[mlir][sparse] Lowering for unary and binary

[jim22k]

Summary

Code for lowering sparse_tensor.unary and sparse_tensor.binary within a linalg.generic block.

[jim22k]

@aartbik Please take a look and let me know your thoughts on my general approach. I also indicated where I am stuck and could use some help, as I don't truly understand the lattice-set theory.

mlir/lib/Dialect/SparseTensor/Utils/Merger.cpp

555

I need help here. I'm not sure what I need to include such that absentVal is added to the output for every missing value in the input. I assume this is something similar to kTensor or kInvariant, as doing v+1 inside linalg.generic will create a dense output. I need something similar to happen here.

614

I'm stuck here. The kBinary below creates one kBinaryRegion and two kUnaryRegions. These are not called/handled for the vector tests. But for the matrix test, the kBinaryRegion is re-evaluated for some reason. The problem is that the origin sparse_tensor.binary operation has already been split apart. I essentially need a no-op here (i.e. it's already working fine. Don't re-evaluate).

I tried calling takeConj again, but that somehow eliminates the disjoint pieces that were added in the kBinary section.

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

107

This test fails with an assertion error:
Assertion failed: (!add || latGT(p0, p1)), function optimizeSet, file Merger.cpp, line 167.

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

95

This test passes.

142

This test passes.

162

This test fails with an assertion error:
Assertion failed: (!add || latGT(p0, p1)), function optimizeSet, file Merger.cpp, line 167.

[aartbik]

mlir/include/mlir/Dialect/SparseTensor/Utils/Merger.h

50

At first reading, I expected the kUnaryRegion and kBinaryRegion, but was a bit surprised with the kUnary and kBinary (since that generalizes a lot of the ops already there). Can you briefly document what the intended differences are here?

Scanning ahead, it seems like one is eventually broken up into several of the others, but I wonder if we can't just keep a single kUnaryRegion and kBinaryRegion, and let the op itself drive the codegen?

99

style: period at end in comment

Also, originally I avoided linking this TensorExpr back to IR of code in MLIR, since during lattice construction, one must be careful to maintain the right 1:1 correspondence between the lattice point and the IR. Obviously, we have little choice for the new binary/unary regions, but let's document that here carefully.

162

ah, I see you solved the issue I alluded to above by passing this explicitly to the merge operations; let's brainstorm later if we can somehow do this using the child nodes only...

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

797–803

This is very surprising? Is this right?

1226

unnecessary format change?

1373

unnecessary format change?

mlir/lib/Dialect/SparseTensor/Utils/Merger.cpp

26

this is not completely in style with the general setup:

only assign the fields that are know to be set, assert on nullptr in all others

61–79

I think kUnary/BinaryRegion should be new cases that set op,
all others assert !op

539

note that this could be a simple cast, not a dyn_cast, but more preferable, it looks like we could simply keep a single kUnaryRegion case, see below

555

This is indeed a bit trickier. For the time being, to focus on the overall logic, let's simply put this case under the same as all the other zero preserving unary ops, and assert that absent() is not set. Then we will enhance the logic later.

614

If you rewrite it as suggested below, I think the no-op comes natural.

631

same, should be a cast here, but it looks like we could simply keep a single kBinaryRegion case by looking at the result of this cast (which then needs to be dynamic again ;-) and decide what to do. I think I prefer that a bit more than artificially introducing the kBinary/kUnary as "handled" cases.

638

it feels this whole block of code, L612 to L639 is really takeDisj with some smart selection of the branches.
Perhaps you can split out the analysis of the MLIR IR (getting the three branches), and then write a new

takeDisj(... , opboth, opleft, opright)

and put the takeDisj close to the other, just so that the actual lattic logic is not so deeply burried inside this huge block

741

At first reading, it was surprising to just see "UnaryOp" here, since that looks like just another arith:: version.
Prefixing it with namespace sparse_tensor:: is of course against styleguide, but would look more clear.

Perhaps we should rename the Unary/BinaryOp of sparse tensor dialect a bit more specific (can be done later).

855–856

what does it mean if we hit this part? an empty block? or an error?

[jim22k]

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

797–803

This is related to my handling of binary[overlap] and unary[present] when those regions are empty. If you look at Merger.cpp:617, it passes a nullptr for the Operation*. That, in turn, manifests in line 886 and returns Value() for the Value output. When that empty Value reaches this code, we skip the insertion.

It works, but it does feel like a hack. Can you think of a better way to indicate to the generating code that no output is needed?

mlir/lib/Dialect/SparseTensor/Utils/Merger.cpp

631

That seems reasonable. I will refactor and check the result of dyn_cast. If it fails, it means the BinaryOp has already been handled. buildLattices has unsigned return type, so what should I return in that case? This is where I want a no-op, meaning "don't add any new lattices points".

638

Good idea to move this up near the other takeDisj().

855–856

This means an empty block, so I want to indicate no value.

[aartbik]

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

797–803

It was unexpected, but am okay with this if (1) it is documented and (2) perhaps we can even add some form of assert when !rhs that verifies that indeed we hit an expected case

mlir/lib/Dialect/SparseTensor/Utils/Merger.cpp

631

If at all possible, I would like the "nop" to be detected at a higher level, i.e. before building the new lattices. Otherwise we would have to somehow return the existing set id (returning a nop value is a bit too intrusive to my taste). But after you have done the restructuring suggested here, I will have another look.

[jim22k]

Updates based on feedback

• Create 2nd takeDisj method specifically for binary
• Eliminate kBinaryRegion/kUnaryRegion in favor of no "handled" cases
• Add better comments
• Eliminate the double handling of kBinary by looking at the loop level

[jim22k]

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

1226

clang-format made this change; not sure why

1373

same as above -- clang-format wanted this changed

1606

This is new and is my attempt to avoid calling kBinary twice so we avoid the need for a "no-op" return.
For matrices, there are two loops, so this piece of code is called twice. If we split apart the binary operation, we run into the need for a no-op. By passing topSort.size() - at, we only split apart the binary in the lowest loop. Previous calls will be a simple takeConj with the binary operation passed unchanged.

mlir/lib/Dialect/SparseTensor/Utils/Merger.cpp

65

This is a weaker assert, but is required because splitting up the binary result in some pieces which are fundamentally unary, so they don't have a y. In this iteration, I am still labeling all the split up chunks of binary as kBinary.

628

I don't know if this is the correct approach, but it makes the matrix tests all pass. We only split up the binary operation once and we never need to return a no-op. At least, that will hold as long as z==1 for the last time we try to buildLattices() on the binary operation.

656

I created the new takeDisj, which is very nice. It does mean that all of the split up pieces of kBinary remain labeled as kBinary, even for left and right which only have a single input argument.

886

These can be combined in the switch because their logic is identical. The binary operation splits up into pieces which may have 1 or 2 input arguments, so they look just like the unary operation being split up.

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

107

This is now passing.

[jim22k]

• Make the absent region of sparse_tensor.unary work

[jim22k]

mlir/lib/Dialect/SparseTensor/Utils/Merger.cpp

555

I figured out how to make the absent region work. I essentially treat it like v + 1 (which is a binary function). Using the same logic, I perform a disjunction. The overlap will be replaced by the present block which only has 1 input argument, so it will sub in the left argument rather than being truly binary.
The rhs is a fixed value, so I create a kInvariant for it and it covers everything which is not a conjunction.

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

162

This test is now passing.

[jim22k]

Merge latest from main

[jim22k]

Add test involving linalg.index

[jim22k]

@aartbik I think this PR is ready for a full review. It's passing all the tests.

• I do need you to look at the if (z != 1) logic and see if that is reasonable. It works for the tests, but I would appreciate your perspective on where that might break.
• I also want to discuss the Kind that is assigned when unary and binary are broken up. I currently have the Kind staying unchanged, but that limits what can be asserted in TensorExps constructor. It also makes other checks strange -- for example, mapSet checks that we only pass in unary operations, but I also have to allow kBinary because some of the split up chunks require mapSet.
• You said you wanted to brainstorm how to pass the blocks around via the children rather than the merge operations directly. I'm open to that if we can figure out a way. Otherwise, I don't think the current implementation is bad.

[aartbik]

Thanks for making such diligent progress with this and apologies for the delay. A few high prio tasks came up but from here on it should be a bit smoother sailing!

mlir/include/mlir/Dialect/SparseTensor/Utils/Merger.h

99

Note that technically this field is only used for unary and binary, but putting this in a union or so seems too convoluted. But just add a comment on when this field is set.

162

I am okay with this for now (and perhaps permanently). It at least makes semantics at top level explicit.

256

"e" and "i" have some designated meaning, so please don't change into im.

Here "z" can use a comment in the method description.
But I am not sure about this approach, see below.

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

797–803

It is a bit strange to refer to "Kinds" here, just rephrase the comment in general terms.
Anything we can assert on rhs or t if not set?

1606

This feels hacky. In the original iteration space theory, building the lattice sets solely depends on the current index, and it has nothing to do with the nesting depth or anything like that. In principle, we need to make this call at every loop nest to determine the remaining expressions.

mlir/lib/Dialect/SparseTensor/Utils/Merger.cpp

63

assert on op at least

65

This feels hacky. I found these asserts very useful to ensure we are truly doing the right thing. If we introduce all sorts of intermediate stages, as in binary but marked unary etc, such important invariants are lost. Can we cleanup the "top level" logic that transforms binary into unary (this is e.g. also done around binary minus into unary minus)

68

&& op

159

period at end, here and below.

640

period at end

741

braces not needed

784

braces not needed

890

period at end

897

period at end

900

here and below, document this shortcut returns

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

56

Period at end

57

Rather than adding and modifying an existing test, I would much rather that you add two new integration tests: unary and binary. Yes, it has some boiler plate to setup inputs, but that way it is more clear where the semi-ring stuff is tested.

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

93

Same here. I would move these into the semi ring integration tests mentioned above, and simply add the novec/vec flags at the top (a test can have more than one RUN/CHECK series)

[jim22k]

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

1606

I agree. This was a quick and dirty approach, but I've convinced myself that it is not just hacky, but actually gives wrong results.
I figured out another way which is giving better results and avoids the need for this hack. I will clean up the code and show it off in the next revision.

[aartbik]

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

1606

Looking forward to that. Thanks!

