A Two-pronged Progress in Structured Dense Matrix Vector Multiplication

Abstract

Matrix-vector multiplication is one of the most fundamental computing primitives. Given a matrix A ∈ F N ×N and a vector b ∈ F N , it is known that in the worst case Θ(N 2 ) operations over F are needed to compute Ab. Many types of structured matrices do admit faster multiplication. However, even given a matrix A that is known to have this property, it is hard in general to recover a representation of A exposing the actual fast multiplication algorithm. Additionally, it is not known in general whether the inverses of such structured matrices can be computed or multiplied quickly. A broad question is thus to identify classes of structured dense matrices that can be represented with O(N ) parameters, and for which matrix-vector multiplication (and ideally other operations such as solvers) can be performed in a sub-quadratic number of operations.One such class of structured matrices that admit nearlinear matrix-vector multiplication are the orthogonal polynomial transforms whose rows correspond to a family of orthogonal polynomials. Other well known classes include the Toeplitz, Hankel, Vandermonde, Cauchy matrices and their extensions (e.g. confluent Cauchy-like matrices) that are all special cases of a low displacement rank property.In this paper, we make progress on two fronts: 1. We introduce the notion of recurrence width of matrices.For matrices A with constant recurrence width, we design algorithms to compute both Ab and A T b with a near-linear number of operations. This notion of width is finer than all the above classes of structured matrices and thus we can compute near-linear matrixvector multiplication for all of them using the same core algorithm. Furthermore, we show that it is possible to solve the harder problems of recovering the structured parameterization of a matrix with low recurrence width, and computing matrix-vector product with its inverse in near-linear time. 2. We additionally adapt our algorithm to a matrix-vector multiplication algorithm for a much more general class of matrices with displacement structure: those with low displacement rank with respect to quasiseparable matrices. This result is a novel connection between matrices with displacement structure and those with rank structure, two large but previously separate classes of structured matrices. This class includes Toeplitzplus-Hankel-like matrices, the Discrete Trigonometric Transforms, and more, and captures all previously known matrices with displacement structure under a unified parameterization and algorithm.Our work unifies, generalizes, and simplifies existing stateof-the-art results in structured matrix-vector multiplication. Finally, we show how applications in areas such as multipoint evaluations of multivariate polynomials can be reduced to problems involving low recurrence width matrices.

On Computing the Resultant of Generic Bivariate Polynomials

“…This shows that for a generic Toeplitz matrix T 0 , a modification of the algorithm of Section 6 using a ΣLU representation of T −1 = (xI n − T 0 ) −1 will compute the characteristic polynomial within the complexity bound of Theorem 1.1 with d = 1. One may expect to have analogous results for more general classes of structured matrices [6,14].…”

Learning Fast Algorithms for Linear Transforms Using Butterfly Factorizations

“…Another class of fast transforms that directly generalize the DFT and DCT are based on orthogonal polynomials [7], which find usage in areas from differential equations to optics. Both orthogonal polynomial transforms [12], and all of the previously introduced matrices with displacement rank structure, were significantly generalized under a single class by De Sa et al [8]. Notably, almost all of the structured matrix classes mentioned here exhibit a form of recursive structure in their construction and superfast algorithms.…”

“…Surprisingly, the converse is also true. The notion of sparse product width (SPW) [8], which roughly corresponds to the total sparsity of a factorization of a matrix, turns out to be equivalent to the length of the shortest straight-line program describing a matrix (up to a constant). Hence it is an optimal descriptor of the algorithmic complexity of matrix-vector multiplication on these types of models [3].…”

“…Following previous experimental settings [5,38,40], we compare our proposed classes to several baselines using dense structured matrices to compress the hidden layer of a single hidden layer neural network. Competing methods include simple low-rank factorizations [9], circulant matrices (equivalent to 1-dimensional convolutions) [6], the adaptive Fastfood transform [45], and low displacement rank methods [38,40] which implicitly define a structured matrix through a displacement equation and admit specialized fast divide-and-conquer algorithms [8]. Our implementation is built on top of the publicly available implementation of Thomas et al [40] with the same hyperparameters, and we report their numbers for the competing baseline methods directly.…”

“…In particular, these approaches search through an exponentially large discrete space using a symbolic form of the matrix [13,14] and recover the solution only up to dimensions 8 × 8. We instead draw two key lessons from the work of De Sa et al [8], who characterize matrices with efficient matrix-vector multiplication algorithms as being factorizable into products of sparse matrices. 1 Thus, the task of learning algorithms can be reduced to finding appropriate sparse matrix product representations of the transforms.…”