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A Probabilistic Basis for Low-Rank Matrix Learning

Simon Segert, Nathan Wycoff

TL;DR

This work provides a rigorous probabilistic foundation for low-rank matrix learning by analyzing the nuclear-norm distribution with density $f(X)\propto e^{-\lambda\|X\|_*}$. It derives the exact normalizing constant, an exact SVD-based stochastic representation, and a tractable approximate surrogate via the Normal Product Distribution, then leverages these results to build efficient proximal-Langevin MCMC and a Gibbs-type sampler for Gaussian likelihoods. The authors also develop a Bayesian scheme to infer the penalty $\lambda$ without grid searches, and demonstrate through matrix denoising and completion experiments that adaptive $\lambda$ attains performance comparable to optimal fixed values. Collectively, the work advances Bayesian low-rank inference by linking fundamental distributions to practical Monte Carlo methods and automatic hyperparameter learning, with implications for denoising, completion, and beyond.

Abstract

Low rank inference on matrices is widely conducted by optimizing a cost function augmented with a penalty proportional to the nuclear norm $\Vert \cdot \Vert_*$. However, despite the assortment of computational methods for such problems, there is a surprising lack of understanding of the underlying probability distributions being referred to. In this article, we study the distribution with density $f(X)\propto e^{-λ\Vert X\Vert_*}$, finding many of its fundamental attributes to be analytically tractable via differential geometry. We use these facts to design an improved MCMC algorithm for low rank Bayesian inference as well as to learn the penalty parameter $λ$, obviating the need for hyperparameter tuning when this is difficult or impossible. Finally, we deploy these to improve the accuracy and efficiency of low rank Bayesian matrix denoising and completion algorithms in numerical experiments.

A Probabilistic Basis for Low-Rank Matrix Learning

TL;DR

This work provides a rigorous probabilistic foundation for low-rank matrix learning by analyzing the nuclear-norm distribution with density . It derives the exact normalizing constant, an exact SVD-based stochastic representation, and a tractable approximate surrogate via the Normal Product Distribution, then leverages these results to build efficient proximal-Langevin MCMC and a Gibbs-type sampler for Gaussian likelihoods. The authors also develop a Bayesian scheme to infer the penalty without grid searches, and demonstrate through matrix denoising and completion experiments that adaptive attains performance comparable to optimal fixed values. Collectively, the work advances Bayesian low-rank inference by linking fundamental distributions to practical Monte Carlo methods and automatic hyperparameter learning, with implications for denoising, completion, and beyond.

Abstract

Low rank inference on matrices is widely conducted by optimizing a cost function augmented with a penalty proportional to the nuclear norm . However, despite the assortment of computational methods for such problems, there is a surprising lack of understanding of the underlying probability distributions being referred to. In this article, we study the distribution with density , finding many of its fundamental attributes to be analytically tractable via differential geometry. We use these facts to design an improved MCMC algorithm for low rank Bayesian inference as well as to learn the penalty parameter , obviating the need for hyperparameter tuning when this is difficult or impossible. Finally, we deploy these to improve the accuracy and efficiency of low rank Bayesian matrix denoising and completion algorithms in numerical experiments.

Paper Structure

This paper contains 38 sections, 6 theorems, 86 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Theoretical Analysis
  4. Basic Properties
  5. Normalizing Constant
  6. Exact Stochastic Representation
  7. Approximate Stochastic Representation: Normal Product Distribution
  8. Simulation and Bayesian Inference
  9. Sampling with Proximal Langevin
  10. Posterior Under Gaussian Likelihood
  11. Bayesian Estimation of $\lambda$
  12. Numerical Experiments
  13. Empirical Behavior of Spatial Frequencies of Images
  14. Effective Sample Size for Normal Sampler
  15. Matrix Denoising
  16. ...and 23 more sections

Key Result

Proposition 3.1

The Nuclear Norm distribution is symmetric under $O(n)\times O(m)$ where $O(n)$ is the general orthogonal group on $\mathbb{R}^n$. That is, if $X\sim \mathrm{NND}(\lambda)$ and $U$ and $V$ are orthogonal matrices, then $UXV\sim \mathrm{NND}(\lambda)$.

Figures (12)

  • Figure 1: Left: Histogram of singular values of $\mathrm{NND}$ variates of size $7\times 2$ and $\lambda=1$. Center: Density of Theorem \ref{['thm:svd']}. Right: A histogram of ordered Gamma(7,1) variates.
  • Figure 2: Conditional Prior Ensures Unimodality. x-axis gives first singular value of matrix $X$, y-axis gives estimated error variance, contours indicate posterior density. See Appendix \ref{['sec:app_uni']}.
  • Figure 3: Empirical distributions of matrix nuclear norm (left) and singular values (right) from MNIST dataset (blue) compared to nuclear norm (green) and iid Gaussian (orange).
  • Figure 4: Example matrix completion image from the sports dataset; from left to right, the original image, the image with half of pixels removed and noise added, the posterior mean with a fixed $\lambda$ at the optimal value, and the posterior mean of the adaptive method. See Appendices \ref{['sec:app_complete']} and \ref{['sec:app_images']} for more.
  • Figure 5: Effective sample size of proximal (prox) versus SVD-Langevin MCMC in (svl); higher is better.
  • ...and 7 more figures

Theorems & Definitions (11)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • Proposition 3.4
  • proof
  • Definition 3.5
  • Theorem 3.6
  • Theorem 4.1
  • ...and 1 more