[jim22k]

Change how repeat calls to unary and binary are handled

• Remove z parameter from buildLattices
• Add mergeOp to TensorExp
• Move tests to their own files

Some things to notice with these changes:

1. There are now two Operation * in TensorExp. This lets us keep the original binary or unary operation as well as the yield operation to be merged after splitting it up into pieces. By keeping the original, it allows us to perform the same logic in further nested loops. In some of the split up pieces (disjoint left or right regions), we do not include the original op. This essentially signals that these pieces have been "handled". Only one region will contain the original and will continue building in the next level down.
2. There is one case where the Kind changes. kBinary keeps the conjoint piece as kBinary, but the disjoint pieces become kUnary. This lets us retain the check in mapSet that requires a unary kind. If you don't like this change of kind, I can change it. It only affects some of the asserts.

[aartbik]

mlir/include/mlir/Dialect/SparseTensor/IR/SparseTensorOps.td

447

Can you actually add a test of this, something like

%0 = linalg.generic #trait_vec_op
   ins(%arga, %argb: tensor<?xi64, #SparseVector>, tensor<?xi64, #SparseVector>)
    outs(%xv: tensor<?xi64, #SparseVector>) {
    ^bb(%a: i64, %b: i64, %x: i64):
      %idx = linalg.index 0 : index
      %cst = arith.index_cast %idx : index to i64
      %1 = sparse_tensor.binary %a, %b : i64, i64 to i64
        overlap={
          ^bb0(%a0: i64, %b0: i64):
            sparse_tensor.yield %cst : i64
        }
        left={}
        right={}
      linalg.yield %1 : i64
} -> tensor<?xi64, #SparseVector>

and then see if we indeed generate the right code? I am not 100% sure how the part outside the new op will interact with codegen

mlir/include/mlir/Dialect/SparseTensor/Utils/Merger.h

99

Please add a bit more description for both, i.e. origOp points to original unary/unary op, mergeOp points to ... before going in detail of what needs to be there

101

This is a better solution than the original nesting depth based one. Still, the danger of a struct is of course that it is easy to add members :-). And although I doubt this is on any memory profile yet, still a bit worrisome to grow the sizeof for all nodes now. So if we keep this, at least add a TODO on improving memory usage in the future.

Furthermore, I am wondering if it perhaps would be better to reflect the state in the type (sort of what you had originally) and just store one op. All of this can be done later, just thinking out loud here.

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

797–803

did you see these comments?

mlir/lib/Dialect/SparseTensor/Utils/Merger.cpp

151

For this first version, I actually prefer that we keep takeDisj intact (other than passing op)
and add a new

takeCombi

method that implements your new logic, just so we don't need to touch other ops in this revision.
We can perhaps merge these methods later into one, but now too much is changing at once.

537

splinter?

540

I patched in your revision, but got compilation errors here.
I think you need to include

#include "mlir/Dialect/SparseTensor/IR/SparseTensor.h"

551

period at end

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

43

Here, and probably for the unary too, scan we add a case where one of the operands is dense (either truly dense, or annotated with "dense"). Just to verify that a lattice with "universal" index is properly tested.

209

looks like these two are never release (thus will fail memsan)

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

41

You have a test with absent, and one with present.
How about also testing when both are set, just for completeness?

Like your

%result = sparse_tensor.unary %a : f64 to i32

present={
  ^bb0(%x: f64):
    %ret = arith.constant 1 : i32
    sparse_tensor.yield %ret : i32
}
absent={
  %ret = arith.constant -1 : i32
  sparse_tensor.yield %ret : i32
}

example?

In fact, even though you test the present case with a matrix, how about also doing that for a vector, just so that all cases are covered for a single loop example. That way, the code also acts like a nice illustration of the feature

113

indentation is off

120

do we want to do the same thing here?
i.e. print values first (to see that we only compute sparse values)
and then print full matrix for stucture?

141

This is never released (and will thus fail our memsan test).

[jim22k]

Change back to one Operation *

• Add new kBinaryHalf kind
• kUnary and kBinary hold their original operation until buildExp
• During buildExp, locate the primary YieldOp and merge that
• Add more tests

[jim22k]

mlir/include/mlir/Dialect/SparseTensor/IR/SparseTensorOps.td

447

This will not work. I get this error:

/Users/jkitchen/Projects/HIVE/llvm-project/mlir/test/Integration/Dialect/SparseTensor/CPU/sparse_binary.mlir:118:16: error: 'arith.index_cast' op operation destroyed but still has uses
        %cst = arith.index_cast %idx : index to i32
               ^
/Users/jkitchen/Projects/HIVE/llvm-project/mlir/test/Integration/Dialect/SparseTensor/CPU/sparse_binary.mlir:118:16: note: see current operation: %0 = "arith.index_cast"(<<NULL VALUE>>) : (<<NULL TYPE>>) -> i32
/Users/jkitchen/Projects/HIVE/llvm-project/mlir/test/Integration/Dialect/SparseTensor/CPU/sparse_binary.mlir:113:10: note: - use: "sparse_tensor.lex_insert"(%2, %21, <<UNKNOWN SSA VALUE>>) : (tensor<?xi32, #sparse_tensor.encoding<{ dimLevelType = [ "compressed" ], pointerBitWidth = 0, indexBitWidth = 0 }>>, memref<?xindex>, i32) -> ()

    %0 = linalg.generic #trait_vec_op
         ^
LLVM ERROR: operation destroyed but still has uses

The only way I have been able to use linalg.index is to pass its result into sparse_tensor.binary as one of the arguments, as seen on line 449. That seems to work fine.

mlir/include/mlir/Dialect/SparseTensor/Utils/Merger.h

101

Alright, I settled on another new approach. We can make this work with only a single Operation * in the struct.

• kUnary: Operation * is the sparse_tensor.unary
• kBinary: Operation * is the sparse_tensor.binary
• kUnaryHandled: Operation * is the YieldOp

When buildLattices encounters kUnary or kBinary, it will build the lattice structure with potentially many regions (i.e. binary always calls takeDisj). The regions which can be considered fully handled will change to kUnaryHandled. The primary region of both unary and binary will remain as kUnary or kBinary and will hold on to the original operation (not the yield). This will let us continue to use the original at every level of the lattice.

When we finally call buildExp, we can dig into the original unary and binary operations and pull out the primary region's YieldOp at that point.

This approach also avoids confusion because kUnary and kBinary always refer to the correct sparse_tensor.unary and sparse_tensor.binary operations. Any disjoint offshoots use a different "Kind" to clarify that they are not the original operation anymore.

I will update the PR with these changes.

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

797–803

I am adding a check to ensure that the kind is either kUnary or kBinary.

mlir/lib/Dialect/SparseTensor/Utils/Merger.cpp

540

That should already be there. See line 11.

[jim22k]

Sorry my comments are out of order with the new PR. They should be read as if they happened before the new PR. I forgot to submit them until now.

[aartbik]

A few last comments. Sorry for being nitpicky on this, rest assured I really like the work, I am just a bit particular on certain things.
Also, can you please make a pass over all my comments and mark them "Done" (or comment on them otherwise).
That makes the re-review a bit easier, since a lot of my past comments still show up as unresolv.ed

mlir/include/mlir/Dialect/SparseTensor/IR/SparseTensorOps.td

440

See below, but in a nutshell, let's not rewrite the example just because the desired form does not work yet.

447

But by passing it as argument, we lose the intersection/union power of the binary op.
Don't you agree that the code above should work, or the code example you had originally?

(okay doing this later, but I would prefer that we show a binary op example in the doc that uses two sparse inputs *and* uses the index in one or more of the branches, and not rewrite the example to match the current implementation status)

The problem is that we probably need some work dealing with the "outside" block computations that are only used in some places (and thus can be placed under the conditional). We can put a TODO if you prefer.

mlir/include/mlir/Dialect/SparseTensor/Utils/Merger.h

49

Maybe add a comment, as in

// semi-ring unary op

67

maybe add same-line comment as in

// semi-ring binary op

101

Looks like you iterated over some naming, since I see kBinaryHalf now.
Since we are bikeshedding, how about kBinaryBranch to show we have one case of a binary now?

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

802

if the if-part has braces, so does the else-part, even when it is a single line stmt

mlir/lib/Dialect/SparseTensor/Utils/Merger.cpp

25

: kind(k), val(v), op(operation) {

and then nothing below, since these are not part of a union so can always be set

Also, it feels weird to have a longer parameter name then field name, so perhaps call the parameter simply "o"
(which is in style with the current conventions)

63

can we assert on y in the new model?

143

does this line apply to L144? If so, please please after that line

350

please follow same order as in enum decl.

535

just use cast (not dyn_cast) and no assert, since this should alway work

546

same here and below, if the cast should work, just cast

799

cast, we assume that verifier has filtered out bad cases

887

The unary and binary codegen blocks are a bit too large for this context (most others are oneliners)
so please move into own method

[jim22k]

mlir/include/mlir/Dialect/SparseTensor/IR/SparseTensorOps.td

447

I would hope that your example test (and the documentation example I originally had) would work. If you are okay with a documentation example that gives an error, but is the desired end goal, I will revert to that. I am just sensitive to introducing a new feature with broken examples, giving people who want to try it out a bad impression of the work, unless you are going to figure out how to make it work before we merge.

mlir/lib/Dialect/SparseTensor/Utils/Merger.cpp

63

We cannot. During the original call, y==-1.
When absentRegion is empty, we call mapSet and y remains equal to -1.
But if absentRegion is not empty, we call takeDisj and both x and y are set.

For this "binary" case handling the absent region, when buildExp is called, we only utilize v0 and completely ignore v1, even though it exists from a lattice perspective.

Would it be helpful to add a comment here about why we can't assert anything about y?

[jim22k]

Updates based on feedback

• Change kBinaryHalf to kBinaryBranch
• Revert binaryop doc example
• Remove dynamic casting

[aartbik]

mlir/include/mlir/Dialect/SparseTensor/IR/SparseTensorOps.td

447

I am okay with that, but please add a small note on that in the doc (this is intended future behavior but still under construction ;-)

mlir/lib/Dialect/SparseTensor/Utils/Merger.cpp

63

Yeah, please document, so that in the future I do not try to put the assert back ;-)

[jim22k]

Okay, adding more comments.

