Approximation of Gram-Schmidt Orthogonalization by Data Matrix

TL;DR

This paper investigates how well normalized data matrices can approximate the Gram-Schmidt orthogonalization process, providing theoretical insights and practical implications for high-dimensional data analysis.

Contribution

It offers a theoretical analysis of approximation errors when using normalized matrices to estimate orthogonalization, explaining phenomena in high-dimensional Gaussian matrices.

Findings

Normalized matrices can effectively approximate orthogonalization in high dimensions

Theoretical error bounds are derived for approximation accuracy

High-dimensional Gaussian matrices closely resemble truncated Haar matrices

Abstract

For a matrix ${\bf A}$ with linearly independent columns, this work studies to use its normalization $\bar{\bf A}$ and ${\bf A}$ itself to approximate its orthonormalization $\bf V$. We theoretically analyze the order of the approximation errors as $\bf A$ and $\bar{\bf A}$ approach ${\bf V}$, respectively. Our conclusion is able to explain the fact that a high dimensional Gaussian matrix can well approximate the corresponding truncated Haar matrix. For applications, this work can serve as a foundation of a wide variety of problems in signal processing such as compressed subspace clustering.