diff --git a/mlir/include/mlir/Dialect/SparseTensor/IR/SparseTensor.h b/mlir/include/mlir/Dialect/SparseTensor/IR/SparseTensor.h
--- a/mlir/include/mlir/Dialect/SparseTensor/IR/SparseTensor.h
+++ b/mlir/include/mlir/Dialect/SparseTensor/IR/SparseTensor.h
@@ -75,6 +75,21 @@
   return isSingletonDLT(getDimLevelType(type, d));
 }
 
+/// Convenience function to test for dense dimension (0 <= d < rank).
+inline bool isDenseDim(SparseTensorEncodingAttr enc, uint64_t d) {
+  return isDenseDLT(getDimLevelType(enc, d));
+}
+
+/// Convenience function to test for compressed dimension (0 <= d < rank).
+inline bool isCompressedDim(SparseTensorEncodingAttr enc, uint64_t d) {
+  return isCompressedDLT(getDimLevelType(enc, d));
+}
+
+/// Convenience function to test for singleton dimension (0 <= d < rank).
+inline bool isSingletonDim(SparseTensorEncodingAttr enc, uint64_t d) {
+  return isSingletonDLT(getDimLevelType(enc, d));
+}
+
 //
 // Dimension level properties.
 //
diff --git a/mlir/lib/Dialect/SparseTensor/Transforms/CodegenUtils.h b/mlir/lib/Dialect/SparseTensor/Transforms/CodegenUtils.h
--- a/mlir/lib/Dialect/SparseTensor/Transforms/CodegenUtils.h
+++ b/mlir/lib/Dialect/SparseTensor/Transforms/CodegenUtils.h
@@ -311,8 +311,282 @@
 }
 
 //===----------------------------------------------------------------------===//
-// SparseTensorLoopEmiter class, manages sparse tensors and helps to generate
-// loop structure to (co)-iterate sparse tensors.
+// SparseTensorDescriptor and helpers, manage the sparse tensor memory layout
+// scheme.
+//
+// Sparse tensor storage scheme for rank-dimensional tensor is organized
+// as a single compound type with the following fields. Note that every
+// memref with ? size actually behaves as a "vector", i.e. the stored
+// size is the capacity and the used size resides in the memSizes array.
+//
+// struct {
+//   memref<rank x index> dimSizes     ; size in each dimension
+//   memref<n x index> memSizes        ; sizes of ptrs/inds/values
+//   ; per-dimension d:
+//   ;  if dense:
+//        <nothing>
+//   ;  if compresed:
+//        memref<? x ptr>  pointers-d  ; pointers for sparse dim d
+//        memref<? x idx>  indices-d   ; indices for sparse dim d
+//   ;  if singleton:
+//        memref<? x idx>  indices-d   ; indices for singleton dim d
+//   memref<? x eltType> values        ; values
+// };
+//
+// The dimSizes array and memSizes array.
+//
+//===----------------------------------------------------------------------===//
+enum class SparseTensorFieldKind {
+  DimSizes,
+  MemSizes,
+  PtrMemRef,
+  IdxMemRef,
+  ValMemRef
+};
+
+constexpr uint64_t dimSizesIdx = 0;
+constexpr uint64_t memSizesIdx = dimSizesIdx + 1;
+constexpr uint64_t dataFieldIdx = memSizesIdx + 1;
+
+/// For each field that will be allocated for the given sparse tensor encoding,
+/// calls the callback with the corresponding field index, field kind, dimension
+/// (for sparse tensor level memrefs) and dimlevelType.
+/// The field index always starts with zero and increments by one between two
+/// callback invocations.
+/// Ideally, all other methods should rely on this function to query a sparse
+/// tensor fields instead of relying ad-hoc index computation.
+void foreachFieldInSparseTensor(
+    SparseTensorEncodingAttr,
+    llvm::function_ref<bool(unsigned /*fieldIdx*/,
+                            SparseTensorFieldKind /*fieldKind*/,
+                            unsigned /*dim (if applicable)*/,
+                            DimLevelType /*DLT (if applicable)*/)>);
+
+/// Same as above, except that it also builds the Type for the corresponding
+/// field.
+void foreachFieldAndTypeInSparseTensor(
+    RankedTensorType,
+    llvm::function_ref<bool(Type /*fieldType*/, unsigned /*fieldIdx*/,
+                            SparseTensorFieldKind /*fieldKind*/,
+                            unsigned /*dim (if applicable)*/,
+                            DimLevelType /*DLT (if applicable)*/)>);
+
+/// Gets the total number of fields for the given sparse tensor encoding.
+unsigned getNumFieldsFromEncoding(SparseTensorEncodingAttr enc);
+
+/// Gets the total number of data fields (index arrays, pointer arrays and a
+/// value array) for the given sparse tensor encoding.
+unsigned getNumDataFieldsFromEncoding(SparseTensorEncodingAttr enc);
+
+/// Get the index of the field in memSizes (only valid for data fields).
+inline unsigned getFieldMemSizesIndex(unsigned fid) {
+  assert(fid >= dataFieldIdx);
+  return fid - dataFieldIdx;
+}
+
+/// A helper class around a array of values that corresponding to a sparse
+/// tensor, provides a set of meaningful APIs to query and update a particular
+/// field in a consistent way.
+/// Users should not make assumption on how a sparse tensor is layouted but
+/// instead relies on this class to access the right value for the right field.
+template <bool mut>
+class SparseTensorDescriptorImpl {
+private:
+  template <bool>
+  struct ArrayStorage;
+
+  template <>
+  struct ArrayStorage<false> {
+    using ValueArray = ValueRange;
+  };
+
+  template <>
+  struct ArrayStorage<true> {
+    using ValueArray = SmallVectorImpl<Value> &;
+  };
+
+  // Uses ValueRange for immuatable descriptors; uses SmallVectorImpl<Value> &
+  // for mutable descriptors.
+  // Using SmallVector for mutable descriptor allows users to reuse it as a tmp
+  // buffers to append value for some special cases, though users should be
+  // responsible to restored the buffer to legal states after their use. It is
+  // probably not a clean way, but it is the most efficient way to avoid copying
+  // the fields into another SmallVector. If a more clear way is wanted, we
+  // should change it to MutableArrayRef instead.
+  using Storage = typename ArrayStorage<mut>::ValueArray;
+
+public:
+  SparseTensorDescriptorImpl(Type tp, Storage fields)
+      : rType(tp.cast<RankedTensorType>()), fields(fields) {
+    assert(getSparseTensorEncoding(tp) &&
+           getNumFieldsFromEncoding(getSparseTensorEncoding(tp)) ==
+               fields.size());
+    // We should make sure the class is trivially copyable (and should be small
+    // enough) such that we can pass it by value.
+    static_assert(
+        std::is_trivially_copyable_v<SparseTensorDescriptorImpl<mut>>);
+  }
+
+  // Implicit (and cheap) type conversion from MutSparseTensorDescriptor to
+  // SparseTensorDescriptor.
+  template <typename T = SparseTensorDescriptorImpl<true>>
+  /*implicit*/ SparseTensorDescriptorImpl(std::enable_if_t<!mut, T> &mDesc)
+      : rType(mDesc.getTensorType()), fields(mDesc.getFields()) {}
+
+  ///
+  /// Getters: get the field index for required field.
+  ///
+
+  unsigned getPtrMemRefIndex(unsigned ptrDim) const {
+    return getFieldIndex(ptrDim, SparseTensorFieldKind::PtrMemRef);
+  }
+
+  unsigned getIdxMemRefIndex(unsigned idxDim) const {
+    return getFieldIndex(idxDim, SparseTensorFieldKind::IdxMemRef);
+  }
+
+  unsigned getDataFieldIndex(unsigned dim, SparseTensorFieldKind kind) const {
+    if (kind == SparseTensorFieldKind::ValMemRef)
+      return getValMemRefIndex();
+    return getFieldIndex(dim, kind);
+  }
+
+  unsigned getValMemRefIndex() const { return fields.size() - 1; }
+
+  unsigned getPtrMemSizesIndex(unsigned dim) const {
+    return getPtrMemRefIndex(dim) - dataFieldIdx;
+  }
+
+  unsigned getIdxMemSizesIndex(unsigned dim) const {
+    return getIdxMemRefIndex(dim) - dataFieldIdx;
+  }
+
+  unsigned getValMemSizesIndex() const {
+    return getValMemRefIndex() - dataFieldIdx;
+  }
+
+  unsigned getFieldMemSizesIndex(unsigned dim,
+                                 SparseTensorFieldKind kind) const {
+    return getFieldMemSizesIndex(getDataFieldIndex(dim, kind));
+  }
+
+  unsigned getNumFields() const { return fields.size(); }
+
+  ///
+  /// Getters: get the value for required field.
+  ///
+
+  Value getDimSizesMemRef() const { return fields[dimSizesIdx]; }
+  Value getMemSizesMemRef() const { return fields[memSizesIdx]; }
+
+  Value getPtrMemRef(unsigned ptrDim) const {
+    return fields[getPtrMemRefIndex(ptrDim)];
+  }
+
+  Value getIdxMemRef(unsigned idxDim) const {
+    return fields[getIdxMemRefIndex(idxDim)];
+  }
+
+  Value getValMemRef() const { return fields[getValMemRefIndex()]; }
+
+  Value getField(unsigned fid) const {
+    assert(fid < fields.size());
+    return fields[fid];
+  }
+
+  ///
+  /// Setters: update the value for required field (only enabled for
+  /// MutSparseTensorDescriptor).
+  ///
+
+  template <typename T = Value>
+  void setDimSizesMemRef(std::enable_if_t<mut, T> v) {
+    fields[dimSizesIdx] = v;
+  }
+
+  template <typename T = Value>
+  void setMemSizesMemRef(std::enable_if_t<mut, T> v) {
+    fields[memSizesIdx] = v;
+  }
+
+  template <typename T = Value>
+  void setPtrMemRef(unsigned ptrDim, std::enable_if_t<mut, T> v) {
+    fields[getPtrMemRefIndex(ptrDim)] = v;
+  }
+
+  template <typename T = Value>
+  void setIdxMemRef(unsigned idxDim, std::enable_if_t<mut, T> v) {
+    fields[getIdxMemRefIndex(idxDim)] = v;
+  }
+
+  template <typename T = Value>
+  void setLvlMemRef(unsigned dim, SparseTensorFieldKind kind,
+                    std::enable_if_t<mut, T> v) {
+    fields[getDataFieldIndex(dim, kind)] = v;
+  }
+
+  template <typename T = Value>
+  void setValMemRef(std::enable_if_t<mut, T> v) {
+    fields[getValMemRefIndex()] = v;
+  }
+
+  template <typename T = Value>
+  void setField(unsigned fid, std::enable_if_t<mut, T> v) {
+    assert(fid < fields.size());
+    fields[fid] = v;
+  }
+
+  RankedTensorType getTensorType() const { return rType; }
+  Storage getFields() const { return fields; }
+
+  Type getElementType(unsigned fidx) const {
+    return fields[fidx].getType().template cast<MemRefType>().getElementType();
+  }
+
+  // TODO: a better places for these functions should be in
+  // SparseTensorEncodingAttr.
+  Type getPtrElementType() const {
+    auto *ctx = rType.getContext();
+    unsigned ptrWidth = getSparseTensorEncoding(rType).getPointerBitWidth();
+    Type indexType = IndexType::get(ctx);
+    return ptrWidth ? IntegerType::get(ctx, ptrWidth) : indexType;
+  }
+
+  Type getIdxElementType() const {
+    auto *ctx = rType.getContext();
+    unsigned idxWidth = getSparseTensorEncoding(rType).getIndexBitWidth();
+    Type indexType = IndexType::get(ctx);
+    return idxWidth ? IntegerType::get(ctx, idxWidth) : indexType;
+  }
+
+private:
+  unsigned getFieldIndex(unsigned dim, SparseTensorFieldKind kind) const {
+    unsigned fieldIdx = -1u;
+    foreachFieldInSparseTensor(
+        getSparseTensorEncoding(rType),
+        [dim, kind, &fieldIdx](unsigned fIdx, SparseTensorFieldKind fKind,
+                               unsigned fDim, DimLevelType dlt) -> bool {
+          if (fDim == dim && kind == fKind) {
+            fieldIdx = fIdx;
+            // Returns false to break the iteration.
+            return false;
+          }
+          return true;
+        });
+    assert(fieldIdx != -1u);
+    return fieldIdx;
+  }
+
+  RankedTensorType rType;
+  Storage fields;
+};
+
+using SparseTensorDescriptor = SparseTensorDescriptorImpl<false>;
+using MutSparseTensorDescriptor = SparseTensorDescriptorImpl<true>;
+
+//===----------------------------------------------------------------------===//
+// SparseTensorLoopEmiter class, manages sparse tensors and helps to
+// generate loop structure to (co)-iterate sparse tensors.
 //
 // An example usage:
 // To generate the following loops over T1<?x?> and T2<?x?>
@@ -345,15 +619,15 @@
   using OutputUpdater = function_ref<Value(OpBuilder &builder, Location loc,
                                            Value memref, Value tensor)>;
 