[jim22k]

Few more comments

[aartbik]

A few last comments, but I am giving you the LGTM, since I think this is ready to go in, so we can work on refining the few remaining issues.
Thanks for your patience during the review, and thanks for this contribution!

mlir/include/mlir/Dialect/SparseTensor/IR/SparseTensorOps.td

447

I don't see the note in the doc (I really meant, the user facing example, so that people trying this out are aware it does not work yet). So something like

Example of A+B in upper triangle, A-B in lower triangle (not working yet, but construct will be available soon).

mlir/lib/Dialect/SparseTensor/Utils/Merger.cpp

792

mark these new helpers as "static" since they are private to the file

[jim22k]

Make helper functions static

[jim22k]

@aartbik Thanks for reviewing and helping me converge on a good solution for lowering.
Once the build passes, I will merge it.

mlir/include/mlir/Dialect/SparseTensor/IR/SparseTensorOps.td

let description = [{
 
       As a convenience, there is also a special token `identity` which can be
       used in place of the left or right region. This token indicates that
       the return value is the input value (i.e. func(%x) => return %x).
       As a practical example, setting `left=identity` and `right=identity`
       would be equivalent to a union operation where non-overlapping values
       in the inputs are copied to the output unchanged.
 
-      Example of isEqual applied to intersecting elements only:
+      Example of isEqual applied to intersecting elements only.
       ```mlir
       %C = sparse_tensor.init...
       %0 = linalg.generic #trait
         ins(%A: tensor<?xf64, #SparseVec>, %B: tensor<?xf64, #SparseVec>)
         outs(%C: tensor<?xi8, #SparseVec>) {
         ^bb0(%a: f64, %b: f64, %c: i8) :
           %result = sparse_tensor.binary %a, %b : f64, f64 to i8
             overlap={
               ^bb0(%arg0: f64, %arg1: f64):
                 %cmp = arith.cmpf "oeq", %arg0, %arg1 : f64
                 %ret_i8 = arith.extui %cmp : i1 to i8
                 sparse_tensor.yield %ret_i8 : i8
             }
             left={}
             right={}
           linalg.yield %result : i8
       } -> tensor<?xi8, #SparseVec>
       ```
 
-      Example of A+B in upper triangle, A-B in lower triangle:
+      Example of replacing every element by its column index. This uses
+      `linalg.index` to get the column index, treating the value like a
+      dense tensor usable in `sparse_tensor.binary`.
       ```mlir
       %C = sparse_tensor.init...
       %1 = linalg.generic #trait
-        ins(%A: tensor<?x?xf64, #CSR>, %B: tensor<?x?xf64, #CSR>
-        outs(%C: tensor<?x?xf64, #CSR> {
-        ^bb0(%a: f64, %b: f64, %c: f64) :
-          %row = linalg.index 0 : index
+        ins(%A: tensor<?x?xf64, #CSR>)
+        outs(%C: tensor<?x?xi32, #CSR>) {
+        ^bb0(%a: f64, %c: i32) :
           %col = linalg.index 1 : index
-          %result = sparse_tensor.binary %a, %b : f64, f64 to f64
+          %col32 = arith.index_cast %col : index to i32
+          %result = sparse_tensor.binary %a, %col32 : f64, i32 to i32
             overlap={
-              ^bb0(%x: f64, %y: f64):
-                %cmp = arith.cmpi "uge", %column, %row : index
-                %upperTriangleResult = arith.addf %x, %y : f64
-                %lowerTriangleResult = arith.subf %x, %y : f64
-                %ret = arith.select %cmp, %upperTriangleResult, %lowerTriangleResult : f64
-                sparse_tensor.yield %ret : f64
-            }
-            left=identity
-            right={
-              ^bb0(%y: f64):
-                %cmp = arith.cmpi "uge", %column, %row : index
-                %lowerTriangleResult = arith.negf %y : f64
-                %ret = arith.select %cmp, %y, %lowerTriangleResult
-                sparse_tensor.yield %ret : f64
+              ^bb0(%x: f64, %y: i32):
+                sparse_tensor.yield %y : i32
             }
+            left={}
+            right={}
           linalg.yield %result : f64
       } -> tensor<?x?xf64, #CSR>
       ```
 
       Example of set difference. Returns a copy of A where its sparse structure
       is *not* overlapped by B. The element type of B can be different than A
       because we never use its values, only its sparse structure.
       ```mlir

mlir/include/mlir/Dialect/SparseTensor/Utils/Merger.h

enum Kind {
   kCastFU, // unsigned
   kCastSF, // signed
   kCastUF, // unsigned
   kCastS,  // signed
   kCastU,  // unsigned
   kCastIdx,
   kTruncI,
   kBitCast,
+  kUnary,
+  kBinaryHalf,
   // Binary operations.
   kMulF,
   kMulI,
   kDivF,
   kDivS, // signed
   kDivU, // unsigned
   kAddF,
   kAddI,
   kSubF,
   kSubI,
   kAndI,
   kOrI,
   kXorI,
   kShrS, // signed
   kShrU, // unsigned
   kShlI,
+  kBinary,
 };
 
 /// Children subexpressions of tensor operations.
 struct Children {
   unsigned e0;
   unsigned e1;
 };
 
 /// Tensor expression. Represents a MLIR expression in tensor index notation.
 struct TensorExp {
-  TensorExp(Kind k, unsigned x, unsigned y, Value v);
+  TensorExp(Kind k, unsigned x, unsigned y, Value v, Operation *operation);
 
   /// Tensor expression kind.
   Kind kind;
 
   union {
     /// Expressions representing tensors simply have a tensor number.
     unsigned tensor;
 
     /// Indices hold the index number.
     unsigned index;
 
     /// Tensor operations hold the indices of their children.
     Children children;
   };
 
   /// Direct link to IR for an invariant or the destination value (to
   /// infer destination type) of a cast operation During code generation,
   /// this field may be used to cache "hoisted" loop invariant tensor loads.
   Value val;
+
+  /// Code blocks used by unary and binary. For the case of kUnary and
+  /// kBinary, this holds the original operation with all regions. For
+  /// kBinaryHalf, this holds the YieldOp for the left or right half
+  /// to be merged into a nested scf loop.
+  Operation *op;
 };
 
 /// 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);
   LatPoint(const BitVector &b, unsigned e);
 
   /// Conjunction of tensor loop indices as bitvector. This represents
   /// all indices involved in the tensor expression
   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.
   BitVector simple;
 
-  /// Index of the tensor expresssion.
+  /// Index of the tensor expression.
   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)
       : outTensor(t - 1), syntheticTensor(t), numTensors(t + 1), numLoops(l),
         hasSparseOut(false), 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 addExp(Kind k, unsigned e, Value v) { return addExp(k, e, -1u, v); }
-  unsigned addExp(Kind k, Value v) { return addExp(k, -1u, -1u, v); }
+  unsigned addExp(Kind k, unsigned e0, unsigned e1 = -1u, Value v = Value(),
+                  Operation *op = nullptr);
+  unsigned addExp(Kind k, unsigned e, Value v, Operation *op = nullptr) {
+    return addExp(k, e, -1u, v, op);
+  }
+  unsigned addExp(Kind k, Value v, Operation *op = nullptr) {
+    return addExp(k, -1u, -1u, v, op);
+  }
 
   /// Adds an iteration lattice point. Returns its index.
   unsigned addLat(unsigned t, unsigned i, unsigned e);
 
   /// Adds a new, initially empty, set. Returns its index.
   unsigned addSet();
 
   /// Computes a single conjunction of two lattice points by taking the "union"
   /// of loop indices (effectively constructing a larger "intersection" of those
   /// indices) with a newly constructed tensor (sub)expression of given kind.
   /// Returns the index of the new lattice point.
-  unsigned conjLatPoint(Kind kind, unsigned p0, unsigned p1);
+  unsigned conjLatPoint(Kind kind, unsigned p0, unsigned p1,
+                        Operation *op = nullptr);
 
   /// 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 takeConj(Kind kind, unsigned s0, unsigned s1,
+                    Operation *op = nullptr);
 
   /// 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 takeDisj(Kind kind, unsigned s0, unsigned s1,
+                    Operation *op = nullptr);
+
+  /// Disjunctive merge of two lattice sets L0 and L1 with custom handling of
+  /// the overlap, left, and right regions. Any region may be left missing in
+  /// the output. Returns the index of the new set.
+  unsigned takeCombi(Kind kind, unsigned s0, unsigned s1, Operation *orig,
+                     bool includeLeft, Kind ltrans, Operation *opleft,
+                     bool includeRight, Kind rtrans, Operation *opright);
 
   /// Maps the unary operator over the lattice set of the operand, i.e. each
   /// lattice point on an expression E is simply copied over, but with OP E
   /// as new expression. Returns the index of the new set.
-  unsigned mapSet(Kind kind, unsigned s0, Value v = Value());
+  unsigned mapSet(Kind kind, unsigned s0, Value v = Value(),
+                  Operation *op = nullptr);
 
   /// 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 optimizeSet(unsigned s0);
 
   /// Simplifies the conditions in a conjunction of a given lattice point
   /// within the given set using just two basic rules:
#ifndef NDEBUG
   void dumpLat(unsigned p) const;
   void dumpSet(unsigned s) const;
   void dumpBits(const BitVector &bits) const;
 #endif
 
   /// Builds the iteration lattices in a bottom-up traversal given the remaining
   /// tensor (sub)expression and the next loop index in the iteration graph.
   /// Returns index of the root expression.
   unsigned buildLattices(unsigned e, unsigned i);
 
   /// Builds a tensor expression from the given Linalg operation.
   /// Returns index of the root expression on success.
   Optional<unsigned> buildTensorExpFromLinalg(linalg::GenericOp op);
 
   /// Rebuilds SSA format from a tensor expression.
   Value buildExp(PatternRewriter &rewriter, Location loc, unsigned e, Value v0,
                  Value v1);

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

static Value genTensorLoad(Merger &merger, CodeGen &codegen,
   if (codegen.curVecLength > 1)
     return genVectorLoad(codegen, rewriter, ptr, args);
   return rewriter.create<memref::LoadOp>(op.getLoc(), ptr, args);
 }
 
