Concentration properties of fractional posterior in 1-bit matrix
  completion
The Tien Mai
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.