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# Almost-Orthogonal Bases for Inner Product Polynomials

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LIPIcs.ITCS.2022.89.pdf
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## Acknowledgements

We would like to thank Goutham Rajendran for discussions and many comments on this work. We also thank Mrinalkanti Ghosh, Fernando Granha Jeronimo, and Madhur Tulsiani for early discussions on the polynomials in the context of Sum-of-Squares.

## Cite As

Chris Jones and Aaron Potechin. Almost-Orthogonal Bases for Inner Product Polynomials. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 89:1-89:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ITCS.2022.89

## 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.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Randomness, geometry and discrete structures
##### Keywords
• Orthogonal polynomials
• Fourier analysis
• combinatorics

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