We consider the problem of collaborative filtering from a channel coding perspective. We model the underlying rating matrix as a finite alphabet matrix with block constant structure. The observations are obtained from this underlying matrix through a discrete memoryless channel with a noisy part representing noisy user behavior and an erasure part representing missing data. Moreover, the clusters over which the underlying matrix is constant are unknown. We establish a threshold result for this model: if the largest cluster size is smaller than C1 log(mn) (where the rating matrix is of size m × n), then the underlying matrix cannot be recovered with any estimator, but if the smallest cluster size is larger than C2 log(mn), then we show a polynomial time estimator with asymptotically vanishing probability of error. In the case of uniform cluster size, not only the order of the threshold, but also the constant is identified.
On the heels of compressed sensing, a new field has very recently emerged. This field addresses a broad range of problems of significant practical interest, namely, the recovery of a data matrix from what appears to be incomplete, and perhaps even corrupted, information. In its simplest form, the problem is to recover a matrix from a small sample of its entries. It comes up in many areas of science and engineering, including collaborative filtering, machine learning, control, remote sensing, and computer vision, to name a few. This paper surveys the novel literature on matrix completion, which shows that under some suitable conditions, one can recover an unknown low-rank matrix from a nearly minimal set of entries by solving a simple convex optimization problem, namely, nuclear-norm minimization subject to data constraints. Further, this paper introduces novel results showing that matrix completion is provably accurate even when the few observed entries are corrupted with a small amount of noise. A typical result is that one can recover an unknown matrix of low rank from just about log noisy samples with an error that is proportional to the noise level. We present numerical results that complement our quantitative analysis and show that, in practice, nuclear-norm minimization accurately fills in the many missing entries of large low-rank matrices from just a few noisy samples. Some analogies between matrix completion and compressed sensing are discussed throughout.