 /// Generates a store on a dense or sparse tensor.
 static void genTensorStore(Merger &merger, CodeGen &codegen,
                            PatternRewriter &rewriter, linalg::GenericOp op,
-                           Value rhs) {
+                           unsigned exp, Value rhs) {
   Location loc = op.getLoc();
   // Test if this is a scalarized reduction.
   if (codegen.redVal) {
     if (codegen.curVecLength > 1)
       rhs = rewriter.create<arith::SelectOp>(loc, codegen.curVecMask, rhs,
                                              codegen.redVal);
     updateReduc(merger, codegen, rhs);
     return;
   }
   // Store during insertion.
   OpOperand *t = op.getOutputOperand(0);
   if (t == codegen.sparseOut) {
+    if (!rhs) {
+      // Only unary and binary are allowed to return uninitialized rhs
+      // to indicate missing output.
+      Kind kind = merger.exp(exp).kind;
+      assert(kind == kUnary || kind == kBinary);
+    } else
       genInsertionStore(codegen, rewriter, op, t, rhs);
     return;
   }
   // Actual store.
   SmallVector<Value, 4> args;
   Value ptr = genSubscript(codegen, rewriter, op, t, args);
   if (codegen.curVecLength > 1)
     genVectorStore(codegen, rewriter, rhs, ptr, args);
   else
if (lhs == t) {
         codegen.redKind = getReduction(last);
         codegen.redExp = exp;
         updateReduc(merger, codegen, load);
       } else {
         Value redVal = codegen.redVal;
         updateReduc(merger, codegen, Value());
         codegen.redExp = -1u;
         codegen.redKind = kNoReduc;
-        genTensorStore(merger, codegen, rewriter, op, redVal);
+        genTensorStore(merger, codegen, rewriter, op, exp, redVal);
       }
     } else {
       // Start or end loop invariant hoisting of a tensor load.
       merger.exp(exp).val =
           atStart ? genTensorLoad(merger, codegen, rewriter, op, exp) : Value();
     }
   } else if (merger.exp(exp).kind != Kind::kInvariant &&
              merger.exp(exp).kind != Kind::kIndex) {
static Operation *genFor(Merger &merger, CodeGen &codegen,
   if (isVector)
     codegen.curVecMask = genVectorMask(codegen, rewriter, iv, lo, hi, step);
   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,
-                           BitVector &indices) {
+                           unsigned idx, bool needsUniv, 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.isDim(b, Dim::kSparse)) {
       unsigned tensor = merger.tensor(b);
       assert(idx == merger.index(b));
rewriter.create<memref::StoreOp>(loc, codegen.loops[idx], codegen.lexIdx,
                                      pos);
   }
 }
 
 /// Generates the induction structure for a while-loop.
 static void genWhileInduction(Merger &merger, CodeGen &codegen,
                               PatternRewriter &rewriter, linalg::GenericOp op,
                               unsigned idx, bool needsUniv,
-                              BitVector &induction,
-                              scf::WhileOp whileOp) {
+                              BitVector &induction, scf::WhileOp whileOp) {
   Location loc = op.getLoc();
   // Finalize each else branch of all if statements.
   if (codegen.redVal || codegen.expValues) {
     while (auto ifOp = dyn_cast_or_null<scf::IfOp>(
                rewriter.getInsertionBlock()->getParentOp())) {
       unsigned y = 0;
       SmallVector<Value, 4> yields;
       if (codegen.redVal) {

 /// 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 ldx = topSort[at - 1];
     Value rhs = genExp(merger, codegen, rewriter, op, exp, ldx);
-    genTensorStore(merger, codegen, rewriter, op, rhs);
+    genTensorStore(merger, codegen, rewriter, op, exp, rhs);
     return;
   }
 
   // Construct iteration lattices for current loop index, with L0 at top.
   unsigned idx = topSort[at];
   unsigned ldx = at == 0 ? -1u : topSort[at - 1];
   unsigned lts = merger.optimizeSet(merger.buildLattices(exp, idx));
 
   // Start a loop sequence.
   bool needsUniv = startLoopSeq(merger, codegen, rewriter, op, topSort, exp, at,
                                 idx, ldx, lts);
 
   // Emit a loop for every lattice point L0 >= Li in this loop sequence.
   unsigned lsize = merger.set(lts).size();
   for (unsigned i = 0; i < lsize; i++) {
     // Start a loop.

mlir/lib/Dialect/SparseTensor/Utils/Merger.cpp

 //===- Merger.cpp - Implementation of iteration lattices ------------------===//
 //
 // Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
 // See https://llvm.org/LICENSE.txt for license information.
 // SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
 //
 //===----------------------------------------------------------------------===//
 
 #include "mlir/Dialect/SparseTensor/Utils/Merger.h"
 #include "mlir/Dialect/Arithmetic/IR/Arithmetic.h"
+#include "mlir/Dialect/SparseTensor/IR/SparseTensor.h"
 
 #include "mlir/IR/Operation.h"
 #include "llvm/Support/Debug.h"
 
 namespace mlir {
 namespace sparse_tensor {
 
 //===----------------------------------------------------------------------===//
 // Constructors.
 //===----------------------------------------------------------------------===//
 
-TensorExp::TensorExp(Kind k, unsigned x, unsigned y, Value v)
+TensorExp::TensorExp(Kind k, unsigned x, unsigned y, Value v,
+                     Operation *operation)
     : kind(k), val(v) {
   switch (kind) {
   case kTensor:
-    assert(x != -1u && y == -1u && !v);
+    assert(x != -1u && y == -1u && !v && !operation);
     tensor = x;
     break;
   case kInvariant:
-    assert(x == -1u && y == -1u && v);
+    assert(x == -1u && y == -1u && v && !operation);
     break;
   case kIndex:
-    assert(x != -1u && y == -1u && !v);
+    assert(x != -1u && y == -1u && !v && !operation);
     index = x;
     break;
   case kAbsF:
   case kCeilF:
   case kFloorF:
   case kNegF:
   case kNegI:
-    assert(x != -1u && y == -1u && !v);
+    assert(x != -1u && y == -1u && !v && !operation);
     children.e0 = x;
     children.e1 = y;
     break;
   case kTruncF:
   case kExtF:
   case kCastFS:
   case kCastFU:
   case kCastSF:
   case kCastUF:
   case kCastS:
   case kCastU:
   case kCastIdx:
   case kTruncI:
   case kBitCast:
-    assert(x != -1u && y == -1u && v);
+    assert(x != -1u && y == -1u && v && !operation);
     children.e0 = x;
     children.e1 = y;
     break;
+  case kUnary:
+    assert(x != -1u && !v && operation);
+    children.e0 = x;
+    children.e1 = y;
+    op = operation;
+    break;
+  case kBinaryHalf:
+    assert(x != -1u && y == -1u && !v && operation);
+    children.e0 = x;
+    children.e1 = y;
+    op = operation;
+    break;
+  case kBinary:
+    assert(x != -1u && y != -1u && !v && operation);
+    children.e0 = x;
+    children.e1 = y;
+    op = operation;
+    break;
   default:
-    assert(x != -1u && y != -1u && !v);
+    assert(x != -1u && y != -1u && !v && !operation);
     children.e0 = x;
     children.e1 = y;
     break;
   }
 }
 
 LatPoint::LatPoint(unsigned n, unsigned e, unsigned b)
     : bits(n, false), simple(), exp(e) {
   bits.set(b);
 }
 
 LatPoint::LatPoint(const BitVector &b, unsigned e)
     : bits(b), simple(), exp(e) {}
 
 //===----------------------------------------------------------------------===//
 // Lattice methods.
 //===----------------------------------------------------------------------===//
 
-unsigned Merger::addExp(Kind k, unsigned e0, unsigned e1, Value v) {
+unsigned Merger::addExp(Kind k, unsigned e0, unsigned e1, Value v,
+                        Operation *op) {
   unsigned e = tensorExps.size();
-  tensorExps.push_back(TensorExp(k, e0, e1, v));
+  tensorExps.push_back(TensorExp(k, e0, e1, v, op));
   return e;
 }
 
 unsigned Merger::addLat(unsigned t, unsigned i, unsigned e) {
   assert(t < numTensors && i < numLoops);
   unsigned p = latPoints.size();
   latPoints.push_back(LatPoint(numLoops * numTensors, e, numTensors * i + t));
   return p;
 }
 
 unsigned Merger::addSet() {
   unsigned s = latSets.size();
   latSets.emplace_back(SmallVector<unsigned, 16>());
   return s;
 }
 
-unsigned Merger::conjLatPoint(Kind kind, unsigned p0, unsigned p1) {
+unsigned Merger::conjLatPoint(Kind kind, unsigned p0, unsigned p1,
+                              Operation *op) {
   unsigned p = latPoints.size();
   BitVector nb = BitVector(latPoints[p0].bits);
   nb |= latPoints[p1].bits;
-  unsigned e = addExp(kind, latPoints[p0].exp, latPoints[p1].exp);
+  unsigned e = addExp(kind, latPoints[p0].exp, latPoints[p1].exp, Value(), op);
   latPoints.push_back(LatPoint(nb, e));
   return p;
 }
 
-unsigned Merger::takeConj(Kind kind, unsigned s0, unsigned s1) {
+unsigned Merger::takeConj(Kind kind, unsigned s0, unsigned s1, Operation *op) {
   unsigned s = addSet();
   for (unsigned p0 : latSets[s0])
     for (unsigned p1 : latSets[s1])
-      latSets[s].push_back(conjLatPoint(kind, p0, p1));
+      latSets[s].push_back(conjLatPoint(kind, p0, p1, op));
   return s;
 }
 
-unsigned Merger::takeDisj(Kind kind, unsigned s0, unsigned s1) {
-  unsigned s = takeConj(kind, s0, s1);
+unsigned Merger::takeDisj(Kind kind, unsigned s0, unsigned s1, Operation *op) {
+  unsigned s = takeConj(kind, s0, s1, op);
   // Followed by all in s0.
   for (unsigned p : latSets[s0])
     latSets[s].push_back(p);
+  // TODO: move this logic into buildLattices
   // Map binary 0-y to unary -y.
   if (kind == kSubF)
     s1 = mapSet(kNegF, s1);
   else if (kind == kSubI)
     s1 = mapSet(kNegI, s1);
   // Followed by all in s1.
   for (unsigned p : latSets[s1])
     latSets[s].push_back(p);
   return s;
 }
 
