[mlir][sparse] refine lexicographic insertion to any tensor

[aartbik]

Summary

First version was vectors only. With some clever "path" insertion,
we now support any d-dimensional tensor. Up next: reductions too

[wrengr]

mlir/lib/ExecutionEngine/SparseTensorUtils.cpp

294–298

style

372

I like removing the need for the sparsity parameter. But for legibility/clarity, add a method bool isCompressedDim(uint64_t) to encapsulate this idiom everywhere it's used. (And/or bool isDenseDim(uint64_t) depending on how you expect Singleton and other DimLevelTypes to go)

421

Excellent! I've been wanting to factor this function out of fromCOO for a while, just hadn't gotten around to it yet :)

435–436

Since the goal of that definition for d is to reverse the iteration order and i is used nowhere else, it would be cleaner to just iterate d directly.

462

dimension.

[aartbik]

mlir/lib/ExecutionEngine/SparseTensorUtils.cpp

372

Agreed, more readable in the long run and it will trigger the singleton work changes needed.

421

We probably can refactor more shared code, but this was rather tricky to get right, so let's tread carefully. ;-)

435–436

I don't disagree, but ending at zero with an unsigned made the iteration a bit awkward....

[aartbik]

addressed comments

[aartbik]

one last comment

[wrengr]

mlir/lib/ExecutionEngine/SparseTensorUtils.cpp

435–436

Doh, I missed that. I still think phrasing things as counting down makes more sense though. Could always rephrase things as a while-loop...

uint64_t rank = getRank();
assert diff < rank;
uint64_t d = rank - 1;
while (true) {
    ...stuff...
    if (d == diff) break;
    --d;
}

Though that does start looking pretty gross too. Actually, while writing that up I just noticed you'll want to have that assertion regardless of whether rephrasing the loop, since otherwise rank - diff will do bad things.

[aartbik]

added asserts on rank

[aartbik]

mlir/lib/ExecutionEngine/SparseTensorUtils.cpp

435–436

Added the assert.

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

static bool isAdmissableTensorExp(Merger &merger, linalg::GenericOp op,
   if (isMaterializing(lhs->get())) {
     // In this first sparse tensor output implementation, this is enforced by
     // rejecting any reduction loops (since the sparse parallel loops give a
     // lexicographically sorted and injective view into that tensor).
     // TODO: generalize to include reductions
     for (auto attr : op.iterator_types())
       if (isReductionIterator(attr))
         return false;
-    // TODO: generalize support lib beyond vectors
-    if (op.iterator_types().size() != 1)
-      return false;
     *sparseOut = lhs;
     return true;
   }
   return false;
 }
 
 //===----------------------------------------------------------------------===//
 // Sparse compiler synthesis methods (reductions).

mlir/lib/ExecutionEngine/SparseTensorUtils.cpp

SparseTensorStorage(const std::vector<uint64_t> &szs, const uint64_t *perm,
     // Prepare sparse pointer structures for all dimensions.
     for (uint64_t r = 0; r < rank; r++)
       if (sparsity[r] == DimLevelType::kCompressed)
         pointers[r].push_back(0);
     // Then assign contents from coordinate scheme tensor if provided.
     if (tensor) {
       uint64_t nnz = tensor->getElements().size();
       values.reserve(nnz);
-      fromCOO(tensor, sparsity, 0, nnz, 0);
-    } else {
-      if (allDense)
-        values.resize(sz, 0);
-      for (uint64_t r = 0; r < rank; r++)
-        idx[r] = -1u;
+      fromCOO(tensor, 0, nnz, 0);
+    } else if (allDense) {
     }
   }
 
   virtual ~SparseTensorStorage() {}
 
   /// Get the rank of the tensor.
   uint64_t getRank() const { return sizes.size(); }
 
public:
   }
   void getIndices(std::vector<I> **out, uint64_t d) override {
     assert(d < getRank());
     *out = &indices[d];
   }
   void getValues(std::vector<V> **out) override { *out = &values; }
 
   /// Partially specialize lexicographic insertions based on template types.
-  // TODO: 1-dim tensors only for now, generalize soon
   void lexInsert(uint64_t *cursor, V val) override {
-    assert((idx[0] == -1u || idx[0] < cursor[0]) && "not lexicographic");
-    indices[0].push_back(cursor[0]);
-    values.push_back(val);
-    idx[0] = cursor[0];
+    // First, wrap up pending insertion path.
+    uint64_t diff = 0;
+    uint64_t top = 0;
+    if (!values.empty()) {
+      diff = lexDiff(cursor);
+      endPath(diff + 1);
+      top = idx[diff] + 1;
+    }
+    // Then continue with insertion path.
+    insPath(cursor, diff, top, val);
   }
 
