Matrix Completion for Structured Observations

Abstract

The need to predict or fill-in missing data, often referred to as matrix completion, is a common challenge in today's data-driven world. Previous strategies typically assume that no structural difference between observed and missing entries exists. Unfortunately, this assumption is woefully unrealistic in many applications. For example, in the classic Netflix challenge, in which one hopes to predict user-movie ratings for unseen films, the fact that the viewer has not watched a given movie may indicate a lack of interest in that movie, thus suggesting a lower rating than otherwise expected. We propose adjusting the standard nuclear norm minimization strategy for matrix completion to account for such structural differences between observed and unobserved entries by regularizing the values of the unobserved entries. We show that the proposed method outperforms nuclear norm minimization in certain settings.

On Inferences from Completed Data

“…Entries of a data matrix could be selected using uniform sampling; that is, each entry could be sampled with equal probability as in [3]. On the other hand, one could employ structured sampling and select entries with probability dependent upon their value as in [7]. The details of these two sampling methods are given in Section I-B.…”

“…for some regularization parameter α > 0. The addition of the 1 -regularization term in the objective of 1 -NNM encourages unobserved entries of the recovered matrix to be near 0, which makes it a natural choice for recovery on an incomplete matrix generated by structured sampling [7].…”

Large Data Analysis and Lyme Disease

“…We see, as expected, that when most of the unobserved entries are small and most of the observed entries are large, there is the most improvement in using the regularizer. Preliminary theoretical results can be found in [12], which are motivated by work in robust principal component analysis [4], but future work is needed to clearly quantify the theoretical gains as a function of the sampling rates. (2), respectively, with 1 regularization on the recovered values for the unobserved entries, we plot ||̃− || /||̂− || .…”

Matrix Completion with Selective Sampling

“…Recently, Molitor and Needell [6] adapted the ordinary nuclear norm minimization method to account for structure in the observed and unobserved entries, but most current methods for matrix completion assume little about the structure of the matrix M and take the observed entries from a uniform random distribution. We propose a situation where the entries need not be observed at random, but can be chosen to account for the relationships between the columns.…”