-unsigned Merger::mapSet(Kind kind, unsigned s0, Value v) {
-  assert(kAbsF <= kind && kind <= kBitCast);
+unsigned Merger::takeCombi(Kind kind, unsigned s0, unsigned s1, Operation *orig,
+                           bool includeLeft, Kind ltrans, Operation *opleft,
+                           bool includeRight, Kind rtrans, Operation *opright) {
+  unsigned s = takeConj(kind, s0, s1, orig);
+  // Left Region.
+  if (includeLeft) {
+    if (opleft)
+      s0 = mapSet(ltrans, s0, Value(), opleft);
+    for (unsigned p : latSets[s0])
+      latSets[s].push_back(p);
+  }
+  // Right Region.
+  if (includeRight) {
+    if (opright)
+      s1 = mapSet(rtrans, s1, Value(), opright);
+    for (unsigned p : latSets[s1])
+      latSets[s].push_back(p);
+  }
+  return s;
+}
+
+unsigned Merger::mapSet(Kind kind, unsigned s0, Value v, Operation *op) {
+  assert(kAbsF <= kind && kind <= kBinaryHalf);
   unsigned s = addSet();
   for (unsigned p : latSets[s0]) {
-    unsigned e = addExp(kind, latPoints[p].exp, v);
+    unsigned e = addExp(kind, latPoints[p].exp, v, op);
     latPoints.push_back(LatPoint(latPoints[p].bits, e));
     latSets[s].push_back(latPoints.size() - 1);
   }
   return s;
 }
 
 unsigned Merger::optimizeSet(unsigned s0) {
   unsigned s = addSet();
static const char *kindToOpSymbol(Kind kind) {
   case kCastSF:
   case kCastUF:
   case kCastS:
   case kCastU:
   case kCastIdx:
   case kTruncI:
   case kBitCast:
     return "cast";
+  case kBinaryHalf:
+    return "binary_half";
+  case kUnary:
+    return "unary";
   case kMulF:
     return "*";
   case kMulI:
     return "*";
   case kDivF:
     return "/";
   case kDivS:
     return "/";
static const char *kindToOpSymbol(Kind kind) {
   case kXorI:
     return "^";
   case kShrS:
     return "a>>";
   case kShrU:
     return ">>";
   case kShlI:
     return "<<";
+  case kBinary:
+    return "binary";
   }
   llvm_unreachable("unexpected kind for symbol");
 }
 
 void Merger::dumpExp(unsigned e) const {
   switch (tensorExps[e].kind) {
   case kTensor:
     if (tensorExps[e].tensor == syntheticTensor)
case kBitCast:
     // A zero preserving operation (viz. f(0) = 0, [Bik96,Ch5]) maps the
     // lattice set of the operand through the operator into a new set.
     //
     //  -y|!y | y |
     //  --+---+---+
     //    | 0 |-y |
     return mapSet(kind, buildLattices(tensorExps[e].children.e0, i),
                   tensorExps[e].val);
+  case kUnary:
+    // A custom unary operation.
+    //
+    //  op y|    !y    |     y      |
+    //  ----+----------+------------+
+    //      | absent() | present(y) |
+    {
+      unsigned child0 = buildLattices(tensorExps[e].children.e0, i);
+      UnaryOp unop = dyn_cast<UnaryOp>(tensorExps[e].op);
+      assert(unop);
+      Region &absentRegion = unop.absentRegion();
+
+      if (absentRegion.empty()) {
+        // Simple mapping over existing values.
+        return mapSet(kind, child0, Value(), unop);
+      } else {
+        // Use a disjunction with `unop` on the left and the absent value as an
+        // invariant on the right.
+        Block &absentBlock = absentRegion.front();
+        YieldOp absentYield = dyn_cast<YieldOp>(absentBlock.getTerminator());
+        Value absentVal = absentYield.result();
+        unsigned rhs = addExp(kInvariant, absentVal);
+        return takeDisj(kind, child0, buildLattices(rhs, i), unop);
+      }
+    }
   case kMulF:
   case kMulI:
   case kAndI:
     // A multiplicative operation only needs to be performed
     // for the conjunction of sparse iteration spaces.
     //
     //  x*y|!y | y |
     //  ---+---+---+
     //  !x | 0 | 0 |
     //   x | 0 |x*y|
     return takeConj(kind, // take binary conjunction
                     buildLattices(tensorExps[e].children.e0, i),
unsigned Merger::buildLattices(unsigned e, unsigned i) {
   case kShlI:
     // A shift operation by an invariant amount (viz. tensor expressions
     // can only occur at the left-hand-side of the operator) can be handled
     // with the conjuction rule.
     assert(isInvariant(tensorExps[e].children.e1));
     return takeConj(kind, // take binary conjunction
                     buildLattices(tensorExps[e].children.e0, i),
                     buildLattices(tensorExps[e].children.e1, i));
+  case kBinaryHalf:
+    // The left or right half of a binary operation which has already
+    // been split into separate operations for each region.
+    return mapSet(kind, buildLattices(tensorExps[e].children.e0, i), Value(),
+                  tensorExps[e].op);
+  case kBinary:
+    // A custom binary operation.
+    //
+    //  x op y|   !y    |       y      |
+    //  ------+---------+--------------+
+    //    !x  |  empty  |   right(y)   |
+    //     x  | left(x) | overlap(x,y) |
+    {
+      unsigned child0 = buildLattices(tensorExps[e].children.e0, i);
+      unsigned child1 = buildLattices(tensorExps[e].children.e1, i);
+      BinaryOp binop = dyn_cast<BinaryOp>(tensorExps[e].op);
+      assert(binop);
+      Region &leftRegion = binop.leftRegion();
+      Region &rightRegion = binop.rightRegion();
+      // Left Region.
+      Operation *leftYield = nullptr;
+      if (!leftRegion.empty()) {
+        Block &leftBlock = leftRegion.front();
+        leftYield = leftBlock.getTerminator();
+      }
+      // Right Region.
+      Operation *rightYield = nullptr;
+      if (!rightRegion.empty()) {
+        Block &rightBlock = rightRegion.front();
+        rightYield = rightBlock.getTerminator();
+      }
+      bool includeLeft = binop.left_identity() || !leftRegion.empty();
+      bool includeRight = binop.right_identity() || !rightRegion.empty();
+      return takeCombi(kBinary, child0, child1, binop, includeLeft, kBinaryHalf,
+                       leftYield, includeRight, kBinaryHalf, rightYield);
+    }
   }
   llvm_unreachable("unexpected expression kind");
 }
 
 Optional<unsigned> Merger::buildTensorExpFromLinalg(linalg::GenericOp op) {
   Operation *yield = op.region().front().getTerminator();
   return buildTensorExp(op, yield->getOperand(0));
 }
 
 /// Only returns false if we are certain this is a nonzero.
 bool Merger::maybeZero(unsigned e) const {
   if (tensorExps[e].kind == kInvariant) {
     if (auto c = tensorExps[e].val.getDefiningOp<arith::ConstantIntOp>())
       return c.value() == 0;
     if (auto c = tensorExps[e].val.getDefiningOp<arith::ConstantFloatOp>())
       return c.value().isZero();
   }
   return true;
if (x.hasValue()) {
       if (isa<arith::ExtUIOp>(def))
         return addExp(kCastU, e, v);
       if (isa<arith::IndexCastOp>(def))
         return addExp(kCastIdx, e, v);
       if (isa<arith::TruncIOp>(def))
         return addExp(kTruncI, e, v);
       if (isa<arith::BitcastOp>(def))
         return addExp(kBitCast, e, v);
+      if (isa<sparse_tensor::UnaryOp>(def))
+        return addExp(kUnary, e, Value(), def);
     }
   }
   // Construct binary operations if subexpressions can be built.
   // See buildLattices() for an explanation of rejecting certain
   // division and shift operations
   if (def->getNumOperands() == 2) {
     auto x = buildTensorExp(op, def->getOperand(0));
     auto y = buildTensorExp(op, def->getOperand(1));
if (x.hasValue() && y.hasValue()) {
       if (isa<arith::XOrIOp>(def))
         return addExp(kXorI, e0, e1);
       if (isa<arith::ShRSIOp>(def) && isInvariant(e1))
         return addExp(kShrS, e0, e1);
       if (isa<arith::ShRUIOp>(def) && isInvariant(e1))
         return addExp(kShrU, e0, e1);
       if (isa<arith::ShLIOp>(def) && isInvariant(e1))
         return addExp(kShlI, e0, e1);
+      if (isa<sparse_tensor::BinaryOp>(def))
+        return addExp(kBinary, e0, e1, Value(), def);
     }
   }
   // Cannot build.
   return None;
 }
 
+Value insertYieldOp(PatternRewriter &rewriter, Location loc, Region &region,
+                    ValueRange vals) {
+  // Make a clone of overlap region.
+  Region tmpRegion;
+  BlockAndValueMapping mapper;
+  region.cloneInto(&tmpRegion, tmpRegion.begin(), mapper);
+  Block &clonedBlock = tmpRegion.front();
+  YieldOp clonedYield = dyn_cast<YieldOp>(clonedBlock.getTerminator());
+  // Merge cloned block and return yield value.
+  Operation *placeholder = rewriter.create<arith::ConstantIndexOp>(loc, 0);
+  rewriter.mergeBlockBefore(&tmpRegion.front(), placeholder, vals);
+  Value val = clonedYield.result();
+  rewriter.eraseOp(clonedYield);
+  rewriter.eraseOp(placeholder);
+  return val;
+}
+
 Value Merger::buildExp(PatternRewriter &rewriter, Location loc, unsigned e,
                        Value v0, Value v1) {
   switch (tensorExps[e].kind) {
   case kTensor:
   case kInvariant:
   case kIndex:
     llvm_unreachable("unexpected non-op");
   // Unary ops.
Value Merger::buildExp(PatternRewriter &rewriter, Location loc, unsigned e,
   case kCastIdx:
     return rewriter.create<arith::IndexCastOp>(loc, inferType(e, v0), v0);
   case kTruncI:
     return rewriter.create<arith::TruncIOp>(loc, inferType(e, v0), v0);
   case kBitCast:
     return rewriter.create<arith::BitcastOp>(loc, inferType(e, v0), v0);
   // Binary ops.
   case kMulF:
     return rewriter.create<arith::MulFOp>(loc, v0, v1);
   case kMulI:
     return rewriter.create<arith::MulIOp>(loc, v0, v1);
   case kDivF:
     return rewriter.create<arith::DivFOp>(loc, v0, v1);
   case kDivS:
     return rewriter.create<arith::DivSIOp>(loc, v0, v1);
   case kDivU:
     return rewriter.create<arith::DivUIOp>(loc, v0, v1);
   case kAddF:
Value Merger::buildExp(PatternRewriter &rewriter, Location loc, unsigned e,
   case kXorI:
     return rewriter.create<arith::XOrIOp>(loc, v0, v1);
   case kShrS:
     return rewriter.create<arith::ShRSIOp>(loc, v0, v1);
   case kShrU:
     return rewriter.create<arith::ShRUIOp>(loc, v0, v1);
   case kShlI:
     return rewriter.create<arith::ShLIOp>(loc, v0, v1);
+  // Set-like ops with custom logic.
+  case kUnary: {
+    if (!v0)
+      // Empty input value must be propagated.
+      return Value();
+    UnaryOp unop = dyn_cast<UnaryOp>(tensorExps[e].op);
+    assert(unop);
+    Region &presentRegion = unop.presentRegion();
+    if (presentRegion.empty())
+      // Uninitialized Value() will be interpreted as missing data in the
+      // output.
+      return Value();
+    return insertYieldOp(rewriter, loc, presentRegion, {v0});
+  }
+  case kBinaryHalf: {
+    Operation *op = tensorExps[e].op;
+    assert(op);
+    return insertYieldOp(rewriter, loc, *op->getBlock()->getParent(), {v0});
+  }
+  case kBinary: {
+    if (!v0 || !v1)
+      // Empty input values must be propagated.
+      return Value();
+    BinaryOp binop = dyn_cast<BinaryOp>(tensorExps[e].op);
+    assert(binop);
+    Region &overlapRegion = binop.overlapRegion();
+    if (overlapRegion.empty())
+      // Uninitialized Value() will be interpreted as missing data in the
+      // output.
+      return Value();
+    return insertYieldOp(rewriter, loc, overlapRegion, {v0, v1});
+  }
   }
   llvm_unreachable("unexpected expression kind in build");
 }
 
