Detection Thresholds in Very Sparse Matrix Completion
نویسندگان
چکیده
We study the matrix completion problem: an underlying $$m \times n$$ P is low rank, with incoherent singular vectors, and a random A equal to on (uniformly) subset of entries size dn. All other are zero. The goal retrieve information from observation A. Let $$A_1$$ be where each entry multiplied by independent $$\{0,1\}$$ -Bernoulli variable parameter 1/2. This paper about when, how why non-Hermitian eigen-spectra matrices $$A_1 (A - A_1)^*$$ $$(A-A_1)^*A_1$$ captures more relevant principal component structure than $$A A^*$$ $$A^* A$$ . show that eigenvalues asymmetric $$A_{1} A_{1})^{*}$$ $$(A-A_{1})^{*} A_{1}$$ modulus greater detection threshold asymptotically $$PP^*$$ $$P^*P$$ associated eigenvectors aligned as well. central surprise intentionally inducing asymmetry additional randomness via matrix, we can extract if had worked value decomposition (SVD) exact non-universal since it explicitly depends element-wise distribution P. reliable, statistically optimal but not perfect recovery, universal data-driven algorithm, possible above this using extracted eigen-decompositions. Averaging left right provably improves estimation accuracy threshold. Our results encompass very sparse regime d order 1 SVD fails or produces unreliable recovery. define another variant analysis procedure bypasses randomization step has smaller constant factor computational cost larger polynomial number observed entries. Both thresholds allow go beyond barrier due well-known theoretical limit $$d \asymp \log for found in literature.
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ژورنال
عنوان ژورنال: Foundations of Computational Mathematics
سال: 2022
ISSN: ['1615-3383', '1615-3375']
DOI: https://doi.org/10.1007/s10208-022-09568-6