   /// Finalizes lexicographic insertions.
-  void endInsert() override { pointers[0].push_back(indices[0].size()); }
+  void endInsert() override {
+    if (values.empty())
+      endDim(0);
+    else
+      endPath(0);
+  }
 
   /// Returns this sparse tensor storage scheme as a new memory-resident
   /// sparse tensor in coordinate scheme with the given dimension order.
   SparseTensorCOO<V> *toCOO(const uint64_t *perm) {
     // Restore original order of the dimension sizes and allocate coordinate
     // scheme with desired new ordering specified in perm.
     uint64_t rank = getRank();
     std::vector<uint64_t> orgsz(rank);
newSparseTensor(uint64_t rank, const uint64_t *sizes, const uint64_t *perm,
     }
     return n;
   }
 
 private:
   /// Initializes sparse tensor storage scheme from a memory-resident sparse
   /// tensor in coordinate scheme. This method prepares the pointers and
   /// indices arrays under the given per-dimension dense/sparse annotations.
-  void fromCOO(SparseTensorCOO<V> *tensor, const DimLevelType *sparsity,
-               uint64_t lo, uint64_t hi, uint64_t d) {
+  void fromCOO(SparseTensorCOO<V> *tensor, uint64_t lo, uint64_t hi,
+               uint64_t d) {
     const std::vector<Element<V>> &elements = tensor->getElements();
     // Once dimensions are exhausted, insert the numerical values.
     assert(d <= getRank());
     if (d == getRank()) {
-      assert(lo >= hi || lo < elements.size());
-      values.push_back(lo < hi ? elements[lo].value : 0);
+      assert(lo < hi && hi <= elements.size());
+      values.push_back(elements[lo].value);
       return;
     }
     // Visit all elements in this interval.
     uint64_t full = 0;
     while (lo < hi) {
       assert(lo < elements.size() && hi <= elements.size());
       // Find segment in interval with same index elements in this dimension.
       uint64_t i = elements[lo].indices[d];
       uint64_t seg = lo + 1;
       while (seg < hi && elements[seg].indices[d] == i)
         seg++;
       // Handle segment in interval for sparse or dense dimension.
-      if (sparsity[d] == DimLevelType::kCompressed) {
+      if (isCompressedDim(d)) {
         indices[d].push_back(i);
       } else {
         // For dense storage we must fill in all the zero values between
         // the previous element (when last we ran this for-loop) and the
         // current element.
         for (; full < i; full++)
-          fromCOO(tensor, sparsity, 0, 0, d + 1); // pass empty
+          endDim(d + 1);
         full++;
       }
-      fromCOO(tensor, sparsity, lo, seg, d + 1);
+      fromCOO(tensor, lo, seg, d + 1);
       // And move on to next segment in interval.
       lo = seg;
     }
     // Finalize the sparse pointer structure at this dimension.
-    if (sparsity[d] == DimLevelType::kCompressed) {
+    if (isCompressedDim(d)) {
       pointers[d].push_back(indices[d].size());
     } else {
       // For dense storage we must fill in all the zero values after
       // the last element.
       for (uint64_t sz = sizes[d]; full < sz; full++)
-        fromCOO(tensor, sparsity, 0, 0, d + 1); // pass empty
+        endDim(d + 1);
     }
   }
 
