# Concentration properties of fractional posterior in 1-bit matrix completion

arXiv: https://arxiv.org/abs/2404.08969 · HTML: https://mlsanity.com/paper/2404.08969/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 62 references that are in the map.
- First posted on arXiv 13 Apr 2024.

## Builds on (most cited first, 24 of 24)

- [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014)
- [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015)
- [A General Framework for Updating Belief Distributions](https://mlsanity.com/paper/1306.6430/) (2013)
- [High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1605.01559/) (2016)
- [User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient](https://mlsanity.com/paper/1710.00095/) (2017)
- [Inconsistency of Bayesian Inference for Misspecified Linear Models, and a Proposal for Repairing It](https://mlsanity.com/paper/1412.3730/) (2014)
- [Concentration of tempered posteriors and of their variational approximations](https://mlsanity.com/paper/1706.09293/) (2017)
- [Robust Generalised Bayesian Inference for Intractable Likelihoods](https://mlsanity.com/paper/2104.07359/) (2021)
- [Inference and Uncertainty Quantification for Noisy Matrix Completion](https://mlsanity.com/paper/1906.04159/) (2019)
- [General Bayesian Updating and the Loss-Likelihood Bootstrap](https://mlsanity.com/paper/1709.07616/) (2017)
- [PU Learning for Matrix Completion](https://mlsanity.com/paper/1411.6081/) (2014)
- [Calibrating general posterior credible regions](https://mlsanity.com/paper/1509.00922/) (2015)
- [A Max-Norm Constrained Minimization Approach to 1-Bit Matrix Completion](https://mlsanity.com/paper/1309.6013/) (2013)
- [A comparison of learning rate selection methods in generalized Bayesian inference](https://mlsanity.com/paper/2012.11349/) (2020)
- [On the Robustness to Misspecification of α-Posteriors and Their Variational Approximations](https://mlsanity.com/paper/2104.08324/) (2021)
- [General Bayesian Loss Function Selection and the use of Improper Models](https://mlsanity.com/paper/2106.01214/) (2021)
- [1-bit Matrix Completion: PAC-Bayesian Analysis of a Variational Approximation](https://mlsanity.com/paper/1604.04191/) (2016)
- [Fast Low-Rank Bayesian Matrix Completion with Hierarchical Gaussian Prior Models](https://mlsanity.com/paper/1708.02455/) (2017)
- [A reduced-rank approach to predicting multiple binary responses through machine learning](https://mlsanity.com/paper/2306.05829/) (2023)
- [Adaptive Multinomial Matrix Completion](https://mlsanity.com/paper/1408.6218/) (2014)
- [Collective Matrix Completion](https://mlsanity.com/paper/1807.09010/) (2018)
- [Adaptation of the Tuning Parameter in General Bayesian Inference with Robust Divergence](https://mlsanity.com/paper/2106.06902/) (2021)
- [From bilinear regression to inductive matrix completion: a quasi-Bayesian analysis](https://mlsanity.com/paper/2210.15290/) (2022)
- [Bayesian matrix completion: prior specification](https://mlsanity.com/paper/1406.1440/) (2014)

## Built on by (most cited first, 7 of 7)

- [High-dimensional Bayesian Tobit regression for censored response with Horseshoe prior](https://mlsanity.com/paper/2505.08288/) (2025)
- [Handling bounded response in high dimensions: a Horseshoe prior Bayesian Beta regression approach](https://mlsanity.com/paper/2505.22211/) (2025)
- [Heavy Lasso: sparse penalized regression under heavy-tailed noise via data-augmented soft-thresholding](https://mlsanity.com/paper/2506.07790/) (2025)
- [Adaptive posterior concentration rates for sparse high-dimensional linear regression with random design and unknown error variance](https://mlsanity.com/paper/2405.19016/) (2024)
- [Impossibility of latent inner product recovery via rate distortion](https://mlsanity.com/paper/2407.11932/) (2024)
- [Robust low-rank estimation with multiple binary responses using pairwise AUC loss](https://mlsanity.com/paper/2601.08618/) (2026)
- [Mixture-of-experts Wishart model for covariance matrices with an application to Cancer drug screening](https://mlsanity.com/paper/2602.13888/) (2026)

## Abstract (arXiv)

The problem of estimating a matrix based on a set of its observed entries is commonly referred to as the matrix completion problem. In this work, we specifically address the scenario of binary observations, often termed as 1-bit matrix completion. While numerous studies have explored Bayesian and frequentist methods for real-value matrix completion, there has been a lack of theoretical exploration regarding Bayesian approaches in 1-bit matrix completion. We tackle this gap by considering a general, non-uniform sampling scheme and providing theoretical assurances on the efficacy of the fractional posterior. Our contributions include obtaining concentration results for the fractional posterior and demonstrating its effectiveness in recovering the underlying parameter matrix. We accomplish this using two distinct types of prior distributions: low-rank factorization priors and a spectral scaled Student prior, with the latter requiring fewer assumptions. Importantly, our results exhibit an adaptive nature by not mandating prior knowledge of the rank of the parameter matrix. Our findings are comparable to those found in the frequentist literature, yet demand fewer restrictive assumptions.