Matrix Completion is the problem of recovering an unknown real-valued low-rank matrix from a subsample of its entries. Important recent results show that the problem can be solved efficiently under the assumption that the unknown matrix is incoherent and the subsample is drawn uniformly at random. Are these assumptions necessary? It is well known that Matrix Completion in its full generality is NP-hard. However, little is known if make additional assumptions such as incoherence and permit the algorithm to output a matrix of slightly higher rank. In this paper we prove that Matrix Completion remains computationally intractable even if the unknown matrix has rank @math but we are allowed to output any constant rank matrix, and even if additionally we assume that the unknown matrix is incoherent and are shown @math of the entries. This result relies on the conjectured hardness of the @math -Coloring problem. We also consider the positive semidefinite Matrix Completion problem. Here we show a similar hardness result under the standard assumption that @math Our results greatly narrow the gap between existing feasibility results and computational lower bounds. In particular, we believe that our results give the first complexity-theoretic justification for why distributional assumptions are needed beyond the incoherence assumption in order to obtain positive results. On the technical side, we contribute several new ideas on how to encode hard combinatorial problems in low-rank optimization problems. We hope that these techniques will be helpful in further understanding the computational limits of Matrix Completion and related problems.
There have been several hardness results for Matrix Completion over finite fields drawing on its connection to problems in coding theory. See, for example, the discussion in @cite_0 @cite_14 . The Matrix Completion problem over the reals seems to behave rather differently and techniques do not seem to transfer from the finite field case.
@cite_0 @cite_14
2034815954 1657130172
Given a matrix whose entries are a mixture of numeric values and symbolic variables, the matrix completion problem is to assign values to the variables so as to maximize the resulting matrix rank. This problem has deep connections to computational complexity and numerous important algorithmic applications. Determining the complexity of this problem is a fundamental open question in computational complexity. Under different settings of parameters, the problem is variously in P, in RP, or NP-hard. We shed new light on this landscape by demonstrating a new region of NP-hard scenarios. As a special case, we obtain the first known hardness result for matrices in which each variable appears only twice.Another particular scenario that we consider is the simultaneous matrix completion problem, where one must simultaneously maximize the rank for several matrices that share variables. This problem has important applications in the field of network coding. Recent work has given a simple, greedy, deterministic algorithm for this problem, assuming that the algorithm works over a sufficiently large field. We show an exact threshold for the field size required to find a simultaneous completion efficiently. This result implies that, surprisingly, the simple greedy algorithm is optimal: finding a simultaneous completion over any smaller field is NP-hard. This paper establishes information-theoretic limits for 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 × n-sensing matrices contain, on average, Ω(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 aforementioned 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 minimum 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.