A Combinatorial Approach to Robust PCA

• URL: http://arxiv.org/abs/2311.16416v1
• Date: Tue, 28 Nov 2023 01:49:51 GMT
• Title: A Combinatorial Approach to Robust PCA
• Authors: Weihao Kong, Mingda Qiao, Rajat Sen
• Abstract summary: We study the problem of recovering Gaussian data under adversarial corruptions. We assume that the Gaussian noises lie in an unknown $k$-dimensional subspace $U subseteq mathbbRd$, and $s$ randomly chosen coordinates of each data point fall into the control of an adversary. Our main result is an efficient algorithm that, when $ks2 = O(d)$, recovers every single data point up to a nearly-optimal error of $tilde O(ks/d)$ in expectation.
• Score: 18.740048806623037
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the problem of recovering Gaussian data under adversarial corruptions when the noises are low-rank and the corruptions are on the coordinate level. Concretely, we assume that the Gaussian noises lie in an unknown $k$-dimensional subspace $U \subseteq \mathbb{R}^d$, and $s$ randomly chosen coordinates of each data point fall into the control of an adversary. This setting models the scenario of learning from high-dimensional yet structured data that are transmitted through a highly-noisy channel, so that the data points are unlikely to be entirely clean. Our main result is an efficient algorithm that, when $ks^2 = O(d)$, recovers every single data point up to a nearly-optimal $\ell_1$ error of $\tilde O(ks/d)$ in expectation. At the core of our proof is a new analysis of the well-known Basis Pursuit (BP) method for recovering a sparse signal, which is known to succeed under additional assumptions (e.g., incoherence or the restricted isometry property) on the underlying subspace $U$. In contrast, we present a novel approach via studying a natural combinatorial problem and show that, over the randomness in the support of the sparse signal, a high-probability error bound is possible even if the subspace $U$ is arbitrary.

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