Estimation of Subspace Arrangements

In the literature of computer vision and image §processing, a fundamental problem in modeling visual §data is that multivariate image or video data tend to §be heterogeneous or multimodal. That is, subsets of §the data may have significantly different geometric §or statistical properties. Recently, subspace §arrangements have become an increasingly popular §class of mathematical objects to use for modeling §multivariate mixed data that are (approximately) §piecewise linear. A subspace arrangement is a union §of multiple subspaces. Each subspace can be used to §model a homogeneous subset of the data. In this work, §we study the problem of segmenting subspace §arrangements. Built on past extensive study of §subspace arrangements in algebraic geometry, we §propose a principled framework that summarizes §important algebraic properties and statistical facts §that are crucial for making the inference of subspace §arrangement models both efficient and robust, even §when the given data are corrupted with noise and/or §contaminated by outliers. The new solutions in many §ways improve and generalize extant methods for §modeling or clustering mixed data.