Xingguo Li

Identifying Outliers in Large Matrices via Randomized Adaptive Compressive Sampling

This paper examines the problem of locating outlier columns in a large, otherwise low-rank, matrix. We propose a simple two-step adaptive sensing and inference approach and establish theoretical guarantees for its performance; our results show that accurate outlier identification is achievable using very few linear summaries of the original data matrix-as few as the squared rank of the low-rank component plus the number of outliers, times constant and logarithmic factors. We demonstrate the performance of our approach experimentally in two stylized applications, one motivated by robust collaborative filtering tasks, and the other by saliency map estimation tasks arising in computer vision and automated surveillance, and also investigate extensions to settings where the data are noisy, or possibly incomplete.

Outlier identification via randomized adaptive compressive sampling

This paper examines the problem of locating outlier columns in a large, otherwise low-rank, matrix. We propose a simple two-step adaptive sensing and inference approach and establish theoretical guarantees for its performance. Our results show that accurate outlier identification is achievable using very few linear summaries of the original data matrix - as few as the squared rank of the low-rank component plus the number of outliers, times constant and logarithmic factors. We demonstrate the performance of our approach experimentally in two stylized applications, one motivated by robust collaborative filtering tasks, and the other by saliency map estimation tasks arising in computer vision and automated surveillance.

Locating salient group-structured image features via adaptive compressive sensing

In this paper we consider the task of locating salient group-structured features in potentially high-dimensional images; the salient feature detection here is modeled as a Robust Principal Component Analysis problem, in which the aim is to locate groups of outlier columns embedded in an otherwise low rank matrix. We adapt an adaptive compressive sensing method from our own previous work (which examined the task of identifying arbitrary sets of outlier columns in large matrices) to settings where the outlier columns occur in groups, and establish theoretical results certifying that accurate group-structured inference is achievable using very few linear measurements of the image, subject to some (arguably) minor structural assumptions on the image itself. We also demonstrate, through extensive numerical simulations, our proposed algorithm in a salient object detection task, and show that it simultaneously achieves low sample and computational complexity, while exhibiting performance comparable to state-of-the-art methods that acquire and process the entire image.

A refined analysis for the sample complexity of adaptive compressive outlier sensing

The Adaptive Compressive Outlier Sensing (ACOS) method, proposed recently in (Li & Haupt, 2015), is a randomized sequential sampling and inference method designed to locate column outliers in large, otherwise low rank, matrices. While the original ACOS established conditions on the sample complexity (i.e., the number of scalar linear measurements) sufficient to enable accurate outlier localization (with high probability), the guarantees required a minimum sample complexity that grew linearly (albeit slowly) in the number of matrix columns. This work presents a refined analysis of the sampling complexity of ACOS that overcomes this limitation; we show that the sample complexity of ACOS is sublinear in both of the matrix dimensions - on the order of the squared rank of the low-rank component plus the number of outliers, times constant and logarithmic factors.

A dictionary based generalization of robust PCA

We analyze the decomposition of a data matrix, assumed to be a superposition of a low-rank component and a component which is sparse in a known dictionary, using a convex demixing method. We provide a unified analysis, encompassing both undercomplete and overcomplete dictionary cases, and show that the constituent components can be successfully recovered under some relatively mild assumptions up to a certain global sparsity level. Further, we corroborate our theoretical results by presenting empirical evaluations in terms of phase transitions in rank and sparsity for various dictionary sizes.

Robust PCA via tensor outlier pursuit

In this paper, we study robust principal component analysis on tensors, in the setting where frame-wise outliers exist. We propose a convex formulation to decompose a tensor into a low rank component and a frame-wise sparse component. Theoretically, we guarantee that exact subspace recovery and outlier identification can be achieved under mild model assumptions. Compared with entry-wise outlier pursuit and naive matricization of tensors with frame-wise outliers, our approach can handle higher ranks and proportion of outliers. Extensive numerical evaluations are provided on both synthetic and real data to support our theory.