Algorithms for structured matrix-vector product of optimal bilinear complexity

TL;DR

This paper introduces explicit algorithms for efficiently computing structured matrix-vector products across various structures, achieving optimal multiplication counts as per Strassen's theoretical minimum, applicable to both simple and complex nested structures.

Contribution

It provides the first explicit algorithms that are optimal in the sense of Strassen for a wide range of structured matrices, including complex nested structures.

Findings

Algorithms achieve minimal multiplications for structured matrices.

Applicable to Toeplitz, Hankel, circulant, and nested structures.

Extends optimal algorithms to complex multilevel structures.

Abstract

We present explicit algorithms for computing structured matrix-vector products that are optimal in the sense of Strassen, i.e., using a provably minimum number of multiplications. These structures include Toeplitz/Hankel/circulant, symmetric, Toeplitz-plus-Hankel, sparse, and multilevel structures. The last category include \textsc{bttb}, \textsc{bhhb}, \textsc{bccb} but also any arbitrarily complicated nested structures built out of other structures.