On Symmetric Factorizations of Hankel Matrices

TL;DR

This paper explores conjectures on faster algorithms for symmetric factorizations of Hankel matrices and their inverses, which could improve solving certain optimization problems, and discusses related sum-of-squares polynomial decompositions.

Contribution

It proposes new conjectures on the running time of symmetric factorizations for Hankel matrices and their inverses, with partial results supporting these conjectures.

Findings

Weaker results on decompositions of the form B B* - C C* for Hankel matrices.

Connections established between Hankel matrix factorizations and sum-of-squares polynomial decompositions.

Discussion of potential implications for faster algorithms in optimization.

Abstract

We present two conjectures regarding the running time of computing symmetric factorizations for a Hankel matrix $\mathbf{H}$ and its inverse $\mathbf{H}^{-1}$ as $\mathbf{B}\mathbf{B}^*$ under fixed-point arithmetic. If solved, these would result in a faster-than-matrix-multiplication algorithm for solving sparse poly-conditioned linear programming problems, a fundamental problem in optimization and theoretical computer science. To justify our proposed conjectures and running times, we show weaker results of computing decompositions of the form $\mathbf{B}\mathbf{B}^* - \mathbf{C}\mathbf{C}^*$ for Hankel matrices and their inverses with the same running time. In addition, to promote our conjectures further, we discuss the connections of Hankel matrices and their symmetric factorizations to sum-of-squares (SoS) decompositions of single-variable polynomials.