DiffRed: Dimensionality Reduction guided by stable rank

TL;DR

DiffRed is a new dimensionality reduction method that combines principal component projection with Gaussian random vectors, providing tighter theoretical bounds and superior empirical performance over existing techniques.

Contribution

It introduces DiffRed, a novel approach that leverages stable rank and combines PCA with random projections, with proven bounds and improved results.

Findings

Achieves near-zero M1 distortion in experiments

Provides tighter theoretical bounds than existing methods

Reduces 6 million dimensional data to 10 dimensions with 54% lower Stress than PCA

Abstract

In this work, we propose a novel dimensionality reduction technique, DiffRed, which first projects the data matrix, A, along first $k_1$ principal components and the residual matrix $A^{*}$ (left after subtracting its $k_1$-rank approximation) along $k_2$ Gaussian random vectors. We evaluate M1, the distortion of mean-squared pair-wise distance, and Stress, the normalized value of RMS of distortion of the pairwise distances. We rigorously prove that DiffRed achieves a general upper bound of $O\left(\sqrt{\frac{1-p}{k_2}}\right)$ on Stress and $O\left(\frac{(1-p)}{\sqrt{k_2*\rho(A^{*})}}\right)$ on M1 where $p$ is the fraction of variance explained by the first $k_1$ principal components and $\rho(A^{*})$ is the stable rank of $A^{*}$. These bounds are tighter than the currently known results for Random maps. Our extensive experiments on a variety of real-world datasets demonstrate that DiffRed achieves near zero M1 and much lower values of Stress as compared to the well-known dimensionality reduction techniques. In particular, DiffRed can map a 6 million dimensional dataset to 10 dimensions with 54% lower Stress than PCA.