   /// Stores the sparse tensor storage scheme into a memory-resident sparse
   /// tensor in coordinate scheme.
   void toCOO(SparseTensorCOO<V> *tensor, std::vector<uint64_t> &reord,
              uint64_t pos, uint64_t d) {
     assert(d <= getRank());
     if (d == getRank()) {
       assert(pos < values.size());
       tensor->add(idx, values[pos]);
-    } else if (pointers[d].empty()) {
+    } else if (isCompressedDim(d)) {
+      // Sparse dimension.
+      for (uint64_t ii = pointers[d][pos]; ii < pointers[d][pos + 1]; ii++) {
+        idx[reord[d]] = indices[d][ii];
+        toCOO(tensor, reord, ii, d + 1);
+      }
+    } else {
       // Dense dimension.
       for (uint64_t i = 0, sz = sizes[d], off = pos * sz; i < sz; i++) {
         idx[reord[d]] = i;
         toCOO(tensor, reord, off + i, d + 1);
       }
+    }
+  }
+
+  /// Ends a deeper, never seen before dimension.
+  void endDim(uint64_t d) {
+    assert(d <= getRank());
+    if (d == getRank()) {
+      values.push_back(0);
+    } else if (isCompressedDim(d)) {
+      pointers[d].push_back(indices[d].size());
     } else {
-      // Sparse dimension.
-      for (uint64_t ii = pointers[d][pos]; ii < pointers[d][pos + 1]; ii++) {
-        idx[reord[d]] = indices[d][ii];
-        toCOO(tensor, reord, ii, d + 1);
-      }
+      for (uint64_t full = 0, sz = sizes[d]; full < sz; full++)
+        endDim(d + 1);
+    }
+  }
+
+  /// Wraps up a single insertion path, inner to outer.
+  void endPath(uint64_t diff) {
+    for (uint64_t i = 0, rank = getRank(); i < rank - diff; i++) {
+      uint64_t d = rank - i - 1;
+      if (isCompressedDim(d)) {
+        pointers[d].push_back(indices[d].size());
+      } else {
+        for (uint64_t full = idx[d] + 1, sz = sizes[d]; full < sz; full++)
+          endDim(d + 1);
+      }
+    }
+
+  /// Continues a single insertion path, outer to inner.
+  void insPath(uint64_t *cursor, uint64_t diff, uint64_t top, V val) {
+    for (uint64_t d = diff, rank = getRank(); d < rank; d++) {
+      uint64_t i = cursor[d];
+      if (isCompressedDim(d)) {
+        indices[d].push_back(i);
+      } else {
+        for (uint64_t full = top; full < i; full++)
+          endDim(d + 1);
       }
+      top = 0;
+      idx[d] = i;
+    }
+    values.push_back(val);
+  }
+
+  /// Finds the lexicographic differing rank.
+  uint64_t lexDiff(uint64_t *cursor) {
+    for (uint64_t r = 0, rank = getRank(); r < rank; r++)
+      if (cursor[r] > idx[r])
+        return r;
+      else
+        assert(cursor[r] == idx[r] && "non-lexicographic insertion");
+    assert(0 && "duplication insertion");
+    return -1u;
+  }
+
+  /// Returns true if dimension is compressed.
+  inline bool isCompressedDim(uint64_t d) const {
+    return (!pointers[d].empty());
   }
 
 private:
   std::vector<uint64_t> sizes; // per-dimension sizes
   std::vector<uint64_t> rev;   // "reverse" permutation
   std::vector<uint64_t> idx;   // index cursor
   std::vector<std::vector<P>> pointers;
   std::vector<std::vector<I>> indices;

mlir/test/Dialect/SparseTensor/sparse_out.mlir

 // NOTE: Assertions have been autogenerated by utils/generate-test-checks.py
 // RUN: mlir-opt %s -sparsification | FileCheck %s
 
 #CSR = #sparse_tensor.encoding<{
   dimLevelType = [ "dense", "compressed" ],
   dimOrdering = affine_map<(i,j) -> (i,j)>
 }>
 
 #DCSR = #sparse_tensor.encoding<{
   dimLevelType = [ "compressed", "compressed" ],
   dimOrdering = affine_map<(i,j) -> (i,j)>
 }>
 
-#trait_scale = {
+#trait_scale_inpl = {
   indexing_maps = [
     affine_map<(i,j) -> (i,j)>   // X (out)
   ],
   iterator_types = ["parallel", "parallel"],
   doc = "X(i,j) = X(i,j) * 2"
 }
 
 // CHECK-LABEL:   func @sparse_simply_dynamic1(

 // CHECK:               memref.store %[[VAL_17]], %[[VAL_8]]{{\[}}%[[VAL_15]]] : memref<?xf32>
 // CHECK:             }
 // CHECK:           }
 // CHECK:           %[[VAL_18:.*]] = sparse_tensor.load %[[VAL_0]] : tensor<32x16xf32, #sparse_tensor.encoding<{{.*}}>>
 // CHECK:           return %[[VAL_18]] : tensor<32x16xf32, #sparse_tensor.encoding<{{.*}}>>
 // CHECK:         }
 func @sparse_simply_dynamic1(%argx: tensor<32x16xf32, #DCSR> {linalg.inplaceable = true}) -> tensor<32x16xf32, #DCSR> {
   %c = arith.constant 2.0 : f32
-  %0 = linalg.generic #trait_scale
+  %0 = linalg.generic #trait_scale_inpl
     outs(%argx: tensor<32x16xf32, #DCSR>) {
       ^bb(%x: f32):
         %1 = arith.mulf %x, %c : f32
         linalg.yield %1 : f32
   } -> tensor<32x16xf32, #DCSR>
   return %0 : tensor<32x16xf32, #DCSR>
 }
 
