Symmetric and sparse tensors arise naturally in many domains including linear algebra, statistics, physics, chemistry, and graph theory. Symmetric tensors are equal to their transposes, so in the n-dimensional case we can save up to a factor of n! by avoiding redundant operations. Sparse tensors, on the other hand, are mostly zero, and we can save asymptotically by processing only nonzeros. Unfortunately, specializing for both symmetry and sparsity at the same time is uniquely challenging. Optimizing for symmetry requires consideration of n! transpositions of a triangular kernel, which can be complex and error prone. Considering multiple transposed iteration orders and triangular loop bounds also complicates iteration through intricate sparse tensor formats. Additionally, since each combination of symmetry and sparse tensor formats requires a specialized implementation, this leads to a combinatorial number of cases. A compiler is needed, but existing compilers cannot take advantage of both symmetry and sparsity within the same kernel. In this paper, we describe the first compiler which can automatically generate symmetry-aware code for sparse or structured tensor kernels. We introduce a taxonomy for symmetry in tensor kernels, and show how to target each kind of symmetry. Our implementation demonstrates significant speedups ranging from 1.36x for SSYMV to 30.4x for a 5-dimensional MTTKRP over the non-symmetric state of the art.

From FORTRAN to NumPy, tensors have revolutionized how we express computation. However, tensors in these, and almost all prominent systems, can only handle dense rectilinear integer grids. Real world tensors often contain underlying structure, such as sparsity, runs of repeated values, or symmetry. Support for structured data is fragmented and incomplete. Existing frameworks limit the tensor structures and program control flow they support to better simplify the problem. In this work, we propose a new programming language, Finch, which supports both flexible control flow and diverse data structures. Finch facilitates a programming model which resolves the challenges of computing over structured tensors by combining control flow and data structures into a common representation where they can be co-optimized. Finch automatically specializes control flow to data so that performance engineers can focus on experimenting with many algorithms. Finch supports a familiar programming language of loops, statements, ifs, breaks, etc., over a wide variety of tensor structures, such as sparsity, run-length-encoding, symmetry, triangles, padding, or blocks. Finch reliably utilizes the key properties of structure, such as structural zeros, repeated values, or clustered non-zeros. We show that this leads to dramatic speedups in operations such as SpMV and SpGEMM, image processing, and graph analytics.