diff --git a/mlir/lib/Dialect/Linalg/IR/LinalgInterfaces.cpp b/mlir/lib/Dialect/Linalg/IR/LinalgInterfaces.cpp
--- a/mlir/lib/Dialect/Linalg/IR/LinalgInterfaces.cpp
+++ b/mlir/lib/Dialect/Linalg/IR/LinalgInterfaces.cpp
@@ -36,6 +36,11 @@
       continue;
     indexingMaps.push_back(linalgOp.getMatchingIndexingMap(&opOperand));
   }
+  if (indexingMaps.empty()) {
+    // If there are no indexing maps, the operand can only be dropped
+    // if the op has no loops.
+    return linalgOp.getNumLoops() == 0;
+  }
   return inversePermutation(concatAffineMaps(indexingMaps)) != AffineMap();
 }
 
diff --git a/mlir/lib/Dialect/Linalg/Transforms/ElementwiseOpFusion.cpp b/mlir/lib/Dialect/Linalg/Transforms/ElementwiseOpFusion.cpp
--- a/mlir/lib/Dialect/Linalg/Transforms/ElementwiseOpFusion.cpp
+++ b/mlir/lib/Dialect/Linalg/Transforms/ElementwiseOpFusion.cpp
@@ -143,10 +143,10 @@
 
 /// Generate the region of the fused tensor operation. The region of the fused
 /// op must be empty.
-static void
-generateFusedElementwiseOpRegion(RewriterBase &rewriter, GenericOp fusedOp,
-                                 AffineMap consumerToProducerLoopsMap,
-                                 OpOperand *fusedOperand, unsigned nloops) {
+static void generateFusedElementwiseOpRegion(
+    RewriterBase &rewriter, GenericOp fusedOp,
+    AffineMap consumerToProducerLoopsMap, OpOperand *fusedOperand,
+    unsigned nloops, llvm::SmallDenseSet<int> &preservedProducerResults) {
   auto producer = cast<GenericOp>(fusedOperand->get().getDefiningOp());
   auto consumer = cast<GenericOp>(fusedOperand->getOwner());
   // Build the region of the fused op.
@@ -202,9 +202,13 @@
     mapper.map(bbArg, fusedBlock->addArgument(bbArg.getType(), bbArg.getLoc()));
 
   // 6. All of the producer's output operands
-  for (BlockArgument bbArg :
-       producerBlock.getArguments().take_back(producer.getNumDpsInits()))
-    mapper.map(bbArg, fusedBlock->addArgument(bbArg.getType(), bbArg.getLoc()));
+  for (auto bbArg : llvm::enumerate(
+           producerBlock.getArguments().take_back(producer.getNumDpsInits()))) {
+    if (!preservedProducerResults.count(bbArg.index()))
+      continue;
+    mapper.map(bbArg.value(), fusedBlock->addArgument(bbArg.value().getType(),
+                                                      bbArg.value().getLoc()));
+  }
 
   // 7. All of consumer's output operands.
   for (BlockArgument bbArg :
@@ -247,8 +251,11 @@
   SmallVector<Value> fusedYieldValues;
   fusedYieldValues.reserve(producerYieldOp.getNumOperands() +
                            consumerYieldOp.getNumOperands());
-  for (auto producerYieldVal : producerYieldOp.getOperands())
-    fusedYieldValues.push_back(mapper.lookupOrDefault(producerYieldVal));
+  for (auto producerYieldVal : llvm::enumerate(producerYieldOp.getOperands())) {
+    if (preservedProducerResults.count(producerYieldVal.index()))
+      fusedYieldValues.push_back(
+          mapper.lookupOrDefault(producerYieldVal.value()));
+  }
   for (auto consumerYieldVal : consumerYieldOp.getOperands())
     fusedYieldValues.push_back(mapper.lookupOrDefault(consumerYieldVal));
   rewriter.create<YieldOp>(fusedOp.getLoc(), fusedYieldValues);
@@ -269,6 +276,18 @@
   // TODO: allow fusing the producer of an output operand.
   assert(consumer.isDpsInput(fusedOperand) &&
          "expected producer of input operand");
+  /// Find the results of the producer that have uses outside of the consumer.
+  llvm::SmallDenseSet<int> preservedProducerResults;
+  for (auto producerResult : llvm::enumerate(producer->getResults())) {
+    auto outputOperand = producer.getDpsInitOperand(producerResult.index());
+    if (producer.payloadUsesValueFromOperand(outputOperand) ||
+        !producer.canOpOperandsBeDropped(outputOperand) ||
+        llvm::any_of(producerResult.value().getUsers(), [&](Operation *user) {
+          return user != consumer.getOperation();
+        })) {
+      preservedProducerResults.insert(producerResult.index());
+    }
+  }
 
   // Compute the fused operands list and indexing maps.
   SmallVector<Value> fusedInputOperands, fusedOutputOperands;
@@ -276,9 +295,9 @@
   SmallVector<AffineMap> fusedIndexMaps;
   fusedInputOperands.reserve(producer.getNumDpsInputs() +
                              consumer.getNumDpsInputs());
-  fusedOutputOperands.reserve(producer.getNumDpsInits() +
+  fusedOutputOperands.reserve(preservedProducerResults.size() +
                               consumer.getNumDpsInits());
-  fusedResultTypes.reserve(producer.getNumDpsInits() +
+  fusedResultTypes.reserve(preservedProducerResults.size() +
                            consumer.getNumDpsInits());
   fusedIndexMaps.reserve(producer->getNumOperands() +
                          consumer->getNumOperands());
@@ -313,13 +332,16 @@
   }
 
   // 6. Collect all of the producer outputs.
-  for (OpOperand *opOperand : producer.getDpsInitOperands()) {
-    fusedOutputOperands.push_back(opOperand->get());
+  for (auto opOperand : llvm::enumerate(producer.getDpsInitOperands())) {
+    if (!preservedProducerResults.count(opOperand.index()))
+      continue;
+
+    fusedOutputOperands.push_back(opOperand.value()->get());
     AffineMap map = getIndexingMapOfProducerOperandsInCoordinatesOfFusedOp(
-        opOperand, producerResultIndexMap,
+        opOperand.value(), producerResultIndexMap,
         consumer.getMatchingIndexingMap(fusedOperand));
     fusedIndexMaps.push_back(map);
-    fusedResultTypes.push_back(opOperand->get().getType());
+    fusedResultTypes.push_back(opOperand.value()->get().getType());
   }
 
   // 7. All of consumer's output operands (skip operands: added by the builder).
@@ -358,9 +380,9 @@
   AffineMap consumerToProducerLoopsMap =
       invProducerResultIndexMap.compose(consumerResultIndexMap);
 
-  generateFusedElementwiseOpRegion(rewriter, fusedOp,
-                                   consumerToProducerLoopsMap, fusedOperand,
-                                   consumer.getNumLoops());
+  generateFusedElementwiseOpRegion(
+      rewriter, fusedOp, consumerToProducerLoopsMap, fusedOperand,
+      consumer.getNumLoops(), preservedProducerResults);
   return fusedOp.getOperation();
 }