Provable Self-Representation Based Outlier Detection in a Union of Subspaces

Abstract

Many computer vision tasks involve processing large amounts of data contaminated by outliers, which need to be detected and rejected. While outlier detection methods based on robust statistics have existed for decades, only recently have methods based on sparse and low-rank representation been developed along with guarantees of correct outlier detection when the inliers lie in one or more lowdimensional subspaces. This paper proposes a new outlier detection method that combines tools from sparse representation with random walks on a graph. By exploiting the property that data points can be expressed as sparse linear combinations of each other, we obtain an asymmetric affinity matrix among data points, which we use to construct a weighted directed graph. By defining a suitable Markov Chain from this graph, we establish a connection between inliers/outliers and essential/inessential states of the Markov chain, which allows us to detect outliers by using random walks. We provide a theoretical analysis that justifies the correctness of our method under geometric and connectivity assumptions. Experimental results on image databases demonstrate its superiority with respect to stateof-the-art sparse and low-rank outlier detection methods.

Learning Competitive and Discriminative Reconstructions for Anomaly Detection

“…We use similar setup as in (You, Robinson, and Vidal 2017), the experimental results are presented in Table 3. The results of the other methods are from (Sabokrou et al 2018) and (Kimura and Yanagihara 2018).…”

“…Each category has at least 80 images. Following previous works (You, Robinson, and Vidal 2017;Sabokrou et al 2018), we randomly select images from n ∈ {1, 3, 5} categories as inliers, and for those categories that have more than 150 images, only the first 150 images are used. A certain number of outliers are randomly selected from the "clutter" category, such that each experiment has exactly 50% outliers.…”

Robust One-Class Kernel Spectral Regression

“…• FMS is a non-convex robust subspace recovery approach [41], designed to be least affected by corruptions in the training set and has been demonstrated to converge to a close vicinity of the correct subspace within few iterations • SRO obtains a weighted directed graph, defines a Markov Chain via self-representation, and identifies outliers via random walks [42]. The SRO method can be considered as one of the leading unsupervised approaches for novelty detection.…”

“…The FMS method is a non-convex robust subspace recovery approach [41], designed to be least affected by corruptions in the training set and has been demonstrated to converge to a close vicinity of the correct subspace within few iterations. A leading unsupervised technique among others is that of SRO [42] which obtains a weighted directed graph, defines a Markov Chain via self-representation, and identifies outliers via random walks.…”

Structured and Unstructured Outlier Identification for Robust PCA: A Fast Parameter Free Algorithm

“…Although the proposed threshold in [22] is independent of the dimension of the subspace r or the number of outliers, the underlying optimization problem is not parameter free and since multiple optimization problems have to be solved, the procedure is also rather complex. Another self-representation based algorithm for detecting outliers from a union of subspaces is proposed in [24], based on random walks in a graph, but it is iterative and requires multiple parameters to be set.…”