This tutorial will give an overview of robust methods for some structured data problems. We will set up the problem with a review of basic ideas in robust statistics. In these problems, one is concerned with estimating a given quantity from data when part of it is corrupted or when some assumptions are not met. In particular, we will look at the problems of robustly estimating means and covariances of datasets. We will then move on to the problems of Robust Subspace Recovery and Robust Principal Component Analysis, which are particularly useful for dimension reduction and outlier filtering for high-dimensional data. In each of these cases, one assumes that a dataset exhibits some form of low-dimensional subspace structure. The goal is to recover these low-dimensional subspaces from corrupted data. We will show that it is possible to leverage the geometry of the space of subspaces to develop efficient algorithms for these problems. We will finish by indicating how these ideas can be applied to other structured data problems, such as the problem of angular synchronization.
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