An Efficient Fill Estimation Algorithm for Sparse Matrices and Tensors in Blocked Formats

Tensors, which are the linear-algebraic extensions of matrices in arbitrary dimensions, have numerous applications to data processing tasks in computer science and computational science. Many tensors used in diverse application domains are sparse, typically containing more than 90% zero entries. Efficient computation with sparse tensors hinges on algorithms that can leverage the sparsity to do less work, but the irregular locations of the nonzero entries pose significant challenges to performance engineers. Many tensor operations such as tensor-vector multiplications can be sped up substantially by breaking the tensor into equally sized blocks (only storing blocks which contain nonzeros) and performing operations in each block using carefully tuned code. However, selecting the best block size from among many possibilities is computationally challenging. Previously, Vuduc et al. defined the fill of a sparse tensor to be the number of stored entries in the blocked format divided by the number of nonzero entries, and showed how the fill can be used as part of an effective, efficient heuristic for evaluating the quality of a particular blocking scheme [1, 2]. In particular, they showed that if the fill could be computed exactly, then the measured performance of their sparse matrix-vector multiply was within 5% of the optimal setting. However, they gave no theoretical accuracy bounds for their method for estimating the fill, and it is vulnerable to several classes of adversarial examples. In this paper, we present a sampling-based method for finding a (1 + ε)-approximation to the fill of an order N tensor for all block sizes less than B, with probability at least 1 − δ, using O(NB log(B/δ)/ε) samples for each block size. We introduce an efficient routine to sample for all B block sizes at once in O(NB ) time. We extend our concentration bounds to a more efficient bound based on sampling without replacement, using the recent Hoeffding-Serfling inequality [3]. We then implement our algorithm and evaluate it on sparse matrices from the University of Florida collection and compare our scheme to that of Vuduc, as implemented in the Optimized Sparse Kernel Interface (OSKI) library, and a brute-force method for obtaining the ground truth. We find that our algorithm provides faster estimates of the fill at all accuracy levels, providing evidence that this is both a theoretical and practical improvement.