Optimal splitting of Parseval frames using Walsh matrices

TL;DR

This paper explores the use of Walsh matrices to construct natural frames in Euclidean space and examines their relation to the Kadison-Singer discrepancy theorem, building on prior probabilistic methods.

Contribution

It introduces a novel approach using Walsh matrices for frame construction and analyzes their connection to the discrepancy theorem, offering a deterministic alternative.

Findings

Walsh matrices can be used to construct frames in Euclidean space.

The constructed frames relate to the discrepancy theorem.

Potential for deterministic methods in frame analysis.

Abstract

In 2014 Adam Marcus, Daniel Spielman and Nikhil Srivastava used random vectors to prove a key discrepancy theorem and in so doing gave a positive answer to the long-standing Kadison-Singer Problem. In this paper we use Walsh matrices to construct a class of natural frames in Euclidean space and discuss how these frames relate to the key discrepancy theorem.