基于随机基追踪的快速近优矩阵完备化
Fast and Near-Optimal Matrix Completion via Randomized Basis Pursuit
摘要：
受压缩感知的原理和巨大成功的启发，从其条目的采样中重构矩阵的问题最近引起了人们的广泛关注。这样的问题可以被看作是一个信息理论的变体，以及研究矩阵完成问题，其主要目标是设计一个有效的算法，可以重建一个矩阵，通过检查只有少量的条目。虽然这在一般情况下是不可能的任务，但Candès和合著者最近证明，在所谓的不相干假设下，在检查$O(nr\log^6n)$的条目后，可以使用半定规划（SDP）重建秩$r$ $n\times n$矩阵。在本文中，我们提出了一种替代方法，这是更有效的，可以重建一个更大的类的矩阵通过检查一个显着较少的条目。具体来说，我们首先介绍了一类所谓的稳定矩阵，并表明它包括所有那些满足不相干假设。然后，我们提出了一个随机基追踪（RBP）算法，并证明了它可以重建一个稳定的秩$r$ $n\times n$矩阵检查后，$O(nr\log n)$的条目。我们的采样界是只有一个对数因子远离信息理论的限制，本质上是最佳的。此外，RBP算法的运行时间由$O(nr^2\log n+n^2r)$限定，这与基于SDP的算法的$Ω(n^4r^2\log^{12}n)$运行时间相比非常有利。也许更重要的是，我们的算法将在多项式时间内提供输入矩阵的精确重构。相比之下，基于SDP的算法只能在多项式时间内提供一个近似值。
英文摘要：
Motivated by the philosophy and phenomenal success of compressed sensing, the problem of reconstructing a matrix from a sampling of its entries has attracted much attention recently. Such a problem can be viewed as an information-theoretic variant of the well-studied matrix completion problem, and the main objective is to design an efficient algorithm that can reconstruct a matrix by inspecting only a small number of its entries. Although this is an impossible task in general, Candès and co-authors have recently shown that under a so-called incoherence assumption, a rank $r$ $n\times n$ matrix can be reconstructed using semidefinite programming (SDP) after one inspects $O(nr\log^6n)$ of its entries. In this paper we propose an alternative approach that is much more efficient and can reconstruct a larger class of matrices by inspecting a significantly smaller number of the entries. Specifically, we first introduce a class of so-called stable matrices and show that it includes all those that satisfy the incoherence assumption. Then, we propose a randomized basis pursuit (RBP) algorithm and show that it can reconstruct a stable rank $r$ $n\times n$ matrix after inspecting $O(nr\log n)$ of its entries. Our sampling bound is only a logarithmic factor away from the information-theoretic limit and is essentially optimal. Moreover, the runtime of the RBP algorithm is bounded by $O(nr^2\log n+n^2r)$, which compares very favorably with the $Ω(n^4r^2\log^{12}n)$ runtime of the SDP-based algorithm. Perhaps more importantly, our algorithm will provide an exact reconstruction of the input matrix in polynomial time. By contrast, the SDP-based algorithm can only provide an approximate one in polynomial time.
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