Sharp Bounds for Sums Associated to Graphs of Matrices

TL;DR

This paper introduces a straightforward algorithm to determine the optimal upper bounds for sums of matrix entry products constrained by graph-based index relations, facilitating analysis of complex matrix sums.

Contribution

The paper presents a novel, simple algorithm that computes sharp bounds for matrix sums with index constraints using graph representations.

Findings

Algorithm efficiently finds optimal bounds for matrix sums.

Bounds are derived from graph structures representing index constraints.

Method simplifies analysis of sums with complex index relations.

Abstract

We provide a simple algorithm for finding the optimal upper bound for sums of products of matrix entries of the form S_pi(N) := sum_{j_1, ..., j_2m = 1}^N t^1_{j_1 j_2} t^2_{j_3 j_4} ... t^m_{j_2m-1 j_2m} where some of the summation indices are constrained to be equal. The upper bound is easily obtained from a graph G associated to the constraints in the sum.