHECK:               scf.if %[[VAL_8]] {
+// CHECK:                 %[[VAL_140:.*]] = load %[[VAL_18]]{{\[}}%[[VAL_34]]] : memref<?xindex>
+// CHECK:                 %[[VAL_141:.*]] = addi %[[VAL_34]], %[[VAL_9]] : index
+// CHECK:                 %[[VAL_142:.*]] = load %[[VAL_18]]{{\[}}%[[VAL_141]]] : memref<?xindex>
+// CHECK:                 %[[VAL_143:.*]] = load %[[VAL_25]]{{\[}}%[[VAL_34]]] : memref<?xf32>
+// CHECK:                 %[[VAL_144:.*]] = scf.for %[[VAL_145:.*]] = %[[VAL_140]] to %[[VAL_142]] step %[[VAL_9]] iter_args(%[[VAL_146:.*]] = %[[VAL_143]]) -> (f32) {
+// CHECK:                   %[[VAL_147:.*]] = load %[[VAL_20]]{{\[}}%[[VAL_145]]] : memref<?xf32>
+// CHECK:                   %[[VAL_148:.*]] = addf %[[VAL_146]], %[[VAL_147]] : f32
+// CHECK:                   scf.yield %[[VAL_148]] : f32
+// CHECK:                 }
+// CHECK:                 store %[[VAL_149:.*]], %[[VAL_25]]{{\[}}%[[VAL_34]]] : memref<?xf32>
+// CHECK:               } else {
+// CHECK:               }
+// CHECK:             }
+// CHECK:             %[[VAL_150:.*]] = cmpi "eq", %[[VAL_35]], %[[VAL_34]] : index
+// CHECK:             %[[VAL_151:.*]] = addi %[[VAL_33]], %[[VAL_9]] : index
+// CHECK:             %[[VAL_152:.*]] = select %[[VAL_150]], %[[VAL_151]], %[[VAL_33]] : index
+// CHECK:             %[[VAL_153:.*]] = addi %[[VAL_34]], %[[VAL_9]] : index
+// CHECK:             scf.yield %[[VAL_152]], %[[VAL_153]] : index, index
+// CHECK:           }
+// CHECK:           scf.for %[[VAL_154:.*]] = %[[VAL_155:.*]]#1 to %[[VAL_24]] step %[[VAL_9]] {
+// CHECK:             %[[VAL_156:.*]] = load %[[VAL_18]]{{\[}}%[[VAL_154]]] : memref<?xindex>
+// CHECK:             %[[VAL_157:.*]] = addi %[[VAL_154]], %[[VAL_9]] : index
+// CHECK:             %[[VAL_158:.*]] = load %[[VAL_18]]{{\[}}%[[VAL_157]]] : memref<?xindex>
+// CHECK:             %[[VAL_159:.*]] = load %[[VAL_25]]{{\[}}%[[VAL_154]]] : memref<?xf32>
+// CHECK:             %[[VAL_160:.*]] = scf.for %[[VAL_161:.*]] = %[[VAL_156]] to %[[VAL_158]] step %[[VAL_9]] iter_args(%[[VAL_162:.*]] = %[[VAL_159]]) -> (f32) {
+// CHECK:               %[[VAL_163:.*]] = load %[[VAL_20]]{{\[}}%[[VAL_161]]] : memref<?xf32>
+// CHECK:               %[[VAL_164:.*]] = addf %[[VAL_162]], %[[VAL_163]] : f32
+// CHECK:               scf.yield %[[VAL_164]] : f32
+// CHECK:             }
+// CHECK:             store %[[VAL_165:.*]], %[[VAL_25]]{{\[}}%[[VAL_154]]] : memref<?xf32>
+// CHECK:           }
+// CHECK:           %[[VAL_166:.*]] = tensor_load %[[VAL_25]] : memref<?xf32>
+// CHECK:           return %[[VAL_166]] : tensor<?xf32>
+// CHECK:         }
+func @sum_kernel_with_inv(%arga: tensor<?x?xf32>,
+                          %argb: tensor<?x?xf32>,
+                          %argc: tensor<?x?xf32>,
+                          %argd: tensor<?xf32>,
+                          %arge: tensor<f32>,
+                          %argx: tensor<?xf32>) -> tensor<?xf32> {
+  %0 = linalg.generic #trait_sum_kernel_with_inv
+    ins(%arga, %argb, %argc, %argd, %arge : tensor<?x?xf32>,
+                                            tensor<?x?xf32>,
+                                            tensor<?x?xf32>,
+                                            tensor<?xf32>,
+                                            tensor<f32>)
+    init(%argx : tensor<?xf32>) {
+      ^bb(%a : f32, %b : f32, %c : f32, %d : f32, %e : f32, %x : f32):
+        %0 = mulf %a, %b  : f32
+        %1 = mulf %0, %d  : f32
+        %2 = mulf %1, %e  : f32
+        %3 = addf %2, %c  : f32
+        %4 = addf %x, %3  : f32
+        linalg.yield %4: f32
+  } -> tensor<?xf32>
+  return %0 : tensor<?xf32>
+}