%0 = linalg.generic #trait_elt_wise_mult
     ins(%arga: tensor<32x16xf32, #CSR>)
     outs(%argx: tensor<32x16xf32, #DCSR>) {
       ^bb(%a: f32, %x: f32):
         %1 = arith.mulf %x, %a : f32
         linalg.yield %1 : f32
   } -> tensor<32x16xf32, #DCSR>
   return %0 : tensor<32x16xf32, #DCSR>
 }
+
+#trait_scale = {
+  indexing_maps = [
+    affine_map<(i,j) -> (i,j)>,  // A
+    affine_map<(i,j) -> (i,j)>   // X (out)
+  ],
+  iterator_types = ["parallel", "parallel"],
+  doc = "X(i,j) = A(i,j) * 2.0"
+}
+
+// CHECK-LABEL:   func @sparse_truly_dynamic(
+// CHECK-SAME:      %[[VAL_0:.*]]: tensor<10x20xf32, #sparse_tensor.encoding<{{.*}}>>
+// CHECK-DAG:       %[[VAL_1:.*]] = arith.constant 2.000000e+00 : f32
+// CHECK-DAG:       %[[VAL_2:.*]] = arith.constant 10 : index
+// CHECK-DAG:       %[[VAL_3:.*]] = arith.constant 20 : index
+// CHECK-DAG:       %[[VAL_4:.*]] = arith.constant 1 : index
+// CHECK-DAG:       %[[VAL_5:.*]] = arith.constant 2 : index
+// CHECK-DAG:       %[[VAL_6:.*]] = arith.constant 0 : index
+// CHECK:           %[[VAL_7:.*]] = sparse_tensor.init{{\[}}%[[VAL_2]], %[[VAL_3]]] : tensor<10x20xf32, #sparse_tensor.encoding<{{.*}}>>
+// CHECK:           %[[VAL_8:.*]] = sparse_tensor.pointers %[[VAL_0]], %[[VAL_4]] : tensor<10x20xf32, #sparse_tensor.encoding<{{.*}}>>
+// CHECK:           %[[VAL_9:.*]] = sparse_tensor.indices %[[VAL_0]], %[[VAL_4]] : tensor<10x20xf32, #sparse_tensor.encoding<{{.*}}>>
+// CHECK:           %[[VAL_10:.*]] = sparse_tensor.values %[[VAL_0]] : tensor<10x20xf32, #sparse_tensor.encoding<{{.*}}>>
+// CHECK:           %[[VAL_11:.*]] = memref.alloca(%[[VAL_5]]) : memref<?xindex>
+// CHECK:           scf.for %[[VAL_12:.*]] = %[[VAL_6]] to %[[VAL_2]] step %[[VAL_4]] {
+// CHECK:             memref.store %[[VAL_12]], %[[VAL_11]]{{\[}}%[[VAL_6]]] : memref<?xindex>
+// CHECK:             %[[VAL_13:.*]] = memref.load %[[VAL_8]]{{\[}}%[[VAL_12]]] : memref<?xindex>
+// CHECK:             %[[VAL_14:.*]] = arith.addi %[[VAL_12]], %[[VAL_4]] : index
+// CHECK:             %[[VAL_15:.*]] = memref.load %[[VAL_8]]{{\[}}%[[VAL_14]]] : memref<?xindex>
+// CHECK:             scf.for %[[VAL_16:.*]] = %[[VAL_13]] to %[[VAL_15]] step %[[VAL_4]] {
+// CHECK:               %[[VAL_17:.*]] = memref.load %[[VAL_9]]{{\[}}%[[VAL_16]]] : memref<?xindex>
+// CHECK:               memref.store %[[VAL_17]], %[[VAL_11]]{{\[}}%[[VAL_4]]] : memref<?xindex>
+// CHECK:               %[[VAL_18:.*]] = memref.load %[[VAL_10]]{{\[}}%[[VAL_16]]] : memref<?xf32>
+// CHECK:               %[[VAL_19:.*]] = arith.mulf %[[VAL_18]], %[[VAL_1]] : f32
+// CHECK:               sparse_tensor.lex_insert %[[VAL_7]], %[[VAL_11]], %[[VAL_19]] : tensor<10x20xf32, #sparse_tensor.encoding<{{.*}}>>
+// CHECK:             }
+// CHECK:           }
+// CHECK:           %[[VAL_20:.*]] = sparse_tensor.load %[[VAL_7]] hasInserts : tensor<10x20xf32, #sparse_tensor.encoding<{{.*}}>>
+// CHECK:           return %[[VAL_20]] : tensor<10x20xf32, #sparse_tensor.encoding<{
+// CHECK:         }
+func @sparse_truly_dynamic(%arga: tensor<10x20xf32, #CSR>) -> tensor<10x20xf32, #DCSR> {
+  %s = arith.constant 2.0 : f32
+  %d10 = arith.constant 10 : index
+  %d20 = arith.constant 20 : index
+  %xm = sparse_tensor.init [%d10, %d20] : tensor<10x20xf32, #DCSR>
+  %0 = linalg.generic #trait_scale
+     ins(%arga: tensor<10x20xf32, #CSR>)
+      outs(%xm: tensor<10x20xf32, #DCSR>) {
+      ^bb(%a: f32, %x: f32):
+        %1 = arith.mulf %a, %s : f32
+        linalg.yield %1 : f32
+  } -> tensor<10x20xf32, #DCSR>
+  return %0 : tensor<10x20xf32, #DCSR>
+}

