Rank Minimization over Finite Fields: Fundamental Limits and Coding-Theoretic Interpretations

This paper establishes information-theoretic limits in estimating a finite field low-rank matrix given random linear measurements of it. These linear measurements are obtained by taking inner products of the low-rank matrix with random sensing matrices. Necessary and sufficient conditions on the number of measurements required are provided. It is shown that these conditions are sharp and the minimum-rank decoder is asymptotically optimal. The reliability function of this decoder is also derived by appealing to de Caen’s lower bound on the probability of a union. The sufficient condition also holds when the sensing matrices are sparse - a scenario that may be amenable to efficient decoding. More precisely, it is shown that if the n\times n-sensing matrices contain, on average, \Omega(nlog n) entries, the number of measurements required is the same as that when the sensing matrices are dense and contain entries drawn uniformly at random from the field. Analogies are drawn between the above results and rank-metric codes in the coding theory literature. In fact, we are also strongly motivated by understanding when minimum rank distance decoding of random rank-metric codes succeeds. To this end, we derive distance properties of equiprobable and sparse rank-metric codes. These distance properties provide a precise geometric interpretation of the fact that the sparse ensemble requires as few measurements as the dense one. Finally, we provide a non-exhaustive procedure to search for the unknown low-rank matrix.

💡 Research Summary

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The paper investigates the fundamental limits of recovering a low‑rank matrix over a finite field from random linear measurements. Let (X\in\mathbb{F}_q^{n\times n}) have rank (r). The measurement process produces (k) scalars (y_i=\langle A_i,X\rangle) where each sensing matrix (A_i) is drawn independently from a prescribed distribution. The authors address two central questions: (i) how many measurements are required to recover (X) with probability approaching one as (n\to\infty); and (ii) how the structure (density) of the sensing matrices influences this requirement.

Using a standard converse based on Fano’s inequality, they derive a necessary condition \