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Concentration properties of fractional posterior in 1-bit matrix completion

The Tien Mai

TL;DR

This work analyzes 1-bit matrix completion under general non-uniform sampling through fractional posterior methods. It proves concentration and recovery guarantees for the parameter matrix $M^*$ using two priors: a low-rank factorization prior and a spectral scaled Student prior, achieving adaptive performance without knowing the true rank. The results yield convergence rates in information and Frobenius norms that match minimax benchmarks under broader sampling schemes and with fewer restrictive assumptions. Notably, the spectral prior enables concentration results without a boundedness assumption on $M^*$ and supports gradient-based sampling methods like Langevin Monte Carlo. Overall, the study advances Bayesian understanding of binary matrix completion and provides practically flexible, adaptive theoretical guarantees.

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.

Concentration properties of fractional posterior in 1-bit matrix completion

TL;DR

This work analyzes 1-bit matrix completion under general non-uniform sampling through fractional posterior methods. It proves concentration and recovery guarantees for the parameter matrix using two priors: a low-rank factorization prior and a spectral scaled Student prior, achieving adaptive performance without knowing the true rank. The results yield convergence rates in information and Frobenius norms that match minimax benchmarks under broader sampling schemes and with fewer restrictive assumptions. Notably, the spectral prior enables concentration results without a boundedness assumption on and supports gradient-based sampling methods like Langevin Monte Carlo. Overall, the study advances Bayesian understanding of binary matrix completion and provides practically flexible, adaptive theoretical guarantees.

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.
Paper Structure (10 sections, 11 theorems, 67 equations)

This paper contains 10 sections, 11 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Notations and method
  3. Notations
  4. 1-bit matrix completion
  5. Main results
  6. Results with a spectral scaled Student prior
  7. Proofs
  8. Proofs for Section \ref{['sc_main1']}
  9. Proofs for Section \ref{['sc_main2']}
  10. Lemma

Key Result

Theorem 1

Assume that Assumption assume_factorbounded holds. Then, there is a small enough $b >0$ such that where for some universal constant $C_{a,B}$ depending only on $a, B$. Specifically, the result remains valid for the selection $b= B^2/[512(nd_1d_2)^4 K^2 \max^2(d_1,d_2)]$.

Theorems & Definitions (30)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Corollary 3
  • Remark 2
  • Theorem 4
  • Remark 3
  • Remark 4
  • Remark 5
  • Theorem 5
  • ...and 20 more