We study the problem of robust subspace recovery (RSR) in the presence of adversarial outliers. That is, we seek a subspace that contains a large portion of a dataset when some fraction of the data points are arbitrarily corrupted. We first examine a theoretical estimator that is intractable to calculate and use it to derive information-theoretic bounds of exact recovery. We then propose two tractable estimators: a variant of RANSAC and a simple relaxation of the theoretical estimator. The two estimators are fast to compute and achieve state-of-the-art theoretical performance in a noiseless RSR setting with adversarial outliers. The former estimator achieves better theoretical guarantees in the noiseless case, while the latter estimator is robust to small noise, and its guarantees significantly improve with non-adversarial models of outliers. We give a complete comparison of guarantees for the adversarial RSR problem, as well as a short discussion on the estimation of affine subspaces.
Some works have given theoretical guarantees for algorithms under the assumption of general position outliers @cite_2 @cite_13 @cite_9 (one possible definition of this notion appears in Definition ). In this case, the outliers can have arbitrary magnitude and may lie close to low-dimensional subspaces, but they are not allowed to have linearly dependent structure. While these distributions can be approximately adversarial, these algorithms cannot deal with arbitrary outlier distributions, which may contain low-dimensional structure.
@cite_9 @cite_13 @cite_2
2775726300 2313824013 2962933442
We consider the RANSAC algorithm in the context of subspace recovery and subspace clustering. We derive some theory and perform some numerical experiments. We also draw some correspondences with the methods of Hardt and Moitra (2013) and Chen and Lerman (2009b). This paper considers the problem of robust subspace recovery: given a set of N points in RD, if many lie in a d-dimensional subspace, then can we recover the underlying subspace? We show that Tyler’s M-estimator can be used to recover the underlying subspace, if the percentage of the inliers is larger than d D and the data points lie in general position. Empirically, Tyler’s M-estimator compares favorably with other convex subspace recovery algorithms in both simulations and experiments on real data sets. We consider a fundamental problem in unsupervised learning called subspace recovery : given a collection of m points in R, if many but not necessarily all of these points are contained in a d-dimensional subspace T can we find it? The points contained in T are called inliers and the remaining points are outliers. This problem has received considerable attention in computer science and in statistics. Yet efficient algorithms from computer science are not robust to adversarial outliers, and the estimators from robust statistics are hard to compute in high dimensions. This is a serious and persistent issue not just in this application, but for many other problems in unsupervised learning. Are there algorithms for subspace recovery that are both robust to outliers and efficient? We give an algorithm that finds T when it contains more than a d n fraction of the points. Hence, for say d = n 2 this estimator is both easy to compute and well-behaved when there are a constant fraction of outliers. We prove that it is small set expansion hard to find T when the fraction of errors is any larger, thus giving evidence that our estimator is an optimal compromise between efficiency and robustness. In fact, this basic problem has a surprising number of connections to other areas including small set expansion, matroid theory and functional analysis that we make use of here.