Deterministic Symmetric Positive Semidefinite Matrix Completion
نویسندگان
چکیده
We consider the problem of recovering a symmetric, positive semidefinite (SPSD) matrix from a subset of its entries, possibly corrupted by noise. In contrast to previous matrix recovery work, we drop the assumption of a random sampling of entries in favor of a deterministic sampling of principal submatrices of the matrix. We develop a set of sufficient conditions for the recovery of a SPSD matrix from a set of its principal submatrices, present necessity results based on this set of conditions and develop an algorithm that can exactly recover a matrix when these conditions are met. The proposed algorithm is naturally generalized to the problem of noisy matrix recovery, and we provide a worst-case bound on reconstruction error for this scenario. Finally, we demonstrate the algorithm’s utility on noiseless and noisy simulated datasets.
منابع مشابه
Appendix for Deterministic Symmetric Positive Semidefinite Matrix Completion
First, note that by assumption rank{A} > 0. Let Ω1 = ρ1 × ρ1 and Ω2 = ρ2 × ρ2 be the two index sets in the theorem. By assumption we have ρ1 × ρ1 ∪ ρ2 × ρ2 = Ω and Ω 6= [n]× [n]. If A1 is not met, then ρ1 ∪ ρ2 6= [n], and from lemma 6 we can conclude recovery of A is impossible. If ρ1 ∪ ρ2 = [n], but A2 is not met then ι2 = |ρ1 ∩ ρ2| < r so it must be that rank{A(ι2, ι2)} < r. Further, by assum...
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