arXiv 2007.094572020-07math.NAcs.NAstat.ML

低秩加稀疏矩阵的压缩感知

Compressed sensing of low-rank plus sparse matrices

Jared Tanner, Simon Vary

备注：: 39 pages, 6 figures

摘要：

将矩阵表示为低秩矩阵加上稀疏矩阵的和是捕获数据中的全局和局部特征的灵活模型，其被推广为Robust PCA（Candes等人，2011年; Amarasekaran等人，2009年）。压缩感知、矩阵补全及其变体（Eldar and Kutyniok，2012; Foucart and Rauhut，2013）已经确定，满足低复杂度模型的数据可以有效地测量，并从与模型复杂度而不是环境维度成比例的多个测量中恢复。该手稿开发了类似的保证，表明可以表示为秩- $r$矩阵和$s$ -稀疏矩阵之和的$m\times n$矩阵可以通过计算上易于处理的方法从$\mathcal{O}(r(m+n-r)+s)\log(mn/s)$线性测量中恢复。更具体地说，我们建立了低秩加稀疏矩阵集是封闭的低秩分量的不相干性是上界为$μ<\sqrt{mn}/(r\sqrt{s})$，随后，上述矩阵的限制等距常数保持有界独立于问题的大小提供$p/mn$，$s/p$，和$r(m+n-r)/p$保持固定。此外，我们表明，半定规划和两个硬阈值梯度下降算法，NIHT和NAHT，收敛到测量矩阵的测量算子的RIC的足够小。这些结果还可证明地解决了凸和非凸的鲁棒PCA的渐近最优分数的腐败$α=\mathcal{O}\left(1/(μr) \right)$，其中$s = α^2 mn$，并提高了以前最好的已知的保证，不需要腐败的分数是由$α$的上限分布在每一列和行。数值实验说明这些结果的合成问题，动态前景/静态背景分离，和多光谱成像。

英文摘要：

Expressing a matrix as the sum of a low-rank matrix plus a sparse matrix is a flexible model capturing global and local features in data popularized as Robust PCA (Candes et al., 2011; Chandrasekaran et al., 2009). Compressed sensing, matrix completion, and their variants (Eldar and Kutyniok, 2012; Foucart and Rauhut, 2013) have established that data satisfying low complexity models can be efficiently measured and recovered from a number of measurements proportional to the model complexity rather than the ambient dimension. This manuscript develops similar guarantees showing that $m\times n$ matrices that can be expressed as the sum of a rank-$r$ matrix and a $s$-sparse matrix can be recovered by computationally tractable methods from $\mathcal{O}(r(m+n-r)+s)\log(mn/s)$ linear measurements. More specifically, we establish that the low-rank plus sparse matrix set is closed provided the incoherence of the low-rank component is upper bounded as $μ<\sqrt{mn}/(r\sqrt{s})$, and subsequently, the restricted isometry constants for the aforementioned matrices remain bounded independent of problem size provided $p/mn$, $s/p$, and $r(m+n-r)/p$ remain fixed. Additionally, we show that semidefinite programming and two hard threshold gradient descent algorithms, NIHT and NAHT, converge to the measured matrix provided the measurement operator's RIC's are sufficiently small. These results also provably solve convex and non-convex formulation of Robust PCA with the asymptotically optimal fraction of corruptions $α=\mathcal{O}\left(1/(μr) \right)$, where $s = α^2 mn$, and improve the previously best known guarantees by not requiring that the fraction of corruptions is spread in every column and row by being upper bounded by $α$. Numerical experiments illustrating these results are shown for synthetic problems, dynamic-foreground/static-background separation, and multispectral imaging.

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