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

Penalized Composite Quasi-Likelihood for Ultrahigh-Dimensional Variable Selection

In high-dimensional model selection problems, penalized simple least-square approaches have been extensively used. This paper addresses the question of both robustness and efficiency of penalized model selection methods, and proposes a data-driven weighted linear combination of convex loss functions, together with weighted $L_1$-penalty. It is completely data-adaptive and does not require prior knowledge of the error distribution. The weighted $L_1$-penalty is used both to ensure the convexity of the penalty term and to ameliorate the bias caused by the $L_1$-penalty. In the setting with dimensionality much larger than the sample size, we establish a strong oracle property of the proposed method that possesses both the model selection consistency and estimation efficiency for the true non-zero coefficients. As specific examples, we introduce a robust method of composite L1-L2, and optimal composite quantile method and evaluate their performance in both simulated and real data examples

Computation of the Structured Singular Value via Moment LMI Relaxations

The Structured Singular Value (SSV) provides a powerful tool to test robust stability and performance of feedback systems subject to structured uncertainties. Unfortunately, computing the SSV is an NP-hard problem, and the polynomial-time algorithms available in the literature are only able to provide, except for some special cases, upper and lower bounds on the exact value of the SSV. In this work, we present a new algorithm to compute an upper bound on the SSV in case of mixed real/complex uncertainties. The underlying idea of the developed approach is to formulate the SSV computation as a (nonconvex) polynomial optimization problem, which is relaxed into a sequence of convex optimization problems through moment-based relaxation techniques. Two heuristics to compute a lower bound on the SSV are also discussed. The analyzed numerical examples show that the developed approach provides tighter bounds than the ones computed by the algorithms implemented in the Robust Control Toolbox in Matlab, and it provides, in most of the cases, coincident lower and upper bounds on the structured singular value

Stable Camera Motion Estimation Using Convex Programming

We study the inverse problem of estimating n locations $t_1, ..., t_n$ (up to global scale, translation and negation) in $R^d$ from noisy measurements of a subset of the (unsigned) pairwise lines that connect them, that is, from noisy measurements of $\pm (t_i - t_j)/\|t_i - t_j\|$ for some pairs (i,j) (where the signs are unknown). This problem is at the core of the structure from motion (SfM) problem in computer vision, where the $t_i$'s represent camera locations in $R^3$. The noiseless version of the problem, with exact line measurements, has been considered previously under the general title of parallel rigidity theory, mainly in order to characterize the conditions for unique realization of locations. For noisy pairwise line measurements, current methods tend to produce spurious solutions that are clustered around a few locations. This sensitivity of the location estimates is a well-known problem in SfM, especially for large, irregular collections of images. In this paper we introduce a semidefinite programming (SDP) formulation, specially tailored to overcome the clustering phenomenon. We further identify the implications of parallel rigidity theory for the location estimation problem to be well-posed, and prove exact (in the noiseless case) and stable location recovery results. We also formulate an alternating direction method to solve the resulting semidefinite program, and provide a distributed version of our formulation for large numbers of locations. Specifically for the camera location estimation problem, we formulate a pairwise line estimation method based on robust camera orientation and subspace estimation. Lastly, we demonstrate the utility of our algorithm through experiments on real images.Comment: 40 pages, 12 figures, 6 tables; notation and some unclear parts updated, some typos correcte

Image Segmentation with Eigenfunctions of an Anisotropic Diffusion Operator

We propose the eigenvalue problem of an anisotropic diffusion operator for image segmentation. The diffusion matrix is defined based on the input image. The eigenfunctions and the projection of the input image in some eigenspace capture key features of the input image. An important property of the model is that for many input images, the first few eigenfunctions are close to being piecewise constant, which makes them useful as the basis for a variety of applications such as image segmentation and edge detection. The eigenvalue problem is shown to be related to the algebraic eigenvalue problems resulting from several commonly used discrete spectral clustering models. The relation provides a better understanding and helps developing more efficient numerical implementation and rigorous numerical analysis for discrete spectral segmentation methods. The new continuous model is also different from energy-minimization methods such as geodesic active contour in that no initial guess is required for in the current model. The multi-scale feature is a natural consequence of the anisotropic diffusion operator so there is no need to solve the eigenvalue problem at multiple levels. A numerical implementation based on a finite element method with an anisotropic mesh adaptation strategy is presented. It is shown that the numerical scheme gives much more accurate results on eigenfunctions than uniform meshes. Several interesting features of the model are examined in numerical examples and possible applications are discussed

Computation of the Structured Singular Value via Moment LMI Relaxations

The Structured Singular Value (SSV) provides a powerful tool to test robust stability and performance of feedback systems subject to structured uncertainties. Unfortunately, computing the SSV is an NP-hard problem, and the polynomial-time algorithms available in the literature are only able to provide, except for some special cases, upper and lower bounds on the exact value of the SSV. In this work, we present a new algorithm to compute an upper bound on the SSV in case of mixed real/complex uncertainties. The underlying idea of the developed approach is to formulate the SSV computation as a (nonconvex) polynomial optimization problem, which is relaxed into a sequence of convex optimization problems through moment-based relaxation techniques. Two heuristics to compute a lower bound on the SSV are also discussed. The analyzed numerical examples show that the developed approach provides tighter bounds than the ones computed by the algorithms implemented in the Robust Control Toolbox in Matlab, and it provides, in most of the cases, coincident lower and upper bounds on the structured singular value