mlir/test/Dialect/Linalg/sparse_3d.mlir

#trait_kernel_3d = {
   ],
   sparse = [
     [ "D", "D", "S" ],  // B
     [ "D", "D" ],       // C
     [ "D", "D" ],       // D
     [ "D", "D" ]        // A
   ],
   iterator_types = ["parallel", "parallel", "reduction", "reduction"],
-  doc = "A(i,j) = SUM_k,l B(i,k,l) * C(k,j) * D(l,j)"
+  doc = "A(i,j) += SUM_k,l B(i,k,l) * C(k,j) * D(l,j)"
 }
 
 // CHECK-LABEL:   func @kernel_3d(
 // CHECK-SAME:                    %[[VAL_0:.*0]]: tensor<?x?xf32>,
 // CHECK-SAME:                    %[[VAL_1:.*1]]: tensor<?x?x?xf32>,
 // CHECK-SAME:                    %[[VAL_2:.*2]]: tensor<?x?xf32>,
 // CHECK-SAME:                    %[[VAL_3:.*3]]: tensor<?x?xf32>) -> tensor<?x?xf32> {
 // CHECK:           %[[VAL_4:.*]] = constant 999 : index
%0 = linalg.generic #trait_kernel_3d
     ^bb(%b: f32, %c: f32, %d : f32, %a : f32):
       %0 = mulf %b, %c : f32
       %1 = mulf %0, %d : f32
       %2 = addf %1, %a : f32
       linalg.yield %2 : f32
   } -> tensor<?x?xf32>
   return %0 : tensor<?x?xf32>
 }
+
 #trait_sum_reduction = {
   indexing_maps = [
-    affine_map<(i,j,k) -> (i,j,k)>,  // a
+    affine_map<(i,j,k) -> (i,j,k)>,  // A
     affine_map<(i,j,k) -> ()>        // x (scalar out)
   ],
   sparse = [
-    [ "S", "S", "S" ],  // a
+    [ "S", "S", "S" ],  // A
     [ ]                 // x
   ],
   iterator_types = ["reduction", "reduction", "reduction"],
-  doc = "x = SUM_ijk a(i,j,k)"
+  doc = "x += SUM_ijk A(i,j,k)"
 }
 
 // CHECK-LABEL:   func @sum_reduction(
 // CHECK-SAME:                        %[[VAL_0:.*]]: tensor<10x20x30xf32>,
 // CHECK-SAME:                        %[[VAL_1:.*]]: tensor<f32>) -> tensor<f32> {
 // CHECK:           %[[VAL_2:.*]] = constant 999 : index
 // CHECK:           %[[VAL_3:.*]] = constant 0 : index
 // CHECK:           %[[VAL_4:.*]] = constant 1 : index