-  /// Constructor: take an array of tensors inputs, on which the generated loops
-  /// will iterate on. The index of the tensor in the array is also the
+  /// Constructor: take an array of tensors inputs, on which the generated
+  /// loops will iterate on. The index of the tensor in the array is also the
   /// tensor id (tid) used in related functions.
   /// If isSparseOut is set, loop emitter assume that the sparse output tensor
   /// is empty, and will always generate loops on it based on the dim sizes.
   /// An optional array could be provided (by sparsification) to indicate the
   /// loop id sequence that will be generated. It is used to establish the
-  /// mapping between affineDimExpr to the corresponding loop index in the loop
-  /// stack that are maintained by the loop emitter.
+  /// mapping between affineDimExpr to the corresponding loop index in the
+  /// loop stack that are maintained by the loop emitter.
   explicit SparseTensorLoopEmitter(ValueRange tensors,
                                    StringAttr loopTag = nullptr,
                                    bool hasOutput = false,
@@ -368,8 +642,8 @@
   /// Generates a list of operations to compute the affine expression.
   Value genAffine(OpBuilder &builder, AffineExpr a, Location loc);
 
-  /// Enters a new loop sequence, the loops within the same sequence starts from
-  /// the break points of previous loop instead of starting over from 0.
+  /// Enters a new loop sequence, the loops within the same sequence starts
+  /// from the break points of previous loop instead of starting over from 0.
   /// e.g.,
   /// {
   ///   // loop sequence start.
@@ -524,10 +798,10 @@
   ///     scf.reduce.return %val
   ///   }
   /// }
-  /// NOTE: only one instruction will be moved into reduce block, transformation
-  /// will fail if multiple instructions are used to compute the reduction
-  /// value.
-  /// Return %ret to user, while %val is provided by users (`reduc`).
+  /// NOTE: only one instruction will be moved into reduce block,
+  /// transformation will fail if multiple instructions are used to compute
+  /// the reduction value. Return %ret to user, while %val is provided by
+  /// users (`reduc`).
   void exitForLoop(RewriterBase &rewriter, Location loc,
                    MutableArrayRef<Value> reduc);
 
@@ -535,9 +809,9 @@
   void exitCoIterationLoop(OpBuilder &builder, Location loc,
                            MutableArrayRef<Value> reduc);
 
