A Channel Coding Perspective of Collaborative Filtering

TL;DR

This paper models collaborative filtering as a channel coding problem, establishing sharp thresholds for when the underlying rating matrix can be reliably recovered based on cluster sizes and noisy observations.

Contribution

It introduces a novel channel coding perspective for collaborative filtering and derives precise thresholds for matrix recoverability depending on cluster sizes.

Findings

Recovery impossible if largest cluster size < C1 log(mn)

Polynomial time estimator achieves low error if smallest cluster size > C2 log(mn)

Exact threshold constants identified for uniform cluster sizes

Abstract

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 {\it unknown}. We establish a sharp threshold result for this model: if the largest cluster size is smaller than $C_1 \log(mn)$ (where the rating matrix is of size $m \times n$), then the underlying matrix cannot be recovered with any estimator, but if the smallest cluster size is larger than $C_2 \log(mn)$, then we show a polynomial time estimator with diminishing probability of error. In the case of uniform cluster size, not only the order of the threshold, but also the constant is identified.