 // CHECK:           scf.for %[[VAL_15:.*]] = %[[VAL_13]] to %[[VAL_14]] step %[[VAL_4]] {
 // CHECK:             %[[VAL_16:.*]] = load %[[VAL_7]]{{\[}}%[[VAL_15]]] : memref<?xindex>
 // CHECK:             %[[VAL_17:.*]] = addi %[[VAL_15]], %[[VAL_4]] : index
 // CHECK:             %[[VAL_18:.*]] = load %[[VAL_7]]{{\[}}%[[VAL_17]]] : memref<?xindex>
 // CHECK:             scf.for %[[VAL_19:.*]] = %[[VAL_16]] to %[[VAL_18]] step %[[VAL_4]] {
 // CHECK:               %[[VAL_20:.*]] = load %[[VAL_9]]{{\[}}%[[VAL_19]]] : memref<?xindex>
 // CHECK:               %[[VAL_21:.*]] = addi %[[VAL_19]], %[[VAL_4]] : index
 // CHECK:               %[[VAL_22:.*]] = load %[[VAL_9]]{{\[}}%[[VAL_21]]] : memref<?xindex>
-// CHECK:               scf.for %[[VAL_23:.*]] = %[[VAL_20]] to %[[VAL_22]] step %[[VAL_4]] {
-// CHECK:                 %[[VAL_24:.*]] = load %[[VAL_12]][] : memref<f32>
-// CHECK:                 %[[VAL_25:.*]] = load %[[VAL_11]]{{\[}}%[[VAL_23]]] : memref<?xf32>
-// CHECK:                 %[[VAL_26:.*]] = addf %[[VAL_24]], %[[VAL_25]] : f32
-// CHECK:                 store %[[VAL_26]], %[[VAL_12]][] : memref<f32>
+// CHECK:               %[[VAL_23:.*]] = load %[[VAL_12]][] : memref<f32>
+// CHECK:               %[[VAL_24:.*]] = scf.for %[[VAL_25:.*]] = %[[VAL_20]] to %[[VAL_22]] step %[[VAL_4]] iter_args(%[[VAL_26:.*]] = %[[VAL_23]]) -> (f32) {
+// CHECK:                 %[[VAL_27:.*]] = load %[[VAL_11]]{{\[}}%[[VAL_25]]] : memref<?xf32>
+// CHECK:                 %[[VAL_28:.*]] = addf %[[VAL_26]], %[[VAL_27]] : f32
+// CHECK:                 scf.yield %[[VAL_28]] : f32
 // CHECK:               }
+// CHECK:               store %[[VAL_29:.*]], %[[VAL_12]][] : memref<f32>
 // CHECK:             }
 // CHECK:           }
-// CHECK:           %[[VAL_27:.*]] = tensor_load %[[VAL_12]] : memref<f32>
-// CHECK:           return %[[VAL_27]] : tensor<f32>
+// CHECK:           %[[VAL_30:.*]] = tensor_load %[[VAL_12]] : memref<f32>
+// CHECK:           return %[[VAL_30]] : tensor<f32>
 // CHECK:         }
 func @sum_reduction(%arga: tensor<10x20x30xf32>, %argx: tensor<f32>) -> tensor<f32> {
   %0 = linalg.generic #trait_sum_reduction
     ins(%arga : tensor<10x20x30xf32>)
     init(%argx : tensor<f32>) {
       ^bb(%a : f32, %x : f32):
         %0 = addf %x, %a  : f32
         linalg.yield %0: f32
   } -> tensor<f32>
   return %0 : tensor<f32>
 }
 