-  /// A optional string attribute that should be attached to the loop generated
-  /// by loop emitter, it might help following passes to identify loops that
-  /// operates on sparse tensors more easily.
+  /// A optional string attribute that should be attached to the loop
+  /// generated by loop emitter, it might help following passes to identify
+  /// loops that operates on sparse tensors more easily.
   StringAttr loopTag;
   /// Whether the loop emitter needs to treat the last tensor as the output
   /// tensor.
@@ -556,7 +830,8 @@
   std::vector<std::vector<Value>> idxBuffer; // to_indices
   std::vector<Value> valBuffer;              // to_value
 
-  // Loop Stack, stores the information of all the nested loops that are alive.
+  // Loop Stack, stores the information of all the nested loops that are
+  // alive.
   std::vector<LoopLevelInfo> loopStack;
 
   // Loop Sequence Stack, stores the unversial index for the current loop
diff --git a/mlir/lib/Dialect/SparseTensor/Transforms/CodegenUtils.cpp b/mlir/lib/Dialect/SparseTensor/Transforms/CodegenUtils.cpp
--- a/mlir/lib/Dialect/SparseTensor/Transforms/CodegenUtils.cpp
+++ b/mlir/lib/Dialect/SparseTensor/Transforms/CodegenUtils.cpp
@@ -90,6 +90,118 @@
   return val;
 }
 
+void sparse_tensor::foreachFieldInSparseTensor(
+    const SparseTensorEncodingAttr enc,
+    llvm::function_ref<bool(unsigned, SparseTensorFieldKind, unsigned,
+                            DimLevelType)>
+        callback) {
+  assert(enc);
+
+#define RETURN_ON_FALSE(idx, kind, dim, dlt)                                   \
+  if (!(callback(idx, kind, dim, dlt)))                                        \
+    return;
+
+  RETURN_ON_FALSE(dimSizesIdx, SparseTensorFieldKind::DimSizes, -1u,
+                  DimLevelType::Undef);
+  RETURN_ON_FALSE(memSizesIdx, SparseTensorFieldKind::MemSizes, -1u,
+                  DimLevelType::Undef);
+
+  static_assert(dataFieldIdx == memSizesIdx + 1);
+  unsigned fieldIdx = dataFieldIdx;
+  // Per-dimension storage.
+  for (unsigned r = 0, rank = enc.getDimLevelType().size(); r < rank; r++) {
+    // Dimension level types apply in order to the reordered dimension.
+    // As a result, the compound type can be constructed directly in the given
+    // order.
+    auto dlt = getDimLevelType(enc, r);
+    if (isCompressedDLT(dlt)) {
+      RETURN_ON_FALSE(fieldIdx++, SparseTensorFieldKind::PtrMemRef, r, dlt);
+      RETURN_ON_FALSE(fieldIdx++, SparseTensorFieldKind::IdxMemRef, r, dlt);
+    } else if (isSingletonDLT(dlt)) {
+      RETURN_ON_FALSE(fieldIdx++, SparseTensorFieldKind::IdxMemRef, r, dlt);
+    } else {
+      assert(isDenseDLT(dlt)); // no fields
+    }
+  }
+  // The values array.
+  RETURN_ON_FALSE(fieldIdx++, SparseTensorFieldKind::ValMemRef, -1u,
+                  DimLevelType::Undef);
+
+#undef RETURN_ON_FALSE
+}
+
+void sparse_tensor::foreachFieldAndTypeInSparseTensor(
+    RankedTensorType rType,
+    llvm::function_ref<bool(Type, unsigned, SparseTensorFieldKind, unsigned,
+                            DimLevelType)>
+        callback) {
+  auto enc = getSparseTensorEncoding(rType);
+  assert(enc);
+  // Construct the basic types.
+  auto *context = rType.getContext();
+  unsigned idxWidth = enc.getIndexBitWidth();
+  unsigned ptrWidth = enc.getPointerBitWidth();
+  Type indexType = IndexType::get(context);
+  Type idxType = idxWidth ? IntegerType::get(context, idxWidth) : indexType;
+  Type ptrType = ptrWidth ? IntegerType::get(context, ptrWidth) : indexType;
+  Type eltType = rType.getElementType();
+  unsigned rank = rType.getShape().size();
+  // memref<rank x index> dimSizes
+  Type dimSizeType = MemRefType::get({rank}, indexType);
+  // memref<n x index> memSizes
+  Type memSizeType =
+      MemRefType::get({getNumDataFieldsFromEncoding(enc)}, indexType);
+  // memref<? x ptr>  pointers
+  Type ptrMemType = MemRefType::get({ShapedType::kDynamic}, ptrType);
+  // memref<? x idx>  indices
+  Type idxMemType = MemRefType::get({ShapedType::kDynamic}, idxType);
+  // memref<? x eltType> values
+  Type valMemType = MemRefType::get({ShapedType::kDynamic}, eltType);
+
+  foreachFieldInSparseTensor(
+      enc,
+      [dimSizeType, memSizeType, ptrMemType, idxMemType, valMemType,
+       callback](unsigned fieldIdx, SparseTensorFieldKind fieldKind,
+                 unsigned dim, DimLevelType dlt) -> bool {
+        switch (fieldKind) {
+        case SparseTensorFieldKind::DimSizes:
+          return callback(dimSizeType, fieldIdx, fieldKind, dim, dlt);
+        case SparseTensorFieldKind::MemSizes:
+          return callback(memSizeType, fieldIdx, fieldKind, dim, dlt);
+        case SparseTensorFieldKind::PtrMemRef:
+          return callback(ptrMemType, fieldIdx, fieldKind, dim, dlt);
+        case SparseTensorFieldKind::IdxMemRef:
+          return callback(idxMemType, fieldIdx, fieldKind, dim, dlt);
+        case SparseTensorFieldKind::ValMemRef:
+          return callback(valMemType, fieldIdx, fieldKind, dim, dlt);
+        };
+      });
+}
+
+unsigned sparse_tensor::getNumFieldsFromEncoding(SparseTensorEncodingAttr enc) {
+  unsigned numFields = 0;
+  foreachFieldInSparseTensor(enc,
+                             [&numFields](unsigned, SparseTensorFieldKind,
+                                          unsigned, DimLevelType) -> bool {
+                               numFields++;
+                               return true;
+                             });
+  return numFields;
+}
+
+unsigned
+sparse_tensor::getNumDataFieldsFromEncoding(SparseTensorEncodingAttr enc) {
+  unsigned numFields = 0; // one value memref
+  foreachFieldInSparseTensor(enc,
+                             [&numFields](unsigned fidx, SparseTensorFieldKind,
+                                          unsigned, DimLevelType) -> bool {
+                               if (fidx >= dataFieldIdx)
+                                 numFields++;
+                               return true;
+                             });
+  assert(numFields == getNumFieldsFromEncoding(enc) - dataFieldIdx);
+  return numFields;
+}
 //===----------------------------------------------------------------------===//
 // Sparse tensor loop emitter class implementations
 //===----------------------------------------------------------------------===//
diff --git a/mlir/lib/Dialect/SparseTensor/Transforms/SparseTensorCodegen.cpp b/mlir/lib/Dialect/SparseTensor/Transforms/SparseTensorCodegen.cpp
--- a/mlir/lib/Dialect/SparseTensor/Transforms/SparseTensorCodegen.cpp
+++ b/mlir/lib/Dialect/SparseTensor/Transforms/SparseTensorCodegen.cpp
@@ -30,11 +30,6 @@
 using namespace mlir::sparse_tensor;
 
 namespace {
-
-static constexpr uint64_t dimSizesIdx = 0;
-static constexpr uint64_t memSizesIdx = 1;
-static constexpr uint64_t fieldsIdx = 2;
-
 //===----------------------------------------------------------------------===//
 // Helper methods.
 //===----------------------------------------------------------------------===//
@@ -44,6 +39,11 @@
   return llvm::cast<UnrealizedConversionCastOp>(tensor.getDefiningOp());
 }
 
