Contamination Estimation via Convex Relaxations

Identifying anomalies and contamination in datasets is important in a wide variety of settings. In this paper, we describe a new technique for estimating contamination in large, discrete valued datasets. Our approach considers the normal condition of the data to be specified by a model consisting of a set of distributions. Our key contribution is in our approach to contamination estimation. Specifically, we develop a technique that identifies the minimum number of data points that must be discarded (i.e., the level of contamination) from an empirical data set in order to match the model to within a specified goodness-of-fit, controlled by a p-value. Appealing to results from large deviations theory, we show a lower bound on the level of contamination is obtained by solving a series of convex programs. Theoretical results guarantee the bound converges at a rate of $O(\sqrt{\log(p)/p})$, where p is the size of the empirical data set.Comment: To appear, ISIT 201

SOCP relaxation bounds for the optimal subset selection problem applied to robust linear regression

This paper deals with the problem of finding the globally optimal subset of h elements from a larger set of n elements in d space dimensions so as to minimize a quadratic criterion, with an special emphasis on applications to computing the Least Trimmed Squares Estimator (LTSE) for robust regression. The computation of the LTSE is a challenging subset selection problem involving a nonlinear program with continuous and binary variables, linked in a highly nonlinear fashion. The selection of a globally optimal subset using the branch and bound (BB) algorithm is limited to problems in very low dimension, tipically d<5, as the complexity of the problem increases exponentially with d. We introduce a bold pruning strategy in the BB algorithm that results in a significant reduction in computing time, at the price of a negligeable accuracy lost. The novelty of our algorithm is that the bounds at nodes of the BB tree come from pseudo-convexifications derived using a linearization technique with approximate bounds for the nonlinear terms. The approximate bounds are computed solving an auxiliary semidefinite optimization problem. We show through a computational study that our algorithm performs well in a wide set of the most difficult instances of the LTSE problem.Comment: 12 pages, 3 figures, 2 table

High Dimensional Semiparametric Scale-Invariant Principal Component Analysis

We propose a new high dimensional semiparametric principal component analysis (PCA) method, named Copula Component Analysis (COCA). The semiparametric model assumes that, after unspecified marginally monotone transformations, the distributions are multivariate Gaussian. COCA improves upon PCA and sparse PCA in three aspects: (i) It is robust to modeling assumptions; (ii) It is robust to outliers and data contamination; (iii) It is scale-invariant and yields more interpretable results. We prove that the COCA estimators obtain fast estimation rates and are feature selection consistent when the dimension is nearly exponentially large relative to the sample size. Careful experiments confirm that COCA outperforms sparse PCA on both synthetic and real-world datasets.Comment: Accepted in IEEE Transactions on Pattern Analysis and Machine Intelligence (TPMAI