mlir/test/Integration/Dialect/SparseTensor/CPU/sparse_matrix_ops.mlir

+// RUN: mlir-opt %s \
+// RUN:   --sparsification --sparse-tensor-conversion \
+// RUN:   --linalg-bufferize --convert-linalg-to-loops \
+// RUN:   --convert-vector-to-scf --convert-scf-to-std \
+// RUN:   --func-bufferize --tensor-constant-bufferize --tensor-bufferize \
+// RUN:   --std-bufferize --finalizing-bufferize --lower-affine \
+// RUN:   --convert-vector-to-llvm --convert-memref-to-llvm --convert-math-to-llvm \
+// RUN:   --convert-std-to-llvm --reconcile-unrealized-casts | \
+// RUN: mlir-cpu-runner \
+// RUN:  -e entry -entry-point-result=void  \
+// RUN:  -shared-libs=%mlir_integration_test_dir/libmlir_c_runner_utils%shlibext | \
+// RUN: FileCheck %s
+
+#DCSR = #sparse_tensor.encoding<{dimLevelType = ["compressed", "compressed"]}>
+
+//
+// Traits for 2-d tensor (aka matrix) operations.
+//
+#trait_scale = {
+  indexing_maps = [
+    affine_map<(i,j) -> (i,j)>,  // A (in)
+    affine_map<(i,j) -> (i,j)>   // X (out)
+  ],
+  iterator_types = ["parallel", "parallel"],
+  doc = "X(i,j) = A(i,j) * 2.0"
+}
+#trait_scale_inpl = {
+  indexing_maps = [
+    affine_map<(i,j) -> (i,j)>   // X (out)
+  ],
+  iterator_types = ["parallel", "parallel"],
+  doc = "X(i,j) *= 2.0"
+}
+#trait_op = {
+  indexing_maps = [
+    affine_map<(i,j) -> (i,j)>,  // A (in)
+    affine_map<(i,j) -> (i,j)>,  // B (in)
+    affine_map<(i,j) -> (i,j)>   // X (out)
+  ],
+  iterator_types = ["parallel", "parallel"],
+  doc = "X(i,j) = A(i,j) OP B(i,j)"
+}
+
+module {
+  // Scales a sparse matrix into a new sparse matrix.
+  func @matrix_scale(%arga: tensor<?x?xf64, #DCSR>) -> tensor<?x?xf64, #DCSR> {
+    %s = arith.constant 2.0 : f64
+    %c0 = arith.constant 0 : index
+    %c1 = arith.constant 1 : index
+    %d0 = tensor.dim %arga, %c0 : tensor<?x?xf64, #DCSR>
+    %d1 = tensor.dim %arga, %c1 : tensor<?x?xf64, #DCSR>
+    %xm = sparse_tensor.init [%d0, %d1] : tensor<?x?xf64, #DCSR>
+    %0 = linalg.generic #trait_scale
+       ins(%arga: tensor<?x?xf64, #DCSR>)
+        outs(%xm: tensor<?x?xf64, #DCSR>) {
+        ^bb(%a: f64, %x: f64):
+          %1 = arith.mulf %a, %s : f64
+          linalg.yield %1 : f64
+    } -> tensor<?x?xf64, #DCSR>
+    return %0 : tensor<?x?xf64, #DCSR>
+  }
+
+  // Scales a sparse matrix in place.
+  func @matrix_scale_inplace(%argx: tensor<?x?xf64, #DCSR>
+                             {linalg.inplaceable = true}) -> tensor<?x?xf64, #DCSR> {
+    %s = arith.constant 2.0 : f64
