1-Bit Matrix Completion

Series: 
Stochastics Seminar
Thursday, February 7, 2013 - 15:05
1 hour (actually 50 minutes)
Location: 
Skyles 006
,  
Georgia Institute of Technology
Organizer: 
In this talk I will describe a theory of matrix completion for the extreme case of noisy 1-bit observations. In this setting, instead of observing a subset of the real-valued entries of a matrix M, we obtain a small number of binary (1-bit) measurements generated according to a probability distribution determined by the real-valued entries of M. The central question I will address is whether or not it is possible to obtain an accurate estimate of M from this data. In general this would seem impossible, but I will show that the maximum likelihood estimate under a suitable constraint returns an accurate estimate of M when $\|M\|_{\infty} \le \alpha$ and $\rank(M) \le r$. If the log-likelihood is a concave function (e.g., the logistic or probit observation models), then we can obtain this maximum likelihood estimate by optimizing a convex program. I will also provide lower bounds showing that this estimate is near-optimal and illustrate the potential of this method with some preliminary numerical simulations.