Robust Kronecker-decomposable component analysis for low-rank modeling

Dictionary learning and component analysis are part of one of the most well-studied and active research fields, at the intersection of signal and image processing, computer vision, and statistical machine learning. In dictionary learning, the current methods of choice are arguably K-SVD and its variants, which learn a dictionary (i.e., a decomposition) for sparse coding via Singular Value Decomposition. In robust component analysis, leading methods derive from Principal Component Pursuit (PCP), which recovers a low-rank matrix from sparse corruptions of unknown magnitude and support. However, K-SVD is sensitive to the presence of noise and outliers in the training set. Additionally, PCP does not provide a dictionary that respects the structure of the data (e.g., images), and requires expensive SVD computations when solved by convex relaxation. In this paper, we introduce a new robust decomposition of images by combining ideas from sparse dictionary learning and PCP. We propose a novel Kronecker-decomposable component analysis which is robust to gross corruption, can be used for low-rank modeling, and leverages separability to solve significantly smaller problems. We design an efficient learning algorithm by drawing links with a restricted form of tensor factorization. The effectiveness of the proposed approach is demonstrated on real-world applications, namely background subtraction and image denoising, by performing a thorough comparison with the current state of the art

Robust Kronecker-decomposable component analysis for low-rank modeling

Dictionary learning and component analysis are part of one of the most well-studied and active research fields, at the intersection of signal and image processing, computer vision, and statistical machine learning. In dictionary learning, the current methods of choice are arguably K-SVD and its variants, which learn a dictionary (i.e., a decomposition) for sparse coding via Singular Value Decomposition. In robust component analysis, leading methods derive from Principal Component Pursuit (PCP), which recovers a low-rank matrix from sparse corruptions of unknown magnitude and support. However, K-SVD is sensitive to the presence of noise and outliers in the training set. Additionally, PCP does not provide a dictionary that respects the structure of the data (e.g., images), and requires expensive SVD computations when solved by convex relaxation. In this paper, we introduce a new robust decomposition of images by combining ideas from sparse dictionary learning and PCP. We propose a novel Kronecker-decomposable component analysis which is robust to gross corruption, can be used for low-rank modeling, and leverages separability to solve significantly smaller problems. We design an efficient learning algorithm by drawing links with a restricted form of tensor factorization. The effectiveness of the proposed approach is demonstrated on real-world applications, namely background subtraction and image denoising, by performing a thorough comparison with the current state of the art

Load curve data cleansing and imputation via sparsity and low rank

The smart grid vision is to build an intelligent power network with an unprecedented level of situational awareness and controllability over its services and infrastructure. This paper advocates statistical inference methods to robustify power monitoring tasks against the outlier effects owing to faulty readings and malicious attacks, as well as against missing data due to privacy concerns and communication errors. In this context, a novel load cleansing and imputation scheme is developed leveraging the low intrinsic-dimensionality of spatiotemporal load profiles and the sparse nature of "bad data.'' A robust estimator based on principal components pursuit (PCP) is adopted, which effects a twofold sparsity-promoting regularization through an $\ell_1$-norm of the outliers, and the nuclear norm of the nominal load profiles. Upon recasting the non-separable nuclear norm into a form amenable to decentralized optimization, a distributed (D-) PCP algorithm is developed to carry out the imputation and cleansing tasks using networked devices comprising the so-termed advanced metering infrastructure. If D-PCP converges and a qualification inequality is satisfied, the novel distributed estimator provably attains the performance of its centralized PCP counterpart, which has access to all networkwide data. Computer simulations and tests with real load curve data corroborate the convergence and effectiveness of the novel D-PCP algorithm.Comment: 8 figures, submitted to IEEE Transactions on Smart Grid - Special issue on "Optimization methods and algorithms applied to smart grid

Foundational principles for large scale inference: Illustrations through correlation mining

When can reliable inference be drawn in the "Big Data" context? This paper presents a framework for answering this fundamental question in the context of correlation mining, with implications for general large scale inference. In large scale data applications like genomics, connectomics, and eco-informatics the dataset is often variable-rich but sample-starved: a regime where the number $n$ of acquired samples (statistical replicates) is far fewer than the number $p$ of observed variables (genes, neurons, voxels, or chemical constituents). Much of recent work has focused on understanding the computational complexity of proposed methods for "Big Data." Sample complexity however has received relatively less attention, especially in the setting when the sample size $n$ is fixed, and the dimension $p$ grows without bound. To address this gap, we develop a unified statistical framework that explicitly quantifies the sample complexity of various inferential tasks. Sampling regimes can be divided into several categories: 1) the classical asymptotic regime where the variable dimension is fixed and the sample size goes to infinity; 2) the mixed asymptotic regime where both variable dimension and sample size go to infinity at comparable rates; 3) the purely high dimensional asymptotic regime where the variable dimension goes to infinity and the sample size is fixed. Each regime has its niche but only the latter regime applies to exa-scale data dimension. We illustrate this high dimensional framework for the problem of correlation mining, where it is the matrix of pairwise and partial correlations among the variables that are of interest. We demonstrate various regimes of correlation mining based on the unifying perspective of high dimensional learning rates and sample complexity for different structured covariance models and different inference tasks