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Experimental recovery regions for robust PCA

机译:强大PCA的实验恢复区域

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The principle of Robust Principal Component Analysis (RPCA) is to additively resolve a matrix into a low-rank and a sparse component. The question that arises in the application of this principle to experimental data is, "when is this resolution an identification of the actual low-rank and sparse components of the data?" We report several experimental findings: (1) while successful recoveries can only be expected when the low-rank component is of low fractional rank and the sparse component is of low fractional sparsity, the subset of matrices that successfully recover is significantly larger than the subset of matrices that satisfy the currently established sufficient conditions; (2) where recovery is unsuccessful, the returned matrices tend to be near half-rank and half-sparsity, suggesting a cross validation principle; (3) the demarcation between the region of consistent recovery and consistent failure is narrow, indicating a phase change in recoverability; and (4) recovery is relatively invariant to matrix distributions, thus synthetic matrices can closely predict recoverability of real matrices. We demonstrate these findings with a variety of synthetic matrices that are faithful to matrices appearing in practice. Furthermore, we apply and verify these results on real-world matrices.
机译:稳健的主成分分析(RPCA)的原理是将矩阵加法分解为低秩和稀疏成分。在将该原理应用于实验数据时出现的问题是:“该分辨率何时可以标识出数据的实际低秩和稀疏成分?”我们报告了一些实验发现:(1)虽然仅当低秩成分的分数阶较低而稀疏成分的分数稀疏度低时才能成功恢复,但成功恢复的矩阵子集明显大于该子集满足当前确定的充分条件的矩阵; (2)在恢复不成功的地方,返回的矩阵往往接近半秩和半稀疏,这表明了交叉验证的原理; (3)一致恢复和一致破坏区域之间的界限很窄,表明可恢复性发生了相变; (4)恢复相对于矩阵分布而言是相对不变的,因此合成矩阵可以紧密预测实际矩阵的可恢复性。我们用忠实于实践中出现的各种合成矩阵来证明这些发现。此外,我们将这些结果应用于实际矩阵。

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