Robust Recovery of Subspace Structures by Low-Rank Representation

In this work we address the subspace recovery problem. Given a set of data samples (vectors) approximately drawn from a union of multiple subspaces, our goal is to segment the samples into their respective subspaces and correct the possible errors as well. To this end, we propose a novel method termed Low-Rank Representation (LRR), which seeks the lowest-rank representation among all the candidates that can represent the data samples as linear combinations of the bases in a given dictionary. It is shown that LRR well solves the subspace recovery problem: when the data is clean, we prove that LRR exactly captures the true subspace structures; for the data contaminated by outliers, we prove that under certain conditions LRR can exactly recover the row space of the original data and detect the outlier as well; for the data corrupted by arbitrary errors, LRR can also approximately recover the row space with theoretical guarantees. Since the subspace membership is provably determined by the row space, these further imply that LRR can perform robust subspace segmentation and error correction, in an efficient way.Comment: IEEE Trans. Pattern Analysis and Machine Intelligenc

Hilbert Functions and Applications to the Estimation of Subspace Arrangements

This paper develops a new mathematical framework for studying the subspace-segmentation problem. We examine some important algebraic properties of subspace arrangements that are closely related to the subspace-segmentation problem. More specifically, we introduce an important class of invariants given by the Hilbert functions. We show that there exist rich relations between subspace arrangements and their corresponding Hilbert functions. We propose a new subspace- segmentation algorithm, and showcase two applications to demonstrate how the new theoretical revelation may solve subspace segmentation and model selection problems under less restrictive conditions with improved results.National Science Foundation / CAREER IIS-0347456, CRS-EHS-0509151, and CCF-TF-0514955ONR YIP N00014-05-1-0633Ope

Estimation of Subspace Arrangements with Applications in Modeling and Segmenting Mixed Data

In recent years, subspace arrangements have become an increasingly popular class of mathematical objects to be used for modeling a multivariate mixed data set that is (approximately) piecewise linear. A subspace arrangement is a union of multiple subspaces. Each subspace can be conveniently used to model a homogeneous subset of the data. Hence, all the subspaces together can capture the heterogeneous structures within the data set. In this paper, we give a comprehensive introduction to one new approach for the estimation of subspace arrangements, known as generalized principal component analysis. We provide a comprehensive summary of important algebraic properties and statistical facts that are crucial for making the inference of subspace arrangements both efficient and robust, even when the given data are corrupted with noise or contaminated by outliers. This new method in many ways improves and generalizes extant methods for modeling or clustering mixed data. There have been successful applications of this new method to many real-world problems in computer vision, image processing, and system identification. In this paper, we will examine a couple of those representative applications.National Science Foundation / NSF CAREER IIS-0347456, NSF CRS-EHS-0509151, NSF CCF-TF-0514955, and NSF CAREER DMS-034901ONR YIP N00014-05-1-0633Ope

Beyond pairwise clustering

We consider the problem of clustering in domains where the affinity relations are not dyadic (pairwise), but rather triadic, tetradic or higher. The problem is an instance of the hypergraph partitioning problem. We propose a two-step algorithm for solving this problem. In the first step we use a novel scheme to approximate the hypergraph using a weighted graph. In the second step a spectral partitioning algorithm is used to partition the vertices of this graph. The algorithm is capable of handling hyperedges of all orders including order two, thus incorporating information of all orders simultaneously. We present a theoretical analysis that relates our algorithm to an existing hypergraph partitioning algorithm and explain the reasons for its superior performance. We report the performance of our algorithm on a variety of computer vision problems and compare it to several existing hypergraph partitioning algorithms

Recovering articulated non-rigid shapes, motions and kinematic chains from video

Recovering articulated shape and motion, especially human body motion, from video is a challenging problem with a wide range of applications in medical study, sport analysis and animation, etc. Previous work on articulated motion recovery generally requires prior knowledge of the kinematic chain and usually does not concern the recovery of the articulated shape. The non-rigidity of some articulated part, e.g. human body motion with non-rigid facial motion, is completely ignored. We propose a factorization-based approach to recover the shape, motion and kinematic chain of an articulated object with non-rigid parts altogether directly from video sequences under a unified framework. The proposed approach is based on our modeling of the articulated non-rigid motion as a set of intersecting motion subspaces. A motion subspace is the linear subspace of the trajectories of an object. It can model a rigid or non-rigid motion. The intersection of two motion subspaces of linked parts models the motion of an articulated joint or axis. Our approach consists of algorithms for motion segmentation, kinematic chain building, and shape recovery. It is robust to outliers and can be automated. We test our approach through synthetic and real experiments and demonstrate how to recover articulated structure with non-rigid parts via a single-view camera without prior knowledge of its kinematic chain

A Grouped Factor Model

In this paper we present a grouped factor model that is designed to explore grouped structures in factor models. We develop an econometric theory consisting of a consistent classification rule to assign variables to their respective groups and a class of consistent model selection criteria to determine the number of groups as well as the number of factors in each group. As a result, we propose a procedure to estimate grouped factor models, in which the unknown number of groups, the unknown relationship between variables to their groups as well as the unknown number of factors in each group are statistically determined based on observed data. The procedure can help to estimate common factor that are pervasive across all groups and group-specific factors that are pervasive only in the respective groups. Simulations show that our proposed estimation procedure has satisfactory finite sample properties