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

Robust subspace learning for static and dynamic affect and behaviour modelling

Machine analysis of human affect and behavior in naturalistic contexts has witnessed a growing attention in the last decade from various disciplines ranging from social and cognitive sciences to machine learning and computer vision. Endowing machines with the ability to seamlessly detect, analyze, model, predict as well as simulate and synthesize manifestations of internal emotional and behavioral states in real-world data is deemed essential for the deployment of next-generation, emotionally- and socially-competent human-centered interfaces. In this thesis, we are primarily motivated by the problem of modeling, recognizing and predicting spontaneous expressions of non-verbal human affect and behavior manifested through either low-level facial attributes in static images or high-level semantic events in image sequences. Both visual data and annotations of naturalistic affect and behavior naturally contain noisy measurements of unbounded magnitude at random locations, commonly referred to as ‘outliers’. We present here machine learning methods that are robust to such gross, sparse noise. First, we deal with static analysis of face images, viewing the latter as a superposition of mutually-incoherent, low-complexity components corresponding to facial attributes, such as facial identity, expressions and activation of atomic facial muscle actions. We develop a robust, discriminant dictionary learning framework to extract these components from grossly corrupted training data and combine it with sparse representation to recognize the associated attributes. We demonstrate that our framework can jointly address interrelated classification tasks such as face and facial expression recognition. Inspired by the well-documented importance of the temporal aspect in perceiving affect and behavior, we direct the bulk of our research efforts into continuous-time modeling of dimensional affect and social behavior. Having identified a gap in the literature which is the lack of data containing annotations of social attitudes in continuous time and scale, we first curate a new audio-visual database of multi-party conversations from political debates annotated frame-by-frame in terms of real-valued conflict intensity and use it to conduct the first study on continuous-time conflict intensity estimation. Our experimental findings corroborate previous evidence indicating the inability of existing classifiers in capturing the hidden temporal structures of affective and behavioral displays. We present here a novel dynamic behavior analysis framework which models temporal dynamics in an explicit way, based on the natural assumption that continuous- time annotations of smoothly-varying affect or behavior can be viewed as outputs of a low-complexity linear dynamical system when behavioral cues (features) act as system inputs. A novel robust structured rank minimization framework is proposed to estimate the system parameters in the presence of gross corruptions and partially missing data. Experiments on prediction of dimensional conflict and affect as well as multi-object tracking from detection validate the effectiveness of our predictive framework and demonstrate that for the first time that complex human behavior and affect can be learned and predicted based on small training sets of person(s)-specific observations.Open Acces

Robust PCA by Manifold Optimization

Robust PCA is a widely used statistical procedure to recover a underlying low-rank matrix with grossly corrupted observations. This work considers the problem of robust PCA as a nonconvex optimization problem on the manifold of low-rank matrices, and proposes two algorithms (for two versions of retractions) based on manifold optimization. It is shown that, with a proper designed initialization, the proposed algorithms are guaranteed to converge to the underlying low-rank matrix linearly. Compared with a previous work based on the Burer-Monterio decomposition of low-rank matrices, the proposed algorithms reduce the dependence on the conditional number of the underlying low-rank matrix theoretically. Simulations and real data examples confirm the competitive performance of our method

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