# 1-bit Matrix Completion: PAC-Bayesian Analysis of a Variational Approximation

arXiv: https://arxiv.org/abs/1604.04191 · HTML: https://mlsanity.com/paper/1604.04191/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Apr 2016.

## Built on by (most cited first, 11 of 11)

- [User-friendly introduction to PAC-Bayes bounds](https://mlsanity.com/paper/2110.11216/) (2021)
- [Concentration properties of fractional posterior in 1-bit matrix completion](https://mlsanity.com/paper/2404.08969/) (2024)
- [Misclassification bounds for PAC-Bayesian sparse deep learning](https://mlsanity.com/paper/2405.01304/) (2024)
- [Improving application performance with biased distributions of quantum states](https://mlsanity.com/paper/2107.07642/) (2021)
- [Impossibility of latent inner product recovery via rate distortion](https://mlsanity.com/paper/2407.11932/) (2024)
- [Simulation comparisons between Bayesian and de-biased estimators in low-rank matrix completion](https://mlsanity.com/paper/2103.11749/) (2021)
- [PAC-Bayesian Matrix Completion with a Spectral Scaled Student Prior](https://mlsanity.com/paper/2104.08191/) (2021)
- [One-Bit Matrix Completion with Differential Privacy](https://mlsanity.com/paper/2110.00719/) (2021)
- [Misclassification excess risk bounds for 1-bit matrix completion](https://mlsanity.com/paper/2312.12945/) (2023)
- [Misclassification excess risk bounds for PAC-Bayesian classification via convexified loss](https://mlsanity.com/paper/2408.08675/) (2024)
- [Empirical Bayes 1-bit matrix completion](https://mlsanity.com/paper/2605.09509/) (2026)

## Abstract (arXiv)

Due to challenging applications such as collaborative filtering, the matrix completion problem has been widely studied in the past few years. Different approaches rely on different structure assumptions on the matrix in hand. Here, we focus on the completion of a (possibly) low-rank matrix with binary entries, the so-called 1-bit matrix completion problem. Our approach relies on tools from machine learning theory: empirical risk minimization and its convex relaxations. We propose an algorithm to compute a variational approximation of the pseudo-posterior. Thanks to the convex relaxation, the corresponding minimization problem is bi-convex, and thus the method behaves well in practice. We also study the performance of this variational approximation through PAC-Bayesian learning bounds. On the contrary to previous works that focused on upper bounds on the estimation error of M with various matrix norms, we are able to derive from this analysis a PAC bound on the prediction error of our algorithm. We focus essentially on convex relaxation through the hinge loss, for which we present the complete analysis, a complete simulation study and a test on the MovieLens data set. However, we also discuss a variational approximation to deal with the logistic loss.