 } // namespace sparse_tensor
 } // namespace mlir

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

+// RUN: mlir-opt %s --sparse-compiler | \
+// 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
+
+#SparseVector = #sparse_tensor.encoding<{dimLevelType = ["compressed"]}>
+#DCSR = #sparse_tensor.encoding<{dimLevelType = ["compressed", "compressed"]}>
+
+//
+// Traits for tensor operations.
+//
+#trait_vec_scale = {
+  indexing_maps = [
+    affine_map<(i) -> (i)>,  // a (in)
+    affine_map<(i) -> (i)>   // x (out)
+  ],
+  iterator_types = ["parallel"]
+}
+#trait_vec_op = {
+  indexing_maps = [
+    affine_map<(i) -> (i)>,  // a (in)
+    affine_map<(i) -> (i)>,  // b (in)
+    affine_map<(i) -> (i)>   // x (out)
+  ],
+  iterator_types = ["parallel"]
+}
+#trait_mat_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 {
+  // Creates a new sparse vector using the minimum values from two input sparse vectors.
+  // When there is no overlap, include the present value in the output.
+  func @vector_min(%arga: tensor<?xf64, #SparseVector>,
+                   %argb: tensor<?xf64, #SparseVector>) -> tensor<?xf64, #SparseVector> {
+    %c = arith.constant 0 : index
+    %d = tensor.dim %arga, %c : tensor<?xf64, #SparseVector>
+    %xv = sparse_tensor.init [%d] : tensor<?xf64, #SparseVector>
+    %0 = linalg.generic #trait_vec_op
+       ins(%arga, %argb: tensor<?xf64, #SparseVector>, tensor<?xf64, #SparseVector>)
+        outs(%xv: tensor<?xf64, #SparseVector>) {
+        ^bb(%a: f64, %b: f64, %x: f64):
+          %1 = sparse_tensor.binary %a, %b : f64, f64 to f64
+            overlap={
+              ^bb0(%a0: f64, %b0: f64):
+                %cmp = arith.cmpf "olt", %a0, %b0 : f64
+                %2 = arith.select %cmp, %a0, %b0: f64
+                sparse_tensor.yield %2 : f64
+            }
+            left=identity
+            right=identity
+          linalg.yield %1 : f64
+    } -> tensor<?xf64, #SparseVector>
+    return %0 : tensor<?xf64, #SparseVector>
+  }
+
+  // Creates a new sparse vector by multiplying a sparse vector with a dense vector.
+  // When there is no overlap, leave the result empty.
+  func @vector_mul(%arga: tensor<?xf64, #SparseVector>,
+                   %argb: tensor<?xf64>) -> tensor<?xf64, #SparseVector> {
+    %c = arith.constant 0 : index
+    %d = tensor.dim %arga, %c : tensor<?xf64, #SparseVector>
+    %xv = sparse_tensor.init [%d] : tensor<?xf64, #SparseVector>
+    %0 = linalg.generic #trait_vec_op
+       ins(%arga, %argb: tensor<?xf64, #SparseVector>, tensor<?xf64>)
+        outs(%xv: tensor<?xf64, #SparseVector>) {
+        ^bb(%a: f64, %b: f64, %x: f64):
+          %1 = sparse_tensor.binary %a, %b : f64, f64 to f64
+            overlap={
+              ^bb0(%a0: f64, %b0: f64):
+                %ret = arith.mulf %a0, %b0 : f64
+                sparse_tensor.yield %ret : f64
+            }
+            left={}
+            right={}
+          linalg.yield %1 : f64
+    } -> tensor<?xf64, #SparseVector>
+    return %0 : tensor<?xf64, #SparseVector>
+  }
+
+  // Take a set difference of two sparse vectors. The result will include only those
+  // sparse elements present in the first, but not the second vector.
+  func @vector_setdiff(%arga: tensor<?xf64, #SparseVector>,
+                       %argb: tensor<?xf64, #SparseVector>) -> tensor<?xf64, #SparseVector> {
+    %c = arith.constant 0 : index
+    %d = tensor.dim %arga, %c : tensor<?xf64, #SparseVector>
+    %xv = sparse_tensor.init [%d] : tensor<?xf64, #SparseVector>
+    %0 = linalg.generic #trait_vec_op
+       ins(%arga, %argb: tensor<?xf64, #SparseVector>, tensor<?xf64, #SparseVector>)
+        outs(%xv: tensor<?xf64, #SparseVector>) {
+        ^bb(%a: f64, %b: f64, %x: f64):
+          %1 = sparse_tensor.binary %a, %b : f64, f64 to f64
+            overlap={}
+            left=identity
+            right={}
+          linalg.yield %1 : f64
+    } -> tensor<?xf64, #SparseVector>
+    return %0 : tensor<?xf64, #SparseVector>
+  }
+
+  // Return the index of each entry
+  func @vector_index(%arga: tensor<?xf64, #SparseVector>) -> tensor<?xi32, #SparseVector> {
+    %c = arith.constant 0 : index
+    %d = tensor.dim %arga, %c : tensor<?xf64, #SparseVector>
+    %xv = sparse_tensor.init [%d] : tensor<?xi32, #SparseVector>
+    %0 = linalg.generic #trait_vec_scale
+       ins(%arga: tensor<?xf64, #SparseVector>)
+        outs(%xv: tensor<?xi32, #SparseVector>) {
+        ^bb(%a: f64, %x: i32):
+          %idx = linalg.index 0 : index
+          %1 = sparse_tensor.binary %a, %idx : f64, index to i32
+            overlap={
+              ^bb0(%x0: f64, %i: index):
+                %ret = arith.index_cast %i : index to i32
+                sparse_tensor.yield %ret : i32
+            }
+            left={}
+            right={}
+          linalg.yield %1 : i32
+    } -> tensor<?xi32, #SparseVector>
+    return %0 : tensor<?xi32, #SparseVector>
+  }
+
+  // Adds two sparse matrices when they intersect. Where they don't intersect,
+  // negate the 2nd argument's values; ignore 1st argument-only values.
+  func @matrix_intersect(%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_mat_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 = sparse_tensor.binary %a, %b: f64, f64 to f64
+            overlap={
+              ^bb0(%x0: f64, %y0: f64):
+                %ret = arith.addf %x0, %y0 : f64
+                sparse_tensor.yield %ret : f64
+            }
+            left={}
+            right={
+              ^bb0(%x1: f64):
+                %lret = arith.negf %x1 : f64
+                sparse_tensor.yield %lret : f64
+            }
+          linalg.yield %1 : f64
+    } -> tensor<?x?xf64, #DCSR>
+    return %0 : tensor<?x?xf64, #DCSR>
+  }
+
+  // Dumps a sparse vector of type f64.
+  func @dump_vec(%arg0: tensor<?xf64, #SparseVector>) {
+    // Dump the values array to verify only sparse contents are stored.
+    %c0 = arith.constant 0 : index
+    %d0 = arith.constant -1.0 : f64
+    %0 = sparse_tensor.values %arg0 : tensor<?xf64, #SparseVector> to memref<?xf64>
+    %1 = vector.transfer_read %0[%c0], %d0: memref<?xf64>, vector<16xf64>
+    vector.print %1 : vector<16xf64>
+    // Dump the dense vector to verify structure is correct.
+    %dv = sparse_tensor.convert %arg0 : tensor<?xf64, #SparseVector> to tensor<?xf64>
+    %2 = bufferization.to_memref %dv : memref<?xf64>
+    %3 = vector.transfer_read %2[%c0], %d0: memref<?xf64>, vector<32xf64>
+    vector.print %3 : vector<32xf64>
+    memref.dealloc %2 : memref<?xf64>
+    return
+  }
+
+  // Dumps a sparse vector of type i32.
+  func @dump_vec_i32(%arg0: tensor<?xi32, #SparseVector>) {
+    // Dump the values array to verify only sparse contents are stored.
+    %c0 = arith.constant 0 : index
+    %d0 = arith.constant -1 : i32
+    %0 = sparse_tensor.values %arg0 : tensor<?xi32, #SparseVector> to memref<?xi32>
+    %1 = vector.transfer_read %0[%c0], %d0: memref<?xi32>, vector<24xi32>
+    vector.print %1 : vector<24xi32>
+    // Dump the dense vector to verify structure is correct.
+    %dv = sparse_tensor.convert %arg0 : tensor<?xi32, #SparseVector> to tensor<?xi32>
+    %2 = bufferization.to_memref %dv : memref<?xi32>
+    %3 = vector.transfer_read %2[%c0], %d0: memref<?xi32>, vector<32xi32>
+    vector.print %3 : vector<32xi32>
+    memref.dealloc %2 : memref<?xi32>
+    return
+  }
+
+  // Dump a sparse matrix.
+  func @dump_mat(%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 = bufferization.to_memref %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 vector kernels.
+  func @entry() {
+    %c0 = arith.constant 0 : index
+
+    // Setup sparse vectors.
+    %v1 = arith.constant sparse<
+       [ [0], [3], [11], [17], [20], [21], [28], [29], [31] ],
+         [ 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0 ]
+    > : tensor<32xf64>
+    %v2 = arith.constant sparse<
+       [ [1], [3], [4], [10], [16], [18], [21], [28], [29], [31] ],
+         [11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0 ]
+    > : tensor<32xf64>
+    %v3 = arith.constant dense<
+      [0., 1., 2., 3., 4., 5., 6., 7., 8., 9., 0., 1., 2., 3., 4., 5., 6., 7., 8., 9.,
+       0., 1., 2., 3., 4., 5., 6., 7., 8., 9., 0., 1.]
+    > : tensor<32xf64>
+    %sv1 = sparse_tensor.convert %v1 : tensor<32xf64> to tensor<?xf64, #SparseVector>
+    %sv2 = sparse_tensor.convert %v2 : tensor<32xf64> to tensor<?xf64, #SparseVector>
+    %dv3 = tensor.cast %v3 : tensor<32xf64> to tensor<?xf64>
+
+    // 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 @vector_min(%sv1, %sv2)
+       : (tensor<?xf64, #SparseVector>,
+          tensor<?xf64, #SparseVector>) -> tensor<?xf64, #SparseVector>
+    %1 = call @vector_mul(%sv1, %dv3)
+      : (tensor<?xf64, #SparseVector>,
+         tensor<?xf64>) -> tensor<?xf64, #SparseVector>
+    %2 = call @vector_setdiff(%sv1, %sv2)
+       : (tensor<?xf64, #SparseVector>,
+          tensor<?xf64, #SparseVector>) -> tensor<?xf64, #SparseVector>
+    %3 = call @vector_index(%sv1)
+       : (tensor<?xf64, #SparseVector>) -> tensor<?xi32, #SparseVector>
+
+    // Call sparse matrix kernels.
+    %5 = call @matrix_intersect(%sm1, %sm2)
+      : (tensor<?x?xf64, #DCSR>, tensor<?x?xf64, #DCSR>) -> tensor<?x?xf64, #DCSR>
+
+    //
+    // Verify the results.
+    //
+    // CHECK:      ( 1, 2, 3, 4, 5, 6, 7, 8, 9, -1, -1, -1, -1, -1, -1, -1 )
+    // CHECK-NEXT: ( 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 4, 0, 0, 5, 6, 0, 0, 0, 0, 0, 0, 7, 8, 0, 9 )
+    // CHECK-NEXT: ( 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, -1, -1, -1, -1, -1, -1 )
+    // CHECK-NEXT: ( 0, 11, 0, 12, 13, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 15, 0, 16, 0, 0, 17, 0, 0, 0, 0, 0, 0, 18, 19, 0, 20 )
+    // CHECK-NEXT: ( 1, 11, 2, 13, 14, 3, 15, 4, 16, 5, 6, 7, 8, 9, -1, -1 )
+    // CHECK-NEXT: ( 1, 11, 0, 2, 13, 0, 0, 0, 0, 0, 14, 3, 0, 0, 0, 0, 15, 4, 16, 0, 5, 6, 0, 0, 0, 0, 0, 0, 7, 8, 0, 9 )
+    // CHECK-NEXT: ( 0, 6, 3, 28, 0, 6, 56, 72, 9, -1, -1, -1, -1, -1, -1, -1 )
+    // CHECK-NEXT: ( 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 28, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 56, 72, 0, 9 )
+    // CHECK-NEXT: ( 1, 3, 4, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 )
+    // CHECK-NEXT: ( 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 4, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 )
+    // CHECK-NEXT: ( 0, 3, 11, 17, 20, 21, 28, 29, 31, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 )
+    // CHECK-NEXT: ( 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 17, 0, 0, 20, 21, 0, 0, 0, 0, 0, 0, 28, 29, 0, 31 )
+    // CHECK-NEXT: ( ( 7, 0, 0, 0, 0, 0, 0, -5 ), ( -4, 0, 0, 0, 0, 0, -3, 0 ), ( 0, -2, 0, 0, 0, 0, 0, 7 ), ( 0, 0, 0, 0, 0, 0, 0, 0 ) )
+    //
+    call @dump_vec(%sv1) : (tensor<?xf64, #SparseVector>) -> ()
+    call @dump_vec(%sv2) : (tensor<?xf64, #SparseVector>) -> ()
+    call @dump_vec(%0) : (tensor<?xf64, #SparseVector>) -> ()
+    call @dump_vec(%1) : (tensor<?xf64, #SparseVector>) -> ()
+    call @dump_vec(%2) : (tensor<?xf64, #SparseVector>) -> ()
+    call @dump_vec_i32(%3) : (tensor<?xi32, #SparseVector>) -> ()
+    call @dump_mat(%5) : (tensor<?x?xf64, #DCSR>) -> ()
+
+    // Release the resources.
+    sparse_tensor.release %sv1 : tensor<?xf64, #SparseVector>
+    sparse_tensor.release %sv2 : tensor<?xf64, #SparseVector>
+    sparse_tensor.release %sm1 : tensor<?x?xf64, #DCSR>
+    sparse_tensor.release %sm2 : tensor<?x?xf64, #DCSR>
+    sparse_tensor.release %0 : tensor<?xf64, #SparseVector>
+    sparse_tensor.release %1 : tensor<?xf64, #SparseVector>
+    sparse_tensor.release %2 : tensor<?xf64, #SparseVector>
+    sparse_tensor.release %3 : tensor<?xi32, #SparseVector>
+    sparse_tensor.release %5 : tensor<?x?xf64, #DCSR>
+    return
+  }
+}
 No newline at end of file

