Principal component analysis is a fundamental operation in computational data analysis, with myriad applications ranging from web search to bioinformatics to computer vision and image analysis. However, its performance and applicability in real scenarios are limited by a lack of robustness to outlying or corrupted observations. This paper considers the idealized "robust principal component analysis" problem of recovering a low rank matrix A from corrupted observations D A E. Here, the error entries E can be arbitrarily large (modeling grossly corrupted observations common in visual and bioinformatic data), but are assumed to be sparse. We prove that most matrices A can be efficiently and exactly recovered from most error sign-and-support patterns, by solving a simple convex program. Our result holds even when the rank of A grows nearly proportionally (up to a logarithmic factor) to the dimensionality of the observation space and the number of errors E grows in proportion to the total number of entries in the matrix.

Second-order maximum-entropy models have recently gained much interest for describing the statistics of binary spike trains. Here, we extend this approach to take continuous stimuli into account as well. By constraining the joint second-order statistics, we obtain a joint Gaussian-Boltzmann distribution of continuous stimuli and binary neural firing patterns, for which we also compute marginal and conditional distributions. This model has the same computational complexity as pure binary models and fitting it to data is a convex problem. We show that the model can be seen as an extension to the classical spike-triggered average/covariance analysis and can be used as a non-linear method for extracting features which a neural population is sensitive to.

Speaker comparison, the process of finding the speaker similarity between two speech signals, occupies a central role in a variety of applications---speaker verification, clustering, and identification. Speaker comparison can be placed in a geometric framework by casting the problem as a model comparison process. For a given speech signal, feature vectors are produced and used to adapt a Gaussian mixture model (GMM). Speaker comparison can then be viewed as the process of compensating and finding metrics on the space of adapted models. We propose a framework, inner product discriminant functions (IPDFs), which extends many common techniques for speaker comparison: support vector machines, joint factor analysis, and linear scoring.

We present a novel and highly effective approach for multi-body motion segmentation. Drawing inspiration from robust statistical model fitting, we estimate putative subspace hypotheses from the data. However, instead of ranking them we encapsulate the hypotheses in a novel Mercer kernel which elicits the potential of two point trajectories to have emerged from the same subspace. The kernel permits the application of well-established statistical learning methods for effective outlier rejection, automatic recovery of the number of motions and accurate segmentation of the point trajectories. The method operates well under severe outliers arising from spurious trajectories or mistracks.

We are often interested in casting classification and clustering problems in a regression framework, because it is feasible to achieve some statistical properties in this framework by imposing some penalty criteria. In this paper we illustrate optimal scoring, which was originally proposed for performing Fisher linear discriminant analysis by regression, in the application of unsupervised learning. In particular, we devise a novel clustering algorithm that we call optimal discriminant clustering (ODC). Thus, our work shows that optimal scoring provides a new approach to the implementation of unsupervised learning.

The principles by which spiking neurons contribute to the astounding computational power of generic cortical microcircuits, and how spike-timing-dependent plasticity (STDP) of synaptic weights could generate and maintain this computational function, are unknown. We show here that STDP, in conjunction with a stochastic soft winner-take-all (WTA) circuit, induces spiking neurons to generate through their synaptic weights implicit internal models for subclasses (or causes") of the high-dimensional spike patterns of hundreds of pre-synaptic neurons. Hence these neurons will fire after learning whenever the current input best matches their internal model. The resulting computational function of soft WTA circuits, a common network motif of cortical microcircuits, could therefore be a drastic dimensionality reduction of information streams, together with the autonomous creation of internal models for the probability distributions of their input patterns. We show that the autonomous generation and maintenance of this computational function can be explained on the basis of rigorous mathematical principles. In particular, we show that STDP is able to approximate a stochastic online Expectation-Maximization (EM) algorithm for modeling the input data. A corresponding result is shown for Hebbian learning in artificial neural networks."

In this paper, we study the manifold regularization for the Sliced Inverse Regression (SIR). The manifold regularization improves the standard SIR in two aspects: 1) it encodes the local geometry for SIR and 2) it enables SIR to deal with transductive and semi-supervised learning problems. We prove that the proposed graph Laplacian based regularization is convergent at rate root-n. The projection directions of the regularized SIR are optimized by using a conjugate gradient method on the Grassmann manifold. Experimental results support our theory.

In this paper we make several contributions towards accelerating approximate Bayesian structural inference for non-decomposable GGMs. Our first contribution is to show how to efficiently compute a BIC or Laplace approximation to the marginal likelihood of non-decomposable graphs using convex methods for precision matrix estimation. This optimization technique can be used as a fast scoring function inside standard Stochastic Local Search (SLS) for generating posterior samples. Our second contribution is a novel framework for efficiently generating large sets of high-quality graph topologies without performing local search. This graph proposal method, which we call Neighborhood Fusion" (NF), samples candidate Markov blankets at each node using sparse regression techniques. Our final contribution is a hybrid method combining the complementary strengths of NF and SLS. Experimental results in structural recovery and prediction tasks demonstrate that NF and hybrid NF/SLS out-perform state-of-the-art local search methods, on both synthetic and real-world datasets, when realistic computational limits are imposed."

In this paper we present an algorithm for separating mixed sounds from a monophonic recording. Our approach makes use of training data which allows us to learn representations of the types of sounds that compose the mixture. In contrast to popular methods that attempt to extract com- pact generalizable models for each sound from training data, we employ the training data itself as a representation of the sources in the mixture. We show that mixtures of known sounds can be described as sparse com- binations of the training data itself, and in doing so produce significantly better separation results as compared to similar systems based on compact statistical models.

The concave-convex procedure (CCCP) is a majorization-minimization algorithm that solves d.c. In machine learning, CCCP is extensively used in many learning algorithms like sparse support vector machines (SVMs), transductive SVMs, sparse principal component analysis, etc. Though widely used in many applications, the convergence behavior of CCCP has not gotten a lot of specific attention. Yuille and Rangarajan analyzed its convergence in their original paper, however, we believe the analysis is not complete. Although the convergence of CCCP can be derived from the convergence of the d.c.