+    %0 = linalg.generic #trait_scale_inpl
+      outs(%argx: tensor<?x?xf64, #DCSR>) {
+        ^bb(%x: f64):
+          %1 = arith.mulf %x, %s : f64
+          linalg.yield %1 : f64
+    } -> tensor<?x?xf64, #DCSR>
+    return %0 : tensor<?x?xf64, #DCSR>
+  }
+
+  // Adds two sparse matrices element-wise into a new sparse matrix.
+  func @matrix_add(%arga: tensor<?x?xf64, #DCSR>,
+                   %argb: tensor<?x?xf64, #DCSR>) -> tensor<?x?xf64, #DCSR> {
+    %c0 = arith.constant 0 : index
+    %c1 = arith.constant 1 : index
+    %d0 = tensor.dim %arga, %c0 : tensor<?x?xf64, #DCSR>
+    %d1 = tensor.dim %arga, %c1 : tensor<?x?xf64, #DCSR>
+    %xv = sparse_tensor.init [%d0, %d1] : tensor<?x?xf64, #DCSR>
+    %0 = linalg.generic #trait_op
+       ins(%arga, %argb: tensor<?x?xf64, #DCSR>, tensor<?x?xf64, #DCSR>)
+        outs(%xv: tensor<?x?xf64, #DCSR>) {
+        ^bb(%a: f64, %b: f64, %x: f64):
+          %1 = arith.addf %a, %b : f64
+          linalg.yield %1 : f64
+    } -> tensor<?x?xf64, #DCSR>
+    return %0 : tensor<?x?xf64, #DCSR>
+  }
+
+  // Multiplies two sparse matrices element-wise into a new sparse matrix.
+  func @matrix_mul(%arga: tensor<?x?xf64, #DCSR>,
+                   %argb: tensor<?x?xf64, #DCSR>) -> tensor<?x?xf64, #DCSR> {
+    %c0 = arith.constant 0 : index
+    %c1 = arith.constant 1 : index
+    %d0 = tensor.dim %arga, %c0 : tensor<?x?xf64, #DCSR>
+    %d1 = tensor.dim %arga, %c1 : tensor<?x?xf64, #DCSR>
+    %xv = sparse_tensor.init [%d0, %d1] : tensor<?x?xf64, #DCSR>
+    %0 = linalg.generic #trait_op
+       ins(%arga, %argb: tensor<?x?xf64, #DCSR>, tensor<?x?xf64, #DCSR>)
+        outs(%xv: tensor<?x?xf64, #DCSR>) {
+        ^bb(%a: f64, %b: f64, %x: f64):
+          %1 = arith.mulf %a, %b : f64
+          linalg.yield %1 : f64
+    } -> tensor<?x?xf64, #DCSR>
+    return %0 : tensor<?x?xf64, #DCSR>
+  }
+
+  // Dump a sparse matrix.
+  func @dump(%arg0: tensor<?x?xf64, #DCSR>) {
+    %d0 = arith.constant 0.0 : f64
+    %c0 = arith.constant 0 : index
+    %dm = sparse_tensor.convert %arg0 : tensor<?x?xf64, #DCSR> to tensor<?x?xf64>
+    %0 = memref.buffer_cast %dm : memref<?x?xf64>
+    %1 = vector.transfer_read %0[%c0, %c0], %d0: memref<?x?xf64>, vector<4x8xf64>
+    vector.print %1 : vector<4x8xf64>
+    memref.dealloc %0 : memref<?x?xf64>
+    return
+  }
+
+  // Driver method to call and verify matrix kernels.
+  func @entry() {
+    %c0 = arith.constant 0 : index
+    %d1 = arith.constant 1.1 : f64
+
+    // Setup sparse matrices.
+    %m1 = arith.constant sparse<
