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DiffRed: Dimensionality Reduction guided by stable rank

Prarabdh Shukla, Gagan Raj Gupta, Kunal Dutta

TL;DR

DiffRed integrates PCA-based subspace preservation with Gaussian random projections applied to the residual, guided by stable rank, to achieve tighter distortion bounds than traditional random maps. The method provides theoretical guarantees: Stress $=O\left(\sqrt{\frac{1-p}{k_2}}\right)$ and $\Lambda_{M1}=O\left(\frac{(1-p)}{\sqrt{k_2\rho(A^*)}}\right)$, with Monte Carlo iterations further reducing failure probabilities. Empirically, DiffRed consistently outperforms PCA, Random Maps, and related approaches across diverse real-world datasets, achieving near-zero M1 and substantially lower Stress, including a notable $6{\times}10^6$-to-$10$ reduction in dimensionality with large Stress gains. The results underscore the practical value of incorporating stable rank into dimensionality reduction, enabling smaller target dimensions while preserving global structure and enabling downstream tasks like visualization and similarity search.

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 $O\left(\frac{(1-p)}{\sqrt{k_2*ρ(A^{*})}}\right)$ on M1 where $p$ is the fraction of variance explained by the first $k_1$ principal components and $ρ(A^{*})$ 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.

DiffRed: Dimensionality Reduction guided by stable rank

TL;DR

DiffRed integrates PCA-based subspace preservation with Gaussian random projections applied to the residual, guided by stable rank, to achieve tighter distortion bounds than traditional random maps. The method provides theoretical guarantees: Stress and , with Monte Carlo iterations further reducing failure probabilities. Empirically, DiffRed consistently outperforms PCA, Random Maps, and related approaches across diverse real-world datasets, achieving near-zero M1 and substantially lower Stress, including a notable -to- reduction in dimensionality with large Stress gains. The results underscore the practical value of incorporating stable rank into dimensionality reduction, enabling smaller target dimensions while preserving global structure and enabling downstream tasks like visualization and similarity search.

Abstract

In this work, we propose a novel dimensionality reduction technique, DiffRed, which first projects the data matrix, A, along first principal components and the residual matrix (left after subtracting its -rank approximation) along 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 on Stress and on M1 where is the fraction of variance explained by the first principal components and is the stable rank of . 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.
Paper Structure (44 sections, 9 theorems, 36 equations, 16 figures, 21 tables, 1 algorithm)

This paper contains 44 sections, 9 theorems, 36 equations, 16 figures, 21 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Formulation
  4. DiffRed Algorithm and Its Analysis
  5. Complexity Analysis
  6. Experiments
  7. Datasets
  8. Experimental Setup
  9. Experiment Results
  10. Insights on M1
  11. Observation 1
  12. Observation 2
  13. Observation 3
  14. Insights on Stress
  15. Observation 1
  16. ...and 29 more sections

Key Result

Lemma 1

There exists a constant $c_1>0$, such that given a random matrix $G$ as defined in the DiffRed Algorithm DiffRed-algo and a data matrix $A$, for all $d \leq D$ and all ${\varepsilon} \in [0,1]$

Figures (16)

  • Figure 1: Stable rank as a measure of "spread" in data in a 3-D example.
  • Figure 2: DiffRed algorithm maps vector $\mathbf{x} \in \mathbb{R}^D$ to $\mathbf{\tilde{x}} \in \mathbb{R}^{k_1+k_2}$ while preserving its component $\mathbf{z}$ in the best-fit-subspace $S_{k_1}$. $\mathbf{r}$ and $\mathbf{y}$ are orthogonal to $\mathbf{z}$.
  • Figure 3: Dependence of M1 for Random Maps on stable rank described in Corollary \ref{['cor:RMapM1Bound']}.($d=10$) [gRS: geneRNASeq, R30k: Reuters30k]
  • Figure 4: Variation of M1 with target dimension for Reuters30k
  • Figure 5: The exponent term $\frac{k_2\rho(A^{*})}{(1-p)^2}$ remains high for different values of $k_1$. ($d=10$)
  • ...and 11 more figures

Theorems & Definitions (20)

  • Definition 1: Data Matrix
  • Definition 2: Embedding Matrix
  • Definition 3: Stable Rank
  • Definition 4: M1 Distortion
  • Definition 5: Stress
  • Definition 6: p
  • Lemma 1
  • Theorem 2: M1 Distortion Bound
  • Corollary 3: M1 bound for RMap
  • Corollary 4: M1 Distortion, Monte Carlo Version
  • ...and 10 more