Improved analysis of the subsampled randomized Hadamard transform

TL;DR

This paper provides a simplified and optimal analysis of the subsampled randomized Hadamard transform, demonstrating its effectiveness in preserving Euclidean geometry of subspaces with improved dimension bounds.

Contribution

It introduces a simpler proof that achieves optimal constants for the embedding dimension in the analysis of the subsampled randomized Hadamard transform.

Findings

Simplified proof of the transform's properties

Optimal constants in embedding dimension estimates

Enhanced understanding of the transform's geometric preservation

Abstract

This paper presents an improved analysis of a structured dimension-reduction map called the subsampled randomized Hadamard transform. This argument demonstrates that the map preserves the Euclidean geometry of an entire subspace of vectors. The new proof is much simpler than previous approaches, and it offers---for the first time---optimal constants in the estimate on the number of dimensions required for the embedding.