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

%0 = linalg.generic #trait_scale
         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>
%0 = linalg.generic #trait_op
           %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 = bufferization.to_memref %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
   }
func @entry() {
     > : 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.
+    // Call sparse matrix 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>

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

+// RUN: mlir-opt %s --sparse-compiler | \
+// 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
+
+#SparseVector = #sparse_tensor.encoding<{dimLevelType = ["compressed"]}>
+#DCSR = #sparse_tensor.encoding<{dimLevelType = ["compressed", "compressed"]}>
+
+//
+// Traits for tensor operations.
+//
+#trait_vec_scale = {
+  indexing_maps = [
+    affine_map<(i) -> (i)>,  // a (in)
+    affine_map<(i) -> (i)>   // x (out)
+  ],
+  iterator_types = ["parallel"]
+}
+#trait_mat_scale = {
+  indexing_maps = [
+    affine_map<(i,j) -> (i,j)>,  // A (in)
+    affine_map<(i,j) -> (i,j)>   // X (out)
+  ],
+  iterator_types = ["parallel", "parallel"]
+}
+
+module {
+  // Invert the structure of a sparse vector. Present values become missing.
+  // Missing values are filled with 1 (i32).
+  func @vector_complement(%arga: tensor<?xf64, #SparseVector>) -> tensor<?xi32, #SparseVector> {
+    %c = arith.constant 0 : index
+    %ci1 = arith.constant 1 : i32
+    %d = tensor.dim %arga, %c : tensor<?xf64, #SparseVector>
+    %xv = sparse_tensor.init [%d] : tensor<?xi32, #SparseVector>
+    %0 = linalg.generic #trait_vec_scale
+       ins(%arga: tensor<?xf64, #SparseVector>)
+        outs(%xv: tensor<?xi32, #SparseVector>) {
+        ^bb(%a: f64, %x: i32):
+          %1 = sparse_tensor.unary %a : f64 to i32
+            present={}
+            absent={
+              sparse_tensor.yield %ci1 : i32
+            }
+          linalg.yield %1 : i32
+    } -> tensor<?xi32, #SparseVector>
+    return %0 : tensor<?xi32, #SparseVector>
+  }
+
+  // Negate existing values. Fill missing ones with +1.
+  func @vector_negation(%arga: tensor<?xf64, #SparseVector>) -> tensor<?xf64, #SparseVector> {
+    %c = arith.constant 0 : index
+    %cf1 = arith.constant 1.0 : f64
+    %d = tensor.dim %arga, %c : tensor<?xf64, #SparseVector>
+    %xv = sparse_tensor.init [%d] : tensor<?xf64, #SparseVector>
+    %0 = linalg.generic #trait_vec_scale
+       ins(%arga: tensor<?xf64, #SparseVector>)
+        outs(%xv: tensor<?xf64, #SparseVector>) {
+        ^bb(%a: f64, %x: f64):
+          %1 = sparse_tensor.unary %a : f64 to f64
+            present={
+              ^bb0(%x0: f64):
+                %ret = arith.negf %x0 : f64
+                sparse_tensor.yield %ret : f64
+            }
+            absent={
+              sparse_tensor.yield %cf1 : f64
+            }
+          linalg.yield %1 : f64
+    } -> tensor<?xf64, #SparseVector>
+    return %0 : tensor<?xf64, #SparseVector>
+  }
+
+  // Clips values to the range [3, 7].
+  func @matrix_clip(%argx: tensor<?x?xf64, #DCSR>) -> tensor<?x?xf64, #DCSR> {
+    %c0 = arith.constant 0 : index
+    %c1 = arith.constant 1 : index
+    %cfmin = arith.constant 3.0 : f64
+    %cfmax = arith.constant 7.0 : f64
+    %d0 = tensor.dim %argx, %c0 : tensor<?x?xf64, #DCSR>
+    %d1 = tensor.dim %argx, %c1 : tensor<?x?xf64, #DCSR>
+    %xv = sparse_tensor.init [%d0, %d1] : tensor<?x?xf64, #DCSR>
+    %0 = linalg.generic #trait_mat_scale
+       ins(%argx: tensor<?x?xf64, #DCSR>)
+        outs(%xv: tensor<?x?xf64, #DCSR>) {
+        ^bb(%a: f64, %x: f64):
+          %1 = sparse_tensor.unary %a: f64 to f64
+            present={
+              ^bb0(%x0: f64):
+                %mincmp = arith.cmpf "ogt", %x0, %cfmin : f64
+                %x1 = arith.select %mincmp, %x0, %cfmin : f64
+                %maxcmp = arith.cmpf "olt", %x1, %cfmax : f64
+                %x2 = arith.select %maxcmp, %x1, %cfmax : f64
+                sparse_tensor.yield %x2 : f64
+            }
+            absent={}
+          linalg.yield %1 : f64
+    } -> tensor<?x?xf64, #DCSR>
+    return %0 : tensor<?x?xf64, #DCSR>
+  }
+
+  // Dumps a sparse vector of type f64.
+  func @dump_vec_f64(%arg0: tensor<?xf64, #SparseVector>) {
+    // Dump the values array to verify only sparse contents are stored.
+    %c0 = arith.constant 0 : index
+    %d0 = arith.constant -1.0 : f64
+    %0 = sparse_tensor.values %arg0 : tensor<?xf64, #SparseVector> to memref<?xf64>
+    %1 = vector.transfer_read %0[%c0], %d0: memref<?xf64>, vector<32xf64>
+    vector.print %1 : vector<32xf64>
+    // Dump the dense vector to verify structure is correct.
+    %dv = sparse_tensor.convert %arg0 : tensor<?xf64, #SparseVector> to tensor<?xf64>
+    %2 = bufferization.to_memref %dv : memref<?xf64>
+    %3 = vector.transfer_read %2[%c0], %d0: memref<?xf64>, vector<32xf64>
+    vector.print %3 : vector<32xf64>
+    memref.dealloc %2 : memref<?xf64>
+    return
+  }
+
+  // Dumps a sparse vector of type i32.
+  func @dump_vec_i32(%arg0: tensor<?xi32, #SparseVector>) {
+    // Dump the values array to verify only sparse contents are stored.
+    %c0 = arith.constant 0 : index
+    %d0 = arith.constant -1 : i32
+    %0 = sparse_tensor.values %arg0 : tensor<?xi32, #SparseVector> to memref<?xi32>
+    %1 = vector.transfer_read %0[%c0], %d0: memref<?xi32>, vector<24xi32>
+    vector.print %1 : vector<24xi32>
+    // Dump the dense vector to verify structure is correct.
+    %dv = sparse_tensor.convert %arg0 : tensor<?xi32, #SparseVector> to tensor<?xi32>
+    %2 = bufferization.to_memref %dv : memref<?xi32>
+    %3 = vector.transfer_read %2[%c0], %d0: memref<?xi32>, vector<32xi32>
+    vector.print %3 : vector<32xi32>
+    memref.dealloc %2 : memref<?xi32>
+    return
+  }
+
+  // Dump a sparse matrix.
+  func @dump_mat(%arg0: tensor<?x?xf64, #DCSR>) {
+    %c0 = arith.constant 0 : index
+    %d0 = arith.constant -1.0 : f64
+    %0 = sparse_tensor.values %arg0 : tensor<?x?xf64, #DCSR> to memref<?xf64>
+    %1 = vector.transfer_read %0[%c0], %d0: memref<?xf64>, vector<16xf64>