+static SparseTensorDescriptor getDescriptorFromTensorTuple(Value tensor) {
+  auto tuple = getTuple(tensor);
+  return SparseTensorDescriptor(tuple.getResultTypes()[0], tuple.getInputs());
+}
+
 /// Packs the given values as a "tuple" value.
 static Value genTuple(OpBuilder &builder, Location loc, Type tp,
                       ValueRange values) {
@@ -51,6 +51,14 @@
       .getResult(0);
 }
 
+static Value genTuple(OpBuilder &builder, Location loc,
+                      SparseTensorDescriptor desc) {
+  return builder
+      .create<UnrealizedConversionCastOp>(loc, desc.getTensorType(),
+                                          desc.getFields())
+      .getResult(0);
+}
+
 /// Flatten a list of operands that may contain sparse tensors.
 static void flattenOperands(ValueRange operands,
                             SmallVectorImpl<Value> &flattened) {
@@ -96,7 +104,7 @@
 
 /// Creates a straightforward counting for-loop.
 static scf::ForOp createFor(OpBuilder &builder, Location loc, Value upper,
-                            SmallVectorImpl<Value> &fields,
+                            MutableArrayRef<Value> fields,
                             Value lower = Value()) {
   Type indexType = builder.getIndexType();
   if (!lower)
@@ -113,81 +121,58 @@
 /// original dimension 'dim'. Returns None if no sparse encoding is
 /// attached to the given tensor type.
 static Optional<Value> sizeFromTensorAtDim(OpBuilder &builder, Location loc,
-                                           RankedTensorType tensorTp,
-                                           Value adaptedValue, unsigned dim) {
-  auto enc = getSparseTensorEncoding(tensorTp);
-  if (!enc)
-    return llvm::None;
-
+                                           SparseTensorDescriptor desc,
+                                           unsigned dim) {
+  RankedTensorType rtp = desc.getTensorType();
   // Access into static dimension can query original type directly.
   // Note that this is typically already done by DimOp's folding.
-  auto shape = tensorTp.getShape();
+  auto shape = rtp.getShape();
   if (!ShapedType::isDynamic(shape[dim]))
     return constantIndex(builder, loc, shape[dim]);
 
   // Any other query can consult the dimSizes array at field DimSizesIdx,
   // accounting for the reordering applied to the sparse storage.
-  auto tuple = getTuple(adaptedValue);
-  Value idx = constantIndex(builder, loc, toStoredDim(tensorTp, dim));
-  return builder
-      .create<memref::LoadOp>(loc, tuple.getInputs()[dimSizesIdx], idx)
+  Value idx = constantIndex(builder, loc, toStoredDim(rtp, dim));
+  return builder.create<memref::LoadOp>(loc, desc.getDimSizesMemRef(), idx)
       .getResult();
 }
 
+//// Gets the dimension size at the given stored dimension 'd', either as a
+//// constant for a static size, or otherwise dynamically descizes.
+// Value sizeAtStoredDim(OpBuilder &builder, Location loc, Rankdesc rtp,
+//                      SmallVectorImpl<Value> &fieldesc d) {
+//  unsigned dim = tdesc, d);
+//  auto shape descpe();
+//  if (!ShapedType::isDyndescim]))
+//    return constantIndex(builder, descim]);
+//  return genLoad(builder, loc, fiedescIdx],
+//                 constandescdesc;
+//}
+
 // Gets the dimension size at the given stored dimension 'd', either as a
 // constant for a static size, or otherwise dynamically through memSizes.
-Value sizeAtStoredDim(OpBuilder &builder, Location loc, RankedTensorType rtp,
-                      SmallVectorImpl<Value> &fields, unsigned d) {
+Value sizeAtStoredDim(OpBuilder &builder, Location loc,
+                      SparseTensorDescriptor desc, unsigned d) {
+  RankedTensorType rtp = desc.getTensorType();
   unsigned dim = toOrigDim(rtp, d);
   auto shape = rtp.getShape();
   if (!ShapedType::isDynamic(shape[dim]))
     return constantIndex(builder, loc, shape[dim]);
-  return genLoad(builder, loc, fields[dimSizesIdx],
-                 constantIndex(builder, loc, d));
-}
 
-/// Translates field index to memSizes index.
-static unsigned getMemSizesIndex(unsigned field) {
-  assert(fieldsIdx <= field);
-  return field - fieldsIdx;
+  return genLoad(builder, loc, desc.getDimSizesMemRef(),
+                 constantIndex(builder, loc, d));
 }
 
-/// Creates a pushback op for given field and updates the fields array
-/// accordingly. This operation also updates the memSizes contents.
 static void createPushback(OpBuilder &builder, Location loc,
-                           SmallVectorImpl<Value> &fields, unsigned field,
+                           MutSparseTensorDescriptor desc, unsigned fidx,
                            Value value, Value repeat = Value()) {
-  assert(fieldsIdx <= field && field < fields.size());
-  Type etp = fields[field].getType().cast<ShapedType>().getElementType();
-  fields[field] = builder.create<PushBackOp>(
-      loc, fields[field].getType(), fields[memSizesIdx], fields[field],
-      toType(builder, loc, value, etp), APInt(64, getMemSizesIndex(field)),
+  Type etp = desc.getElementType(fidx);
+  Value field = desc.getField(fidx);
+  Value newField = builder.create<PushBackOp>(
+      loc, field.getType(), desc.getMemSizesMemRef(), field,
+      toType(builder, loc, value, etp), APInt(64, getFieldMemSizesIndex(fidx)),
       repeat);
-}
-
-/// Returns field index of sparse tensor type for pointers/indices, when set.
-static unsigned getFieldIndex(Type type, unsigned ptrDim, unsigned idxDim) {
-  assert(getSparseTensorEncoding(type));
-  RankedTensorType rType = type.cast<RankedTensorType>();
-  unsigned field = fieldsIdx; // start past header
-  for (unsigned r = 0, rank = rType.getShape().size(); r < rank; r++) {
-    if (isCompressedDim(rType, r)) {
-      if (r == ptrDim)
-        return field;
-      field++;
-      if (r == idxDim)
-        return field;
-      field++;
-    } else if (isSingletonDim(rType, r)) {
-      if (r == idxDim)
-        return field;
-      field++;
-    } else {
-      assert(isDenseDim(rType, r)); // no fields
-    }
-  }
-  assert(ptrDim == -1u && idxDim == -1u);
-  return field + 1; // return values field index
+  desc.setField(fidx, newField);
 }
 
