# A Probabilistic Basis for Low-Rank Matrix Learning

arXiv: https://arxiv.org/abs/2510.05447 · HTML: https://mlsanity.com/paper/2510.05447/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 64 references that are in the map.
- First posted on arXiv 6 Oct 2025.

## Builds on (most cited first, 9 of 9)

- [Image-to-Image Translation with Conditional Adversarial Networks](https://mlsanity.com/paper/1611.07004/) (2016)
- [Proximal Markov chain Monte Carlo algorithms](https://mlsanity.com/paper/1306.0187/) (2013)
- [Low-Rank Matrix Completion: A Contemporary Survey](https://mlsanity.com/paper/1907.11705/) (2019)
- [Proximal Algorithms in Statistics and Machine Learning](https://mlsanity.com/paper/1502.03175/) (2015)
- [Low Rank Regularization: A Review](https://mlsanity.com/paper/1808.04521/) (2018)
- [Fast Low-Rank Bayesian Matrix Completion with Hierarchical Gaussian Prior Models](https://mlsanity.com/paper/1708.02455/) (2017)
- [Bayesian methods for low-rank matrix estimation: short survey and theoretical study](https://mlsanity.com/paper/1306.3862/) (2013)
- [Bayesian Uncertainty Quantification for Low-Rank Matrix Completion](https://mlsanity.com/paper/2101.01299/) (2021)
- [Flat Minima in Linear Estimation and an Extended Gauss Markov Theorem](https://mlsanity.com/paper/2311.11093/) (2023)

## Built on by (most cited first, 1 of 1)

- [Exact Schur-Sylvester Dimensionality Reductions for Non-Smooth Stochastic Complexity and Manifold Sampling](https://mlsanity.com/paper/2606.23867/) (2026)

## Abstract (arXiv)

Low rank inference on matrices is widely conducted by optimizing a cost function augmented with a penalty proportional to the nuclear norm Vert · 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-λVert XVert_*, 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.