Concentration properties of fractional posterior in 1-bit matrix completion
arXiv:2404.08969 · doi:10.1007/s10994-024-06691-z
Abstract
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.
References in corpus (16)
- Rényi Divergence and Kullback-Leibler Divergence
- Recovering low-rank matrices from few coefficients in any basis
- Matrix estimation by Universal Singular Value Thresholding
- A General Framework for Updating Belief Distributions
- Inference and Uncertainty Quantification for Noisy Matrix Completion
- Noisy low-rank matrix completion with general sampling distribution
- Sparse Regression Learning by Aggregation and Langevin Monte-Carlo
- Calibrating general posterior credible regions
- Fast Low-Rank Bayesian Matrix Completion with Hierarchical Gaussian Prior Models
- 1-bit Matrix Completion: PAC-Bayesian Analysis of a Variational Approximation
- A comparison of learning rate selection methods in generalized Bayesian inference
- Mirror averaging with sparsity priors
- Pseudo-Bayesian Quantum Tomography with Rank-adaptation
- Exponential weights in multivariate regression and a low-rankness favoring prior
- From bilinear regression to inductive matrix completion: a quasi-Bayesian analysis
- A reduced-rank approach to predicting multiple binary responses through machine learning