 /// Maps a sparse tensor type to the appropriate compounded buffers.
@@ -196,66 +181,24 @@
   auto enc = getSparseTensorEncoding(type);
   if (!enc)
     return llvm::None;
-  // Construct the basic types.
-  auto *context = type.getContext();
-  unsigned idxWidth = enc.getIndexBitWidth();
-  unsigned ptrWidth = enc.getPointerBitWidth();
+
   RankedTensorType rType = type.cast<RankedTensorType>();
-  Type indexType = IndexType::get(context);
-  Type idxType = idxWidth ? IntegerType::get(context, idxWidth) : indexType;
-  Type ptrType = ptrWidth ? IntegerType::get(context, ptrWidth) : indexType;
-  Type eltType = rType.getElementType();
-  //
-  // Sparse tensor storage scheme for rank-dimensional tensor is organized
-  // as a single compound type with the following fields. Note that every
-  // memref with ? size actually behaves as a "vector", i.e. the stored
-  // size is the capacity and the used size resides in the memSizes array.
-  //
-  // struct {
-  //   memref<rank x index> dimSizes     ; size in each dimension
-  //   memref<n x index> memSizes        ; sizes of ptrs/inds/values
-  //   ; per-dimension d:
-  //   ;  if dense:
-  //        <nothing>
-  //   ;  if compresed:
-  //        memref<? x ptr>  pointers-d  ; pointers for sparse dim d
-  //        memref<? x idx>  indices-d   ; indices for sparse dim d
-  //   ;  if singleton:
-  //        memref<? x idx>  indices-d   ; indices for singleton dim d
-  //   memref<? x eltType> values        ; values
-  // };
-  //
-  unsigned rank = rType.getShape().size();
-  unsigned lastField = getFieldIndex(type, -1u, -1u);
-  // The dimSizes array and memSizes array.
-  fields.push_back(MemRefType::get({rank}, indexType));
-  fields.push_back(MemRefType::get({getMemSizesIndex(lastField)}, indexType));
-  // Per-dimension storage.
-  for (unsigned r = 0; r < rank; r++) {
-    // Dimension level types apply in order to the reordered dimension.
-    // As a result, the compound type can be constructed directly in the given
-    // order. Clients of this type know what field is what from the sparse
-    // tensor type.
-    if (isCompressedDim(rType, r)) {
-      fields.push_back(MemRefType::get({ShapedType::kDynamic}, ptrType));
-      fields.push_back(MemRefType::get({ShapedType::kDynamic}, idxType));
-    } else if (isSingletonDim(rType, r)) {
-      fields.push_back(MemRefType::get({ShapedType::kDynamic}, idxType));
-    } else {
-      assert(isDenseDim(rType, r)); // no fields
-    }
-  }
-  // The values array.
-  fields.push_back(MemRefType::get({ShapedType::kDynamic}, eltType));
-  assert(fields.size() == lastField);
+  foreachFieldAndTypeInSparseTensor(
+      rType,
+      [&fields](Type fieldType, unsigned fieldIdx,
+                SparseTensorFieldKind /*fieldKind*/, unsigned /*dim*/,
+                DimLevelType /*dlt*/) -> bool {
+        assert(fieldIdx == fields.size());
+        fields.push_back(fieldType);
+        return true;
+      });
   return success();
 }
 
 /// Generates code that allocates a sparse storage scheme for given rank.
 static void allocSchemeForRank(OpBuilder &builder, Location loc,
-                               RankedTensorType rtp,
-                               SmallVectorImpl<Value> &fields, unsigned field,
-                               unsigned r0) {
+                               MutSparseTensorDescriptor desc, unsigned r0) {
+  RankedTensorType rtp = desc.getTensorType();
   unsigned rank = rtp.getShape().size();
   Value linear = constantIndex(builder, loc, 1);
   for (unsigned r = r0; r < rank; r++) {
@@ -263,36 +206,35 @@
       // Append linear x pointers, initialized to zero. Since each compressed
       // dimension initially already has a single zero entry, this maintains
       // the desired "linear + 1" length property at all times.
-      unsigned ptrWidth = getSparseTensorEncoding(rtp).getPointerBitWidth();
-      Type indexType = builder.getIndexType();
-      Type ptrType = ptrWidth ? builder.getIntegerType(ptrWidth) : indexType;
+      Type ptrType = desc.getPtrElementType();
       Value ptrZero = constantZero(builder, loc, ptrType);
-      createPushback(builder, loc, fields, field, ptrZero, linear);
+      unsigned fidx = desc.getPtrMemRefIndex(r);
+      createPushback(builder, loc, desc, fidx, ptrZero, linear);
       return;
     }
     if (isSingletonDim(rtp, r)) {
       return; // nothing to do
     }         // Keep compounding the size, but nothing needs to be initialized
-      // at this level. We will eventually reach a compressed level or
-      // otherwise the values array for the from-here "all-dense" case.
-      assert(isDenseDim(rtp, r));
-      Value size = sizeAtStoredDim(builder, loc, rtp, fields, r);
-      linear = builder.create<arith::MulIOp>(loc, linear, size);
+    // at this level. We will eventually reach a compressed level or
+    // otherwise the values array for the from-here "all-dense" case.
+    assert(isDenseDim(rtp, r));
+    Value size = sizeAtStoredDim(builder, loc, desc, r);
+    linear = builder.create<arith::MulIOp>(loc, linear, size);
   }
   // Reached values array so prepare for an insertion.
   Value valZero = constantZero(builder, loc, rtp.getElementType());
-  createPushback(builder, loc, fields, field, valZero, linear);
-  assert(fields.size() == ++field);
+  createPushback(builder, loc, desc, desc.getValMemRefIndex(), valZero, linear);
 }
 
