Collaborative Filtering in a Non-Uniform World: Learning with the Weighted Trace Norm

We show that matrix completion with trace-norm regularization can be significantly hurt when entries of the matrix are sampled non-uniformly. We introduce a weighted version of the trace-norm regularizer that works well also with non-uniform sampling. Our experimental results demonstrate that the weighted trace-norm regularization indeed yields significant gains on the (highly non-uniformly sampled) Netflix dataset.

💡 Research Summary

The paper addresses a fundamental limitation of trace‑norm (nuclear‑norm) regularization in matrix completion when the observed entries are sampled non‑uniformly—a situation that is common in real‑world collaborative‑filtering systems such as movie‑rating platforms. While trace‑norm regularization enjoys strong theoretical guarantees under the assumption of uniform random sampling, the authors demonstrate both analytically and empirically that these guarantees break down when some rows (users) or columns (items) are observed far more frequently than others. In such cases, the standard trace‑norm penalty under‑regularizes the poorly sampled portions of the matrix, leading to high reconstruction error and over‑fitting on densely sampled entries.

To remedy this, the authors propose a weighted trace‑norm regularizer that incorporates the sampling probabilities of each row and column. Let (p_i) be the marginal probability that row (i) is observed and (q_j) the marginal probability for column (j). Define diagonal weighting matrices (D_r = \text{diag}(p_1,\dots,p_n)) and (D_c = \text{diag}(q_1,\dots,q_m)). The weighted regularizer is then the trace norm of the scaled matrix
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