+       [ [0,0], [0,1], [1,7], [2,2], [2,4], [2,7], [3,0], [3,2], [3,3] ],
+         [ 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0 ]
+    > : tensor<4x8xf64>
+    %m2 = arith.constant sparse<
+       [ [0,0], [0,7], [1,0], [1,6], [2,1], [2,7] ],
+         [6.0, 5.0, 4.0, 3.0, 2.0, 1.0 ]
+    > : tensor<4x8xf64>
+    %sm1 = sparse_tensor.convert %m1 : tensor<4x8xf64> to tensor<?x?xf64, #DCSR>
+    %sm2 = sparse_tensor.convert %m2 : tensor<4x8xf64> to tensor<?x?xf64, #DCSR>
+
+    // Call sparse vector kernels.
+    %0 = call @matrix_scale(%sm1)
+      : (tensor<?x?xf64, #DCSR>) -> tensor<?x?xf64, #DCSR>
+    %1 = call @matrix_scale_inplace(%sm1)
+      : (tensor<?x?xf64, #DCSR>) -> tensor<?x?xf64, #DCSR>
+    %2 = call @matrix_add(%sm1, %sm2)
+      : (tensor<?x?xf64, #DCSR>, tensor<?x?xf64, #DCSR>) -> tensor<?x?xf64, #DCSR>
+    %3 = call @matrix_mul(%sm1, %sm2)
+      : (tensor<?x?xf64, #DCSR>, tensor<?x?xf64, #DCSR>) -> tensor<?x?xf64, #DCSR>
+
+    //
+    // Verify the results.
+    //
+    // CHECK:      ( ( 2, 4, 0, 0, 0, 0, 0, 0 ), ( 0, 0, 0, 0, 0, 0, 0, 6 ), ( 0, 0, 8, 0, 10, 0, 0, 12 ), ( 14, 0, 16, 18, 0, 0, 0, 0 ) )
+    // CHECK-NEXT: ( ( 6, 0, 0, 0, 0, 0, 0, 5 ), ( 4, 0, 0, 0, 0, 0, 3, 0 ), ( 0, 2, 0, 0, 0, 0, 0, 1 ), ( 0, 0, 0, 0, 0, 0, 0, 0 ) )
+    // CHECK-NEXT: ( ( 2, 4, 0, 0, 0, 0, 0, 0 ), ( 0, 0, 0, 0, 0, 0, 0, 6 ), ( 0, 0, 8, 0, 10, 0, 0, 12 ), ( 14, 0, 16, 18, 0, 0, 0, 0 ) )
+    // CHECK-NEXT: ( ( 2, 4, 0, 0, 0, 0, 0, 0 ), ( 0, 0, 0, 0, 0, 0, 0, 6 ), ( 0, 0, 8, 0, 10, 0, 0, 12 ), ( 14, 0, 16, 18, 0, 0, 0, 0 ) )
+    // CHECK-NEXT: ( ( 8, 4, 0, 0, 0, 0, 0, 5 ), ( 4, 0, 0, 0, 0, 0, 3, 6 ), ( 0, 2, 8, 0, 10, 0, 0, 13 ), ( 14, 0, 16, 18, 0, 0, 0, 0 ) )
+    // CHECK-NEXT: ( ( 12, 0, 0, 0, 0, 0, 0, 0 ), ( 0, 0, 0, 0, 0, 0, 0, 0 ), ( 0, 0, 0, 0, 0, 0, 0, 12 ), ( 0, 0, 0, 0, 0, 0, 0, 0 ) )
+    //
+    call @dump(%sm1) : (tensor<?x?xf64, #DCSR>) -> ()
+    call @dump(%sm2) : (tensor<?x?xf64, #DCSR>) -> ()
+    call @dump(%0) : (tensor<?x?xf64, #DCSR>) -> ()
+    call @dump(%1) : (tensor<?x?xf64, #DCSR>) -> ()
+    call @dump(%2) : (tensor<?x?xf64, #DCSR>) -> ()
+    call @dump(%3) : (tensor<?x?xf64, #DCSR>) -> ()
+
+    // Release the resources.
+    sparse_tensor.release %sm1 : tensor<?x?xf64, #DCSR>
+    sparse_tensor.release %sm2 : tensor<?x?xf64, #DCSR>
+    sparse_tensor.release %0 : tensor<?x?xf64, #DCSR>
+    sparse_tensor.release %2 : tensor<?x?xf64, #DCSR>
+    sparse_tensor.release %3 : tensor<?x?xf64, #DCSR>
+    return
+  }
+}