 /// Creates allocation operation.
-static Value createAllocation(OpBuilder &builder, Location loc, Type type,
-                              Value sz, bool enableInit) {
-  auto memType = MemRefType::get({ShapedType::kDynamic}, type);
-  Value buffer = builder.create<memref::AllocOp>(loc, memType, sz);
+static Value createAllocation(OpBuilder &builder, Location loc,
+                              MemRefType memRefType, Value sz,
+                              bool enableInit) {
+  Value buffer = builder.create<memref::AllocOp>(loc, memRefType, sz);
+  Type elemType = memRefType.getElementType();
   if (enableInit) {
-    Value fillValue =
-        builder.create<arith::ConstantOp>(loc, type, builder.getZeroAttr(type));
+    Value fillValue = builder.create<arith::ConstantOp>(
+        loc, elemType, builder.getZeroAttr(elemType));
     builder.create<linalg::FillOp>(loc, fillValue, buffer);
   }
   return buffer;
@@ -308,72 +250,69 @@
 static void createAllocFields(OpBuilder &builder, Location loc, Type type,
                               ValueRange dynSizes, bool enableInit,
                               SmallVectorImpl<Value> &fields) {
-  auto enc = getSparseTensorEncoding(type);
-  assert(enc);
-  // Construct the basic types.
-  unsigned idxWidth = enc.getIndexBitWidth();
-  unsigned ptrWidth = enc.getPointerBitWidth();
   RankedTensorType rtp = type.cast<RankedTensorType>();
-  Type indexType = builder.getIndexType();
-  Type idxType = idxWidth ? builder.getIntegerType(idxWidth) : indexType;
-  Type ptrType = ptrWidth ? builder.getIntegerType(ptrWidth) : indexType;
-  Type eltType = rtp.getElementType();
-  auto shape = rtp.getShape();
-  unsigned rank = shape.size();
   Value heuristic = constantIndex(builder, loc, 16);
+
+  foreachFieldAndTypeInSparseTensor(
+      rtp,
+      [&](Type fType, unsigned fIdx, SparseTensorFieldKind fKind,
+          unsigned /*dim*/, DimLevelType /*dlt*/) -> bool {
+        assert(fields.size() == fIdx);
+        auto memRefTp = fType.cast<MemRefType>();
+        Value field;
+        switch (fKind) {
+        case SparseTensorFieldKind::DimSizes:
+        case SparseTensorFieldKind::MemSizes:
+          field = builder.create<memref::AllocOp>(loc, memRefTp);
+          break;
+        case SparseTensorFieldKind::PtrMemRef:
+        case SparseTensorFieldKind::IdxMemRef:
+        case SparseTensorFieldKind::ValMemRef:
+          field =
+              createAllocation(builder, loc, memRefTp, heuristic, enableInit);
+          break;
+        }
+        assert(field);
+        fields.push_back(field);
+        // Returns true to ontinue the iteration.
+        return true;
+      });
+
+  MutSparseTensorDescriptor desc(rtp, fields);
+
   // Build original sizes.
   SmallVector<Value> sizes;
+  auto shape = rtp.getShape();
+  unsigned rank = shape.size();
   for (unsigned r = 0, o = 0; r < rank; r++) {
     if (ShapedType::isDynamic(shape[r]))
       sizes.push_back(dynSizes[o++]);
     else
       sizes.push_back(constantIndex(builder, loc, shape[r]));
   }
-  // The dimSizes array and memSizes array.
-  unsigned lastField = getFieldIndex(type, -1u, -1u);
-  Value dimSizes =
-      builder.create<memref::AllocOp>(loc, MemRefType::get({rank}, indexType));
-  Value memSizes = builder.create<memref::AllocOp>(
-      loc, MemRefType::get({getMemSizesIndex(lastField)}, indexType));
-  fields.push_back(dimSizes);
-  fields.push_back(memSizes);
-  // Per-dimension storage.
-  for (unsigned r = 0; r < rank; r++) {
-    if (isCompressedDim(rtp, r)) {
-      fields.push_back(
-          createAllocation(builder, loc, ptrType, heuristic, enableInit));
-      fields.push_back(
-          createAllocation(builder, loc, idxType, heuristic, enableInit));
-    } else if (isSingletonDim(rtp, r)) {
-      fields.push_back(
-          createAllocation(builder, loc, idxType, heuristic, enableInit));
-    } else {
-      assert(isDenseDim(rtp, r)); // no fields
-    }
-  }
-  // The values array.
-  fields.push_back(
-      createAllocation(builder, loc, eltType, heuristic, enableInit));
-  assert(fields.size() == lastField);
   // Initialize the storage scheme to an empty tensor. Initialized memSizes
   // to all zeros, sets the dimSizes to known values and gives all pointer
   // fields an initial zero entry, so that it is easier to maintain the
   // "linear + 1" length property.
   builder.create<linalg::FillOp>(
-      loc, ValueRange{constantZero(builder, loc, indexType)},
-      ValueRange{memSizes}); // zero memSizes
-  Value ptrZero = constantZero(builder, loc, ptrType);
-  for (unsigned r = 0, field = fieldsIdx; r < rank; r++) {
+      loc, constantZero(builder, loc, builder.getIndexType()),
+      desc.getMemSizesMemRef()); // zero memSizes
+
+  Value ptrZero = constantZero(builder, loc, desc.getPtrElementType());
+  for (unsigned r = 0; r < rank; r++) {
     unsigned ro = toOrigDim(rtp, r);
-    genStore(builder, loc, sizes[ro], dimSizes, constantIndex(builder, loc, r));
+    // Fills dim sizes array.
+    genStore(builder, loc, sizes[ro], desc.getDimSizesMemRef(),
+             constantIndex(builder, loc, r));
+
+    // Pushes a leading zero to pointers memref.
     if (isCompressedDim(rtp, r)) {
-      createPushback(builder, loc, fields, field, ptrZero);
-      field += 2;
-    } else if (isSingletonDim(rtp, r)) {
-      field += 1;
+      unsigned fidx =
+          desc.getDataFieldIndex(r, SparseTensorFieldKind::PtrMemRef);
+      createPushback(builder, loc, desc, fidx, ptrZero);
     }
   }
-  allocSchemeForRank(builder, loc, rtp, fields, fieldsIdx, /*rank=*/0);
+  allocSchemeForRank(builder, loc, desc, /*rank=*/0);
 }
 
 /// Helper method that generates block specific to compressed case:
@@ -397,19 +336,22 @@
 ///  }
 ///  pos[d] = next
 static Value genCompressed(OpBuilder &builder, Location loc,
-                           RankedTensorType rtp, SmallVectorImpl<Value> &fields,
+                           MutSparseTensorDescriptor desc,
                            SmallVectorImpl<Value> &indices, Value value,
-                           Value pos, unsigned field, unsigned d) {
+                           Value pos, unsigned d) {
+  RankedTensorType rtp = desc.getTensorType();
   unsigned rank = rtp.getShape().size();
   SmallVector<Type> types;
   Type indexType = builder.getIndexType();
   Type boolType = builder.getIntegerType(1);
+  unsigned idxIndex = desc.getIdxMemRefIndex(d);
+  unsigned ptrIndex = desc.getPtrMemRefIndex(d);
   Value one = constantIndex(builder, loc, 1);
   Value pp1 = builder.create<arith::AddIOp>(loc, pos, one);
-  Value plo = genLoad(builder, loc, fields[field], pos);
-  Value phi = genLoad(builder, loc, fields[field], pp1);
-  Value psz = constantIndex(builder, loc, getMemSizesIndex(field + 1));
-  Value msz = genLoad(builder, loc, fields[memSizesIdx], psz);
+  Value plo = genLoad(builder, loc, desc.getField(ptrIndex), pos);
+  Value phi = genLoad(builder, loc, desc.getField(ptrIndex), pp1);
+  Value psz = constantIndex(builder, loc, getFieldMemSizesIndex(idxIndex));
+  Value msz = genLoad(builder, loc, desc.getMemSizesMemRef(), psz);
   Value phim1 = builder.create<arith::SubIOp>(
       loc, toType(builder, loc, phi, indexType), one);
   // Conditional expression.
@@ -419,49 +361,55 @@
   scf::IfOp ifOp1 = builder.create<scf::IfOp>(loc, types, lt, /*else*/ true);
   types.pop_back();
   builder.setInsertionPointToStart(&ifOp1.getThenRegion().front());
-  Value crd = genLoad(builder, loc, fields[field + 1], phim1);
+  Value crd = genLoad(builder, loc, desc.getField(idxIndex), phim1);
   Value eq = builder.create<arith::CmpIOp>(loc, arith::CmpIPredicate::eq,
                                            toType(builder, loc, crd, indexType),
                                            indices[d]);
   builder.create<scf::YieldOp>(loc, eq);
   builder.setInsertionPointToStart(&ifOp1.getElseRegion().front());
   if (d > 0)
-    genStore(builder, loc, msz, fields[field], pos);
+    genStore(builder, loc, msz, desc.getField(ptrIndex), pos);
   builder.create<scf::YieldOp>(loc, constantI1(builder, loc, false));
   builder.setInsertionPointAfter(ifOp1);
   Value p = ifOp1.getResult(0);
-  // If present construct. Note that for a non-unique dimension level, we simply
-  // set the condition to false and rely on CSE/DCE to clean up the IR.
+  // If present construct. Note that for a non-unique dimension level, we
+  // simply set the condition to false and rely on CSE/DCE to clean up the IR.
   //
   // TODO: generate less temporary IR?
   //
-  for (unsigned i = 0, e = fields.size(); i < e; i++)
-    types.push_back(fields[i].getType());
+  for (unsigned i = 0, e = desc.getNumFields(); i < e; i++)
+    types.push_back(desc.getField(i).getType());
   types.push_back(indexType);
   if (!isUniqueDim(rtp, d))
     p = constantI1(builder, loc, false);
   scf::IfOp ifOp2 = builder.create<scf::IfOp>(loc, types, p, /*else*/ true);
   // If present (fields unaffected, update next to phim1).
   builder.setInsertionPointToStart(&ifOp2.getThenRegion().front());
-  fields.push_back(phim1);
-  builder.create<scf::YieldOp>(loc, fields);
-  fields.pop_back();
+
+  // FIXME: This does not looks like a clean way, but probably the most
+  // efficient way.
+  desc.getFields().push_back(phim1);
+  builder.create<scf::YieldOp>(loc, desc.getFields());
+  desc.getFields().pop_back();
+
   // If !present (changes fields, update next).
   builder.setInsertionPointToStart(&ifOp2.getElseRegion().front());
   Value mszp1 = builder.create<arith::AddIOp>(loc, msz, one);
-  genStore(builder, loc, mszp1, fields[field], pp1);
-  createPushback(builder, loc, fields, field + 1, indices[d]);
+  genStore(builder, loc, mszp1, desc.getField(ptrIndex), pp1);
+  createPushback(builder, loc, desc, idxIndex, indices[d]);
   // Prepare the next dimension "as needed".
   if ((d + 1) < rank)
-    allocSchemeForRank(builder, loc, rtp, fields, field + 2, d + 1);
-  fields.push_back(msz);
-  builder.create<scf::YieldOp>(loc, fields);
-  fields.pop_back();
+    allocSchemeForRank(builder, loc, desc, d + 1);
+
+  desc.getFields().push_back(msz);
+  builder.create<scf::YieldOp>(loc, desc.getFields());
+  desc.getFields().pop_back();
+
   // Update fields and return next pos.
   builder.setInsertionPointAfter(ifOp2);
   unsigned o = 0;
-  for (unsigned i = 0, e = fields.size(); i < e; i++)
-    fields[i] = ifOp2.getResult(o++);
+  for (unsigned i = 0, e = desc.getNumFields(); i < e; i++)
+    desc.setField(i, ifOp2.getResult(o++));
   return ifOp2.getResult(o);
 }
 
