Almost-Orthogonal Bases for Inner Product Polynomials

TL;DR

This paper introduces an almost orthogonal basis for low-degree inner product polynomials across Gaussian, spherical, and Boolean vectors, with a combinatorial graph-based description and analysis of polynomial expectations.

Contribution

It provides a novel basis for these polynomial spaces with a combinatorial topological interpretation and analyzes their expected products, revealing non-negativity in Gaussian and Boolean cases.

Findings

Basis admits a combinatorial description based on graph topology.

Expected product values are non-negative in Gaussian and Boolean cases.

Conjecture: planarity of the graph implies non-negativity in the spherical case.

Abstract

In this paper, we consider low-degree polynomials of inner products between a collection of random vectors. We give an almost orthogonal basis for this vector space of polynomials when the random vectors are Gaussian, spherical, or Boolean. In all three cases, our basis admits an interesting combinatorial description based on the topology of the underlying graph of inner products. We also analyze the expected value of the product of two polynomials in our basis. In all three cases, we show that this expected value can be expressed in terms of collections of matchings on the underlying graph of inner products. In the Gaussian and Boolean cases, we show that this expected value is always non-negative. In the spherical case, we show that this expected value can be negative but we conjecture that if the underlying graph of inner products is planar then this expected value will always be non-negative.