+#trait_sum_reduction_inv = {
+  indexing_maps = [
+    affine_map<(i,j,k) -> (i,j,k)>,  // A
+    affine_map<(i,j,k) -> (i)>,      // b
+    affine_map<(i,j,k) -> ()>        // x (scalar out)
+  ],
+  sparse = [
+    [ "D", "D", "D" ], // A
+    [ "D" ],           // b
+    [ ]                // x
+  ],
+  iterator_types = ["reduction", "reduction", "reduction"],
+  doc = "x += SUM_i A(i,j,k) * b(i)"
+}
+
+// CHECK-LABEL:   func @sum_reduction_inv(
+// CHECK-SAME:                            %[[VAL_0:.*]]: tensor<?x?x?xf32>,
+// CHECK-SAME:                            %[[VAL_1:.*]]: tensor<?xf32>,
+// CHECK-SAME:                            %[[VAL_2:.*]]: tensor<f32>) -> tensor<f32> {
+// CHECK:           %[[VAL_3:.*]] = constant 2 : index
+// CHECK:           %[[VAL_4:.*]] = constant 0 : index
+// CHECK:           %[[VAL_5:.*]] = constant 1 : index
+// CHECK:           %[[VAL_6:.*]] = dim %[[VAL_0]], %[[VAL_4]] : tensor<?x?x?xf32>
+// CHECK:           %[[VAL_7:.*]] = dim %[[VAL_0]], %[[VAL_5]] : tensor<?x?x?xf32>
+// CHECK:           %[[VAL_8:.*]] = dim %[[VAL_0]], %[[VAL_3]] : tensor<?x?x?xf32>
+// CHECK:           %[[VAL_9:.*]] = alloca(%[[VAL_6]], %[[VAL_7]], %[[VAL_8]]) : memref<?x?x?xf32>
+// CHECK:           %[[VAL_10:.*]] = dim %[[VAL_1]], %[[VAL_4]] : tensor<?xf32>
+// CHECK:           %[[VAL_11:.*]] = alloca(%[[VAL_10]]) : memref<?xf32>
+// CHECK:           %[[VAL_12:.*]] = alloca() : memref<f32>
+// CHECK:           scf.for %[[VAL_13:.*]] = %[[VAL_4]] to %[[VAL_10]] step %[[VAL_5]] {
+// CHECK:             %[[VAL_14:.*]] = load %[[VAL_11]]{{\[}}%[[VAL_13]]] : memref<?xf32>
+// CHECK:             scf.for %[[VAL_15:.*]] = %[[VAL_4]] to %[[VAL_7]] step %[[VAL_5]] {
+// CHECK:               %[[VAL_16:.*]] = load %[[VAL_12]][] : memref<f32>
+// CHECK:               %[[VAL_17:.*]] = scf.for %[[VAL_18:.*]] = %[[VAL_4]] to %[[VAL_8]] step %[[VAL_5]] iter_args(%[[VAL_19:.*]] = %[[VAL_16]]) -> (f32) {
+// CHECK:                 %[[VAL_20:.*]] = load %[[VAL_9]]{{\[}}%[[VAL_13]], %[[VAL_15]], %[[VAL_18]]] : memref<?x?x?xf32>
+// CHECK:                 %[[VAL_21:.*]] = mulf %[[VAL_20]], %[[VAL_14]] : f32
+// CHECK:                 %[[VAL_22:.*]] = addf %[[VAL_19]], %[[VAL_21]] : f32
+// CHECK:                 scf.yield %[[VAL_22]] : f32
+// CHECK:               }
+// CHECK:               store %[[VAL_23:.*]], %[[VAL_12]][] : memref<f32>
+// CHECK:             }
+// CHECK:           }
+// CHECK:           %[[VAL_24:.*]] = tensor_load %[[VAL_12]] : memref<f32>
+// CHECK:           return %[[VAL_24]] : tensor<f32>
+// CHECK:         }
+func @sum_reduction_inv(%arga: tensor<?x?x?xf32>,
+                        %argb: tensor<?xf32>,
+            %argx: tensor<f32>) -> tensor<f32> {
+  %0 = linalg.generic #trait_sum_reduction_inv
+    ins(%arga, %argb : tensor<?x?x?xf32>, tensor<?xf32>)
+    init(%argx : tensor<f32>) {
+      ^bb(%a : f32, %b : f32, %x : f32):
+        %0 = mulf %a, %b  : f32
+        %1 = addf %x, %0  : f32
+        linalg.yield %1: f32
+  } -> tensor<f32>
+  return %0 : tensor<f32>
+}
+
 #trait_invariants = {
   indexing_maps = [
     affine_map<(i,j,k) -> (i)>,      // a
     affine_map<(i,j,k) -> (j)>,      // b
     affine_map<(i,j,k) -> (k)>,      // c
-    affine_map<(i,j,k) -> (i,j,k)>   // x
+    affine_map<(i,j,k) -> (i,j,k)>   // X (out)
   ],
   sparse = [
     [ "D" ],           // a
     [ "D" ],           // b
     [ "D" ],           // c
-    [ "D", "D", "D" ]  // x
+    [ "D", "D", "D" ]  // X
   ],
   iterator_types = ["parallel", "parallel", "parallel"],
-  doc = "x(i,j,k) = a(i) * b(j) * c(k)"
+  doc = "X(i,j,k) = a(i) * b(j) * c(k)"
 }
 
 // CHECK-LABEL:   func @invariants(
 // CHECK-SAME:                     %[[VAL_0:.*]]: tensor<10xf32>,
 // CHECK-SAME:                     %[[VAL_1:.*]]: tensor<20xf32>,
 // CHECK-SAME:                     %[[VAL_2:.*]]: tensor<30xf32>) -> tensor<10x20x30xf32> {
 // CHECK:           %[[VAL_3:.*]] = constant 10 : index
 // CHECK:           %[[VAL_4:.*]] = constant 20 : index