@@ -476,12 +424,12 @@
 ///
 /// TODO: better unord/not-unique; also generalize, optimize, specialize!
 ///
-static void genInsert(OpBuilder &builder, Location loc, RankedTensorType rtp,
-                      SmallVectorImpl<Value> &fields,
+static void genInsert(OpBuilder &builder, Location loc,
+                      MutSparseTensorDescriptor desc,
                       SmallVectorImpl<Value> &indices, Value value) {
+  RankedTensorType rtp = desc.getTensorType();
   unsigned rank = rtp.getShape().size();
   assert(rank == indices.size());
-  unsigned field = fieldsIdx; // start past header
   Value pos = constantZero(builder, loc, builder.getIndexType());
   // Generate code for every dimension.
   for (unsigned d = 0; d < rank; d++) {
@@ -493,67 +441,67 @@
       //   }
       //   pos[d] = indices.size() - 1
       //   <insert @ pos[d] at next dimension d + 1>
-      pos = genCompressed(builder, loc, rtp, fields, indices, value, pos, field,
-                          d);
-      field += 2;
+      pos = genCompressed(builder, loc, desc, indices, value, pos, d);
     } else if (isSingletonDim(rtp, d)) {
       // Create:
       //   indices[d].push_back(i[d])
       //   pos[d] = pos[d-1]
       //   <insert @ pos[d] at next dimension d + 1>
-      createPushback(builder, loc, fields, field, indices[d]);
-      field += 1;
+      unsigned fidx = desc.getIdxMemRefIndex(d);
+      createPushback(builder, loc, desc, fidx, indices[d]);
     } else {
       assert(isDenseDim(rtp, d));
       // Construct the new position as:
       //   pos[d] = size * pos[d-1] + i[d]
       //   <insert @ pos[d] at next dimension d + 1>
-      Value size = sizeAtStoredDim(builder, loc, rtp, fields, d);
+      Value size = sizeAtStoredDim(builder, loc, desc, d);
       Value mult = builder.create<arith::MulIOp>(loc, size, pos);
       pos = builder.create<arith::AddIOp>(loc, mult, indices[d]);
     }
   }
   // Reached the actual value append/insert.
-  if (!isDenseDim(rtp, rank - 1))
-    createPushback(builder, loc, fields, field++, value);
-  else
-    genStore(builder, loc, value, fields[field++], pos);
-  assert(fields.size() == field);
+  unsigned valIdx = desc.getValMemRefIndex();
+  if (!isDenseDim(rtp, rank - 1)) {
+    createPushback(builder, loc, desc, valIdx, value);
+  } else
+    genStore(builder, loc, value, desc.getValMemRef(), pos);
 }
 
 /// Generations insertion finalization code.
-static void genEndInsert(OpBuilder &builder, Location loc, RankedTensorType rtp,
-                         SmallVectorImpl<Value> &fields) {
+static void genEndInsert(OpBuilder &builder, Location loc,
+                         MutSparseTensorDescriptor desc) {
+  RankedTensorType rtp = desc.getTensorType();
   unsigned rank = rtp.getShape().size();
-  unsigned field = fieldsIdx; // start past header
   for (unsigned d = 0; d < rank; d++) {
     if (isCompressedDim(rtp, d)) {
       // Compressed dimensions need a pointer cleanup for all entries
       // that were not visited during the insertion pass.
       //
-      // TODO: avoid cleanup and keep compressed scheme consistent at all times?
+      // TODO: avoid cleanup and keep compressed scheme consistent at all
+      // times?
       //
       if (d > 0) {
         unsigned ptrWidth = getSparseTensorEncoding(rtp).getPointerBitWidth();
         Type indexType = builder.getIndexType();
         Type ptrType = ptrWidth ? builder.getIntegerType(ptrWidth) : indexType;
-        Value mz = constantIndex(builder, loc, getMemSizesIndex(field));
-        Value hi = genLoad(builder, loc, fields[memSizesIdx], mz);
+        Value ptrMemRef = desc.getPtrMemRef(d);
+        Value mz = constantIndex(builder, loc, desc.getPtrMemSizesIndex(d));
+        Value hi = genLoad(builder, loc, desc.getMemSizesMemRef(), mz);
         Value zero = constantIndex(builder, loc, 0);
         Value one = constantIndex(builder, loc, 1);
         // Vector of only one, but needed by createFor's prototype.
-        SmallVector<Value, 1> inits{genLoad(builder, loc, fields[field], zero)};
+        SmallVector<Value, 1> inits{genLoad(builder, loc, ptrMemRef, zero)};
         scf::ForOp loop = createFor(builder, loc, hi, inits, one);
         Value i = loop.getInductionVar();
         Value oldv = loop.getRegionIterArg(0);
-        Value newv = genLoad(builder, loc, fields[field], i);
+        Value newv = genLoad(builder, loc, ptrMemRef, i);
         Value ptrZero = constantZero(builder, loc, ptrType);
         Value cond = builder.create<arith::CmpIOp>(
             loc, arith::CmpIPredicate::eq, newv, ptrZero);
         scf::IfOp ifOp = builder.create<scf::IfOp>(loc, TypeRange(ptrType),
                                                    cond, /*else*/ true);
         builder.setInsertionPointToStart(&ifOp.getThenRegion().front());
-        genStore(builder, loc, oldv, fields[field], i);
+        genStore(builder, loc, oldv, ptrMemRef, i);
         builder.create<scf::YieldOp>(loc, oldv);
         builder.setInsertionPointToStart(&ifOp.getElseRegion().front());
         builder.create<scf::YieldOp>(loc, newv);
@@ -561,14 +509,10 @@
         builder.create<scf::YieldOp>(loc, ifOp.getResult(0));
         builder.setInsertionPointAfter(loop);
       }
-      field += 2;
-    } else if (isSingletonDim(rtp, d)) {
-      field++;
     } else {
-      assert(isDenseDim(rtp, d));
+      assert(isDenseDim(rtp, d) || isSingletonDim(rtp, d));
     }
   }
-  assert(fields.size() == ++field);
 }
 
 //===----------------------------------------------------------------------===//
@@ -659,12 +603,12 @@
   matchAndRewrite(tensor::DimOp op, OpAdaptor adaptor,
                   ConversionPatternRewriter &rewriter) const override {
     Optional<int64_t> index = op.getConstantIndex();
-    if (!index)
+    if (!index || !getSparseTensorEncoding(adaptor.getSource().getType()))
       return failure();
-    auto sz =
-        sizeFromTensorAtDim(rewriter, op.getLoc(),
-                            op.getSource().getType().cast<RankedTensorType>(),
-                            adaptor.getSource(), *index);
+
+    auto desc = getDescriptorFromTensorTuple(adaptor.getSource());
+    auto sz = sizeFromTensorAtDim(rewriter, op.getLoc(), desc, *index);
+
     if (!sz)
       return failure();
 
