Information-theoretic limits of matrix completion

We propose an information-theoretic framework for matrix completion. The theory goes beyond the low-rank structure and applies to general matrices of “low description complexity”. Specifically, we consider random matrices X ∈ ℝ^m×n of arbitrary distribution (continuous, discrete, discrete-continuous mixture, or even singular). With S ⊆ ℝ^m×n an ε-support set of X, i.e., P[X ∈ S] ≥ 1 - ε, and equation denoting the lower Minkowski dimension of S, we show that equation measurements of the form 〈A_i,X〉, with A_i denoting the measurement matrices, suffice to recover X with probability of error at most ε. The result holds for Lebesgue a.a. A_i and does not need incoherence between the A_i and the unknown matrix X. We furthermore show that equation measurements also suffice to recover the unknown matrix X from measurements taken with rank-one A_i, again this applies to a.a. rank-one A_i. Rank-one measurement matrices are attractive as they require less storage space than general measurement matrices and can be applied faster. Particularizing our results to the recovery of low-rank matrices, we find that k > (m+n-r)r measurements are sufficient to recover matrices of rank at most r. Finally, we construct a class of rank-r matrices that can be recovered with arbitrarily small probability of error from k <; (m + n - r)r measurements.