diff --git a/mlir/include/mlir/IR/AffineMap.h b/mlir/include/mlir/IR/AffineMap.h
--- a/mlir/include/mlir/IR/AffineMap.h
+++ b/mlir/include/mlir/IR/AffineMap.h
@@ -224,9 +224,9 @@
 
 /// Returns a map of codomain to domain dimensions such that the first codomain
 /// dimension for a particular domain dimension is selected.
-/// Returns an empty map if the input map is empty or if `map` is not invertible
-/// (i.e. `map` does not contain a subset that is a permutation of full domain
-/// rank).
+/// Returns an empty map if the input map is empty.
+/// Returns null map (not empty map) if `map` is not invertible (i.e. `map` does
+/// not contain a subset that is a permutation of full domain rank).
 ///
 /// Prerequisites:
 ///   1. `map` has no symbols.
diff --git a/mlir/lib/Dialect/Linalg/Transforms/Fusion.cpp b/mlir/lib/Dialect/Linalg/Transforms/Fusion.cpp
--- a/mlir/lib/Dialect/Linalg/Transforms/Fusion.cpp
+++ b/mlir/lib/Dialect/Linalg/Transforms/Fusion.cpp
@@ -416,6 +416,8 @@
   AffineMap consumerIndexMap = consumerOp.getIndexingMap(consumerIdx);
   AffineMap invProducerResultIndexMap =
       inversePermutation(producerOp.getOutputIndexingMap(0));
+  if (!invProducerResultIndexMap)
+    return {};
 
   // Compute the fused op operandslist by replacing the operand corresponding to
   // the result of the producer, with the operands of the producer.
@@ -559,6 +561,9 @@
       if (!fusedOp)
         continue;
       rewriter.replaceOp(op, fusedOp.getValue().getOperation()->getResults());
+      if (llvm::all_of(definingOp.getResults(),
+                       [](Value val) -> bool { return val.use_empty(); }))
+        rewriter.eraseOp(definingOp);
       return success();
     }
     return failure();
diff --git a/mlir/lib/Dialect/Linalg/Transforms/LinalgToLoops.cpp b/mlir/lib/Dialect/Linalg/Transforms/LinalgToLoops.cpp
--- a/mlir/lib/Dialect/Linalg/Transforms/LinalgToLoops.cpp
+++ b/mlir/lib/Dialect/Linalg/Transforms/LinalgToLoops.cpp
@@ -652,6 +652,8 @@
   auto maps = llvm::to_vector<8>(
       llvm::map_range(mapsRange, [](AffineMapAttr a) { return a.getValue(); }));
   AffineMap invertedMap = inversePermutation(concatAffineMaps(maps));
+  if (!invertedMap)
+    return {};
   if (invertedMap.isEmpty()) {
     LinalgScopedEmitter<IndexedValueTy, ConcreteOpTy>::emitScalarImplementation(
         {}, linalgOp);
diff --git a/mlir/lib/Dialect/Linalg/Transforms/LinalgTransforms.cpp b/mlir/lib/Dialect/Linalg/Transforms/LinalgTransforms.cpp
--- a/mlir/lib/Dialect/Linalg/Transforms/LinalgTransforms.cpp
+++ b/mlir/lib/Dialect/Linalg/Transforms/LinalgTransforms.cpp
@@ -291,11 +291,13 @@
   auto linOp = cast<LinalgOp>(op);
   auto permutationMap = inversePermutation(
       AffineMap::getPermutationMap(permutation, rewriter.getContext()));
+  assert(permutationMap && "expected permutation to be invertible");
   SmallVector<AffineMap, 4> newIndexingMap;
   auto indexingMaps = linOp.indexing_maps().getValue();
   for (unsigned i = 0, e = linOp.getNumInputsAndOutputs(); i != e; ++i) {
-    AffineMap m = indexingMaps[i].cast<AffineMapAttr>().getValue().compose(
-        permutationMap);
+    AffineMap m = indexingMaps[i].cast<AffineMapAttr>().getValue();
+    if (!permutationMap.isEmpty())
+      m = m.compose(permutationMap);
     newIndexingMap.push_back(m);
   }
   auto itTypes = linOp.iterator_types().getValue();
diff --git a/mlir/lib/Dialect/Linalg/Transforms/Tiling.cpp b/mlir/lib/Dialect/Linalg/Transforms/Tiling.cpp
--- a/mlir/lib/Dialect/Linalg/Transforms/Tiling.cpp
+++ b/mlir/lib/Dialect/Linalg/Transforms/Tiling.cpp
@@ -349,6 +349,8 @@
   if (!permutation.empty())
     invPermutationMap = inversePermutation(
         AffineMap::getPermutationMap(permutation, ScopedContext::getContext()));
+  if (!invPermutationMap)
+    return llvm::None;
 
   OpBuilder::InsertionGuard g(b);
   b.setInsertionPoint(op);
@@ -361,7 +363,8 @@
   auto maps = llvm::to_vector<8>(
       llvm::map_range(mapsRange, [](AffineMapAttr a) { return a.getValue(); }));
   auto viewSizesToLoopsMap = inversePermutation(concatAffineMaps(maps));
-  assert(viewSizesToLoopsMap && "expected invertible map");
+  if (!viewSizesToLoopsMap)
+    return llvm::None;
 
   SmallVector<SubViewOp::Range, 4> loopRanges;
   LoopIndexToRangeIndexMap loopIndexToRangeIndex;
diff --git a/mlir/test/Dialect/Linalg/fusion-tensor.mlir b/mlir/test/Dialect/Linalg/fusion-tensor.mlir
--- a/mlir/test/Dialect/Linalg/fusion-tensor.mlir
+++ b/mlir/test/Dialect/Linalg/fusion-tensor.mlir
@@ -83,7 +83,6 @@
 
 // CHECK-DAG: [[MAP0:#[a-zA-Z0-9_]*]] = affine_map<(d0, d1) -> (d0, d1)>
 // CHECK-DAG: [[MAP1:#[a-zA-Z0-9_]*]] = affine_map<(d0, d1) -> (d0)>
-// CHECK-DAG: [[MAP2:#[a-zA-Z0-9_]*]] = affine_map<(d0) -> (d0)>
 #map0 = affine_map<(d0, d1) -> (d0, d1)>
 #map1 = affine_map<(d0, d1) -> (d0)>
 #map2 = affine_map<(d0) -> (d0)>
@@ -119,7 +118,7 @@
       %1 = addf %arg3, %arg4 : f32
       linalg.yield %1 : f32
   }: tensor<f32>, tensor<f32> -> tensor<f32>
-  // CHECK: linalg.generic {args_in = 2 : i64, args_out = 1 : i64
+  // CHECK: linalg.generic {args_in = 3 : i64, args_out = 1 : i64
   // CHECK: addf
   // CHECK: mulf
   %1 = linalg.generic {args_in = 2 : i64, args_out = 1 : i64, indexing_maps = [#map0, #map0, #map0], iterator_types = []} %0, %arg2 {