+    vector.print %1 : vector<16xf64>
+    %dm = sparse_tensor.convert %arg0 : tensor<?x?xf64, #DCSR> to tensor<?x?xf64>
+    %2 = bufferization.to_memref %dm : memref<?x?xf64>
+    %3 = vector.transfer_read %2[%c0, %c0], %d0: memref<?x?xf64>, vector<4x8xf64>
+    vector.print %3 : vector<4x8xf64>
+    memref.dealloc %2 : memref<?x?xf64>
+    return
+  }
+
+  // Driver method to call and verify vector kernels.
+  func @entry() {
+    %c0 = arith.constant 0 : index
+
+    // Setup sparse vectors.
+    %v1 = arith.constant sparse<
+       [ [0], [3], [11], [17], [20], [21], [28], [29], [31] ],
+         [ 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0 ]
+    > : tensor<32xf64>
+    %sv1 = sparse_tensor.convert %v1 : tensor<32xf64> to tensor<?xf64, #SparseVector>
+
+    // 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>
+    %sm1 = sparse_tensor.convert %m1 : tensor<4x8xf64> to tensor<?x?xf64, #DCSR>
+
+    // Call sparse vector kernels.
+    %0 = call @vector_complement(%sv1)
+       : (tensor<?xf64, #SparseVector>) -> tensor<?xi32, #SparseVector>
+    %1 = call @vector_negation(%sv1)
+       : (tensor<?xf64, #SparseVector>) -> tensor<?xf64, #SparseVector>
+
+
+    // Call sparse matrix kernels.
+    %2 = call @matrix_clip(%sm1)
+      : (tensor<?x?xf64, #DCSR>) -> tensor<?x?xf64, #DCSR>
+
+    //
+    // Verify the results.
+    //
+    // CHECK:      ( 1, 2, 3, 4, 5, 6, 7, 8, 9, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 )
+    // CHECK-NEXT: ( 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 4, 0, 0, 5, 6, 0, 0, 0, 0, 0, 0, 7, 8, 0, 9 )
+    // CHECK-NEXT: ( 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1 )
+    // CHECK-NEXT: ( 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0 )
+    // CHECK-NEXT: ( -1, 1, 1, -2, 1, 1, 1, 1, 1, 1, 1, -3, 1, 1, 1, 1, 1, -4, 1, 1, -5, -6, 1, 1, 1, 1, 1, 1, -7, -8, 1, -9 )
+    // CHECK-NEXT: ( -1, 1, 1, -2, 1, 1, 1, 1, 1, 1, 1, -3, 1, 1, 1, 1, 1, -4, 1, 1, -5, -6, 1, 1, 1, 1, 1, 1, -7, -8, 1, -9 )
+    // CHECK-NEXT: ( 3, 3, 3, 4, 5, 6, 7, 7, 7, -1, -1, -1, -1, -1, -1, -1 )
+    // CHECK-NEXT: ( ( 3, 3, 0, 0, 0, 0, 0, 0 ), ( 0, 0, 0, 0, 0, 0, 0, 3 ), ( 0, 0, 4, 0, 5, 0, 0, 6 ), ( 7, 0, 7, 7, 0, 0, 0, 0 ) )
+    //
+    call @dump_vec_f64(%sv1) : (tensor<?xf64, #SparseVector>) -> ()
+    call @dump_vec_i32(%0) : (tensor<?xi32, #SparseVector>) -> ()
+    call @dump_vec_f64(%1) : (tensor<?xf64, #SparseVector>) -> ()
+    call @dump_mat(%2) : (tensor<?x?xf64, #DCSR>) -> ()
+
+    // Release the resources.
+    sparse_tensor.release %sv1 : tensor<?xf64, #SparseVector>
+    sparse_tensor.release %sm1 : tensor<?x?xf64, #DCSR>
+    sparse_tensor.release %0 : tensor<?xi32, #SparseVector>
+    sparse_tensor.release %1 : tensor<?xf64, #SparseVector>
+    sparse_tensor.release %2 : tensor<?x?xf64, #DCSR>
+    return
+  }
+}
 No newline at end of file

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

%0 = linalg.generic #trait_op
         outs(%xv: tensor<?xf64, #SparseVector>) {
         ^bb(%a: f64, %b: f64, %x: f64):
           %1 = arith.addf %a, %b : f64
           linalg.yield %1 : f64
     } -> tensor<?xf64, #SparseVector>
     return %0 : tensor<?xf64, #SparseVector>
   }
 
   // Multiplies two sparse vectors into a new sparse vector.
   func @vector_mul(%arga: tensor<?xf64, #SparseVector>,
                    %argb: tensor<?xf64, #SparseVector>) -> tensor<?xf64, #SparseVector> {
     %c = arith.constant 0 : index
     %d = tensor.dim %arga, %c : tensor<?xf64, #SparseVector>
     %xv = sparse_tensor.init [%d] : tensor<?xf64, #SparseVector>
     %0 = linalg.generic #trait_op
        ins(%arga, %argb: tensor<?xf64, #SparseVector>, tensor<?xf64, #SparseVector>)
         outs(%xv: tensor<?xf64, #SparseVector>) {
         ^bb(%a: f64, %b: f64, %x: f64):
           %1 = arith.mulf %a, %b : f64
%0 = linalg.generic #trait_op
           linalg.yield %1 : f64
     } -> tensor<?xf64, #DenseVector>
     return %0 : tensor<?xf64, #DenseVector>
   }
 
   // Sum reduces dot product of two sparse vectors.
   func @vector_dotprod(%arga: tensor<?xf64, #SparseVector>,
                        %argb: tensor<?xf64, #SparseVector>,
-           %argx: tensor<f64> {linalg.inplaceable = true}) -> tensor<f64> {
+                       %argx: tensor<f64> {linalg.inplaceable = true}) -> tensor<f64> {
     %0 = linalg.generic #trait_dot
        ins(%arga, %argb: tensor<?xf64, #SparseVector>, tensor<?xf64, #SparseVector>)
         outs(%argx: tensor<f64>) {
         ^bb(%a: f64, %b: f64, %x: f64):
           %1 = arith.mulf %a, %b : f64
-    %2 = arith.addf %x, %1 : f64
+          %2 = arith.addf %x, %1 : f64
           linalg.yield %2 : f64
     } -> tensor<f64>
     return %0 : tensor<f64>
   }
 
-  // Dumps a sparse vector.
+  // Dumps a sparse vector of type f64.
   func @dump(%arg0: tensor<?xf64, #SparseVector>) {
     // Dump the values array to verify only sparse contents are stored.
     %c0 = arith.constant 0 : index
     %d0 = arith.constant -1.0 : f64
     %0 = sparse_tensor.values %arg0 : tensor<?xf64, #SparseVector> to memref<?xf64>
     %1 = vector.transfer_read %0[%c0], %d0: memref<?xf64>, vector<16xf64>
     vector.print %1 : vector<16xf64>
     // Dump the dense vector to verify structure is correct.
     %dv = sparse_tensor.convert %arg0 : tensor<?xf64, #SparseVector> to tensor<?xf64>
     %2 = bufferization.to_memref %dv : memref<?xf64>
     %3 = vector.transfer_read %2[%c0], %d0: memref<?xf64>, vector<32xf64>
     vector.print %3 : vector<32xf64>
     memref.dealloc %2 : memref<?xf64>
     return
   }
 
   // Driver method to call and verify vector kernels.
   func @entry() {
     %c0 = arith.constant 0 : index
     %d1 = arith.constant 1.1 : f64
 
     // Setup sparse vectors.
     %v1 = arith.constant sparse<
        [ [0], [3], [11], [17], [20], [21], [28], [29], [31] ],
          [ 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0 ]
     > : tensor<32xf64>
     %v2 = arith.constant sparse<
        [ [1], [3], [4], [10], [16], [18], [21], [28], [29], [31] ],
          [11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0 ]
     > : tensor<32xf64>