mlir/test/Integration/Dialect/SparseTensor/CPU/sparse_tensor_ops.mlir

+// RUN: mlir-opt %s \
+// RUN:   --sparsification --sparse-tensor-conversion \
+// RUN:   --linalg-bufferize --convert-linalg-to-loops \
+// RUN:   --convert-vector-to-scf --convert-scf-to-std \
+// RUN:   --func-bufferize --tensor-constant-bufferize --tensor-bufferize \
+// RUN:   --std-bufferize --finalizing-bufferize --lower-affine \
+// RUN:   --convert-vector-to-llvm --convert-memref-to-llvm --convert-math-to-llvm \
+// RUN:   --convert-std-to-llvm --reconcile-unrealized-casts | \
+// RUN: mlir-cpu-runner \
+// RUN:  -e entry -entry-point-result=void  \
+// RUN:  -shared-libs=%mlir_integration_test_dir/libmlir_c_runner_utils%shlibext | \
+// RUN: FileCheck %s
+
+#ST1 = #sparse_tensor.encoding<{dimLevelType = ["compressed", "compressed", "compressed"]}>
+#ST2 = #sparse_tensor.encoding<{dimLevelType = ["compressed", "compressed", "dense"]}>
+
+//
+// Trait for 3-d tensor operation.
+//
+#trait_scale = {
+  indexing_maps = [
+    affine_map<(i,j,k) -> (i,j,k)>,  // A (in)
+    affine_map<(i,j,k) -> (i,j,k)>   // X (out)
+  ],
+  iterator_types = ["parallel", "parallel", "parallel"],
+  doc = "X(i,j,k) = A(i,j,k) * 2.0"
+}
+
+module {
+  // Scales a sparse tensor into a new sparse tensor.
+  func @tensor_scale(%arga: tensor<?x?x?xf64, #ST1>) -> tensor<?x?x?xf64, #ST2> {
+    %s = arith.constant 2.0 : f64
+    %c0 = arith.constant 0 : index
+    %c1 = arith.constant 1 : index
+    %c2 = arith.constant 2 : index
+    %d0 = tensor.dim %arga, %c0 : tensor<?x?x?xf64, #ST1>
+    %d1 = tensor.dim %arga, %c1 : tensor<?x?x?xf64, #ST1>
+    %d2 = tensor.dim %arga, %c2 : tensor<?x?x?xf64, #ST1>
+    %xm = sparse_tensor.init [%d0, %d1, %d2] : tensor<?x?x?xf64, #ST2>
+    %0 = linalg.generic #trait_scale
+       ins(%arga: tensor<?x?x?xf64, #ST1>)
+        outs(%xm: tensor<?x?x?xf64, #ST2>) {
+        ^bb(%a: f64, %x: f64):
+          %1 = arith.mulf %a, %s : f64
+          linalg.yield %1 : f64
+    } -> tensor<?x?x?xf64, #ST2>
+    return %0 : tensor<?x?x?xf64, #ST2>
+  }
+
+  // Driver method to call and verify tensor kernel.
+  func @entry() {
+    %c0 = arith.constant 0 : index
+    %d1 = arith.constant -1.0 : f64
+
+    // Setup sparse tensor.
+    %t = arith.constant dense<
+      [ [ [1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 ],
+          [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 ],
+          [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 ],
+          [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0 ] ],
+        [ [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 ],
+          [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 ],
+          [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 ],
+          [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 ] ],
+        [ [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 ],
+          [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 ],
+          [0.0, 3.0, 4.0, 0.0, 0.0, 0.0, 0.0, 5.0 ],
+          [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 ] ] ]> : tensor<3x4x8xf64>
+    %st = sparse_tensor.convert %t : tensor<3x4x8xf64> to tensor<?x?x?xf64, #ST1>
+
+    // Call sparse vector kernels.
+    %0 = call @tensor_scale(%st) : (tensor<?x?x?xf64, #ST1>) -> tensor<?x?x?xf64, #ST2>
+
+    // Sanity check on stored values.
+    //
+    // CHECK:      ( 1, 2, 3, 4, 5, -1, -1, -1 )
+    // CHECK-NEXT: ( 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 6, 8, 0, 0, 0, 0, 10, -1, -1, -1, -1, -1, -1, -1, -1 )
+    %m1 = sparse_tensor.values %st : tensor<?x?x?xf64, #ST1> to memref<?xf64>
+    %m2 = sparse_tensor.values %0  : tensor<?x?x?xf64, #ST2> to memref<?xf64>
+    %v1 = vector.transfer_read %m1[%c0], %d1: memref<?xf64>, vector<8xf64>
+    %v2 = vector.transfer_read %m2[%c0], %d1: memref<?xf64>, vector<32xf64>
+    vector.print %v1 : vector<8xf64>
+    vector.print %v2 : vector<32xf64>
+
+    // Release the resources.
+    sparse_tensor.release %st : tensor<?x?x?xf64, #ST1>
+    sparse_tensor.release %0  : tensor<?x?x?xf64, #ST2>
+    return
+  }
+}