Precise expressions for random projections: Low-rank approximation and randomized Newton

TL;DR

This paper derives precise spectral expressions for the expected outcomes of random projection methods in data reduction, improving understanding of their performance in machine learning tasks.

Contribution

It introduces novel spectral analysis techniques that accurately predict the expected behavior of various sketching methods, bridging theory and practical performance.

Findings

Expressions match empirical results closely

Applicable to Gaussian and Rademacher sketches

Enhances understanding of sketching in machine learning

Abstract

It is often desirable to reduce the dimensionality of a large dataset by projecting it onto a low-dimensional subspace. Matrix sketching has emerged as a powerful technique for performing such dimensionality reduction very efficiently. Even though there is an extensive literature on the worst-case performance of sketching, existing guarantees are typically very different from what is observed in practice. We exploit recent developments in the spectral analysis of random matrices to develop novel techniques that provide provably accurate expressions for the expected value of random projection matrices obtained via sketching. These expressions can be used to characterize the performance of dimensionality reduction in a variety of common machine learning tasks, ranging from low-rank approximation to iterative stochastic optimization. Our results apply to several popular sketching methods, including Gaussian and Rademacher sketches, and they enable precise analysis of these methods in terms of spectral properties of the data. Empirical results show that the expressions we derive reflect the practical performance of these sketching methods, down to lower-order effects and even constant factors.