Exact Completion of Rectangular Matrices Using Ramanujan Bigraphs

Shantanu Prasad Burnwal, M. Vidyasagar
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Abstract

In this paper, we study the matrix completion problem: Suppose $X \in {\mathbb{R}^{{n_r} \times {n_c}}}$ is unknown except for an upper bound r on its rank. By measuring a small number m ≪ nrnc of elements of X, is it possible to recover X exactly, or at least, to construct a reasonable approximation of X? At present, there are two approaches to choosing the sample set, namely probabilistic and deterministic. Probabilistic methods can guarantee exact recovery of the unknown matrix, but only with high probability. In this approach, samples are taken uniformly at random. Therefore we need to start sampling for every new matrix afresh. In the deterministic approach, sampling points can be kept fixed. At present, there are very few deterministic methods, and they mostly apply only to square matrices. In this paper, we present a deterministic method for selecting the sample set that can guarantee the exact recovery of the unknown matrix. This approach works for the recovery of rectangular as well as square matrices. We achieve this by choosing the elements to be sampled as the edge set of a Ramanujan bigraph. If samples are the edge set of a Ramanujan bigraph, then we can recover the unknown matrix from that sample set using nuclear norm minimization. A companion paper discusses the explicit construction of Ramanujan bigraphs. We provide a sufficient condition, that is if the samples taken are of the order of r3 then we can recover the unknown entries exactly if the unknown matrix satisfies some coherence condition. We believe this the first sufficient condition available using deterministic sampling technique and nuclear norm minimization.
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矩形矩阵用拉马努金图的精确补全
本文研究了矩阵补全问题:假设$X \ In {\mathbb{R}^{{n_r} \乘以{n_c}} $除了秩上有上界R外是未知的。通过测量少量的X元素的m < nrnc,是否有可能精确地得到X,或者至少构造出X的合理近似值?目前,有两种选择样本集的方法,即概率方法和确定性方法。概率方法可以保证未知矩阵的精确恢复,但只有在高概率的情况下。在这种方法中,样本是均匀随机抽取的。因此,我们需要对每个新矩阵重新开始采样。在确定性方法中,采样点可以保持固定。目前,确定性方法很少,而且大多只适用于方阵。本文提出了一种确定的样本集选择方法,保证了未知矩阵的精确恢复。这种方法既适用于矩形矩阵的恢复,也适用于方阵的恢复。我们通过选择要采样的元素作为拉马努金图的边集来实现这一点。如果样本是拉马努金图的边集,那么我们可以使用核范数最小化从该样本集恢复未知矩阵。另一篇论文讨论了拉马努金图的显式构造。我们提供了一个充分条件,如果所取的样本是r3阶的,那么如果未知矩阵满足相干性条件,我们可以准确地恢复未知项。我们认为这是采用确定性抽样技术和核范数最小化方法得到的第一个充分条件。
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