@@ -760,10 +704,11 @@
     // Prepare fields.
     SmallVector<Value> fields(tuple.getInputs());
     // Generate optional insertion finalization code.
+    MutSparseTensorDescriptor desc(srcType, fields);
     if (op.getHasInserts())
-      genEndInsert(rewriter, op.getLoc(), srcType, fields);
+      genEndInsert(rewriter, op.getLoc(), desc);
     // Replace operation with resulting memrefs.
-    rewriter.replaceOp(op, genTuple(rewriter, op.getLoc(), srcType, fields));
+    rewriter.replaceOp(op, genTuple(rewriter, op.getLoc(), desc));
     return success();
   }
 };
@@ -775,7 +720,10 @@
   LogicalResult
   matchAndRewrite(ExpandOp op, OpAdaptor adaptor,
                   ConversionPatternRewriter &rewriter) const override {
+    if (!getSparseTensorEncoding(op.getTensor().getType()))
+      return failure();
     Location loc = op->getLoc();
+    auto desc = getDescriptorFromTensorTuple(adaptor.getTensor());
     RankedTensorType srcType =
         op.getTensor().getType().cast<RankedTensorType>();
     Type eltType = srcType.getElementType();
@@ -787,8 +735,7 @@
     // dimension size, translated back to original dimension). Note that we
     // recursively rewrite the new DimOp on the **original** tensor.
     unsigned innerDim = toOrigDim(srcType, srcType.getRank() - 1);
-    auto sz = sizeFromTensorAtDim(rewriter, loc, srcType, adaptor.getTensor(),
-                                  innerDim);
+    auto sz = sizeFromTensorAtDim(rewriter, loc, desc, innerDim);
     assert(sz); // This for sure is a sparse tensor
     // Generate a memref for `sz` elements of type `t`.
     auto genAlloc = [&](Type t) {
@@ -839,6 +786,7 @@
     // Prepare fields and indices.
     SmallVector<Value> fields(tuple.getInputs());
     SmallVector<Value> indices(adaptor.getIndices());
+    MutSparseTensorDescriptor desc(dstType, fields);
     // If the innermost dimension is ordered, we need to sort the indices
     // in the "added" array prior to applying the compression.
     unsigned rank = dstType.getShape().size();
@@ -859,17 +807,17 @@
     //      filled[index] = false;
     //      yield new_memrefs
     //    }
-    scf::ForOp loop = createFor(rewriter, loc, count, fields);
+    scf::ForOp loop = createFor(rewriter, loc, count, desc.getFields());
     Value i = loop.getInductionVar();
     Value index = genLoad(rewriter, loc, added, i);
     Value value = genLoad(rewriter, loc, values, index);
     indices.push_back(index);
     // TODO: faster for subsequent insertions?
-    genInsert(rewriter, loc, dstType, fields, indices, value);
+    genInsert(rewriter, loc, desc, indices, value);
     genStore(rewriter, loc, constantZero(rewriter, loc, eltType), values,
              index);
     genStore(rewriter, loc, constantI1(rewriter, loc, false), filled, index);
-    rewriter.create<scf::YieldOp>(loc, fields);
+    rewriter.create<scf::YieldOp>(loc, desc.getFields());
     rewriter.setInsertionPointAfter(loop);
     Value result = genTuple(rewriter, loc, dstType, loop->getResults());
     // Deallocate the buffers on exit of the full loop nest.
@@ -897,11 +845,12 @@
     // Prepare fields and indices.
     SmallVector<Value> fields(tuple.getInputs());
     SmallVector<Value> indices(adaptor.getIndices());
+    MutSparseTensorDescriptor desc(dstType, fields);
     // Generate insertion.
     Value value = adaptor.getValue();
-    genInsert(rewriter, op->getLoc(), dstType, fields, indices, value);
+    genInsert(rewriter, op->getLoc(), desc, indices, value);
     // Replace operation with resulting memrefs.
-    rewriter.replaceOp(op, genTuple(rewriter, op.getLoc(), dstType, fields));
+    rewriter.replaceOp(op, genTuple(rewriter, op.getLoc(), desc));
     return success();
   }
 };
@@ -918,11 +867,9 @@
     // Replace the requested pointer access with corresponding field.
     // The cast_op is inserted by type converter to intermix 1:N type
     // conversion.
-    auto tuple = getTuple(adaptor.getTensor());
-    unsigned idx = Base::getIndexForOp(tuple, op);
-    auto fields = tuple.getInputs();
-    assert(idx < fields.size());
-    rewriter.replaceOp(op, fields[idx]);
+    auto desc = getDescriptorFromTensorTuple(adaptor.getTensor());
+    Value field = Base::getFieldForOp(desc, op);
+    rewriter.replaceOp(op, field);
     return success();
   }
 };
@@ -933,10 +880,10 @@
 public:
   using SparseGetterOpConverter::SparseGetterOpConverter;
   // Callback for SparseGetterOpConverter.
-  static unsigned getIndexForOp(UnrealizedConversionCastOp /*tuple*/,
-                                ToPointersOp op) {
+  static Value getFieldForOp(const SparseTensorDescriptor &desc,
+                             ToPointersOp op) {
     uint64_t dim = op.getDimension().getZExtValue();
-    return getFieldIndex(op.getTensor().getType(), /*ptrDim=*/dim, -1u);
+    return desc.getPtrMemRef(dim);
   }
 };
 
@@ -946,10 +893,10 @@
 public:
   using SparseGetterOpConverter::SparseGetterOpConverter;
   // Callback for SparseGetterOpConverter.
-  static unsigned getIndexForOp(UnrealizedConversionCastOp /*tuple*/,
-                                ToIndicesOp op) {
+  static Value getFieldForOp(const SparseTensorDescriptor &desc,
+                             ToIndicesOp op) {
     uint64_t dim = op.getDimension().getZExtValue();
-    return getFieldIndex(op.getTensor().getType(), -1u, /*idxDim=*/dim);
+    return desc.getIdxMemRef(dim);
   }
 };
 
@@ -959,10 +906,9 @@
 public:
   using SparseGetterOpConverter::SparseGetterOpConverter;
   // Callback for SparseGetterOpConverter.
-  static unsigned getIndexForOp(UnrealizedConversionCastOp tuple,
-                                ToValuesOp /*op*/) {
-    // The last field holds the value buffer.
-    return tuple.getInputs().size() - 1;
+  static Value getFieldForOp(const SparseTensorDescriptor &desc,
+                             ToValuesOp /*op*/) {
+    return desc.getValMemRef();
   }
 };
 
@@ -994,12 +940,11 @@
   matchAndRewrite(NumberOfEntriesOp op, OpAdaptor adaptor,
                   ConversionPatternRewriter &rewriter) const override {
     // Query memSizes for the actually stored values size.
-    auto tuple = getTuple(adaptor.getTensor());
-    auto fields = tuple.getInputs();
-    unsigned lastField = fields.size() - 1;
+    auto desc = getDescriptorFromTensorTuple(adaptor.getTensor());
     Value field =
-        constantIndex(rewriter, op.getLoc(), getMemSizesIndex(lastField));
-    rewriter.replaceOpWithNewOp<memref::LoadOp>(op, fields[memSizesIdx], field);
+        constantIndex(rewriter, op.getLoc(), desc.getValMemSizesIndex());
+    rewriter.replaceOpWithNewOp<memref::LoadOp>(op, desc.getMemSizesMemRef(),
+                                                field);
     return success();
   }
 };