Abstract: Robust diffusion adaptive estimation algorithms based on the maximum correntropy criterion (MCC), including adaptation to combination MCC and combination to adaptation MCC, are developed to deal with the distributed estimation over network in impulsive (long-tailed) noise environments. The cost functions used in distributed estimation are in general based on the mean square error (MSE) criterion, which is desirable when the measurement noise is Gaussian. In non-Gaussian situations, such as the impulsive-noise case, MCC based methods may achieve much better performance than the MSE methods as they take into account higher order statistics of error distribution. The proposed methods can also outperform the robust diffusion least mean p-power(DLMP) and diffusion minimum error entropy (DMEE) algorithms. The mean and mean square convergence analysis of the new algorithms are also carried out.

With the increasing size of today's data sets, finding the right parameter configuration in model selection via cross-validation can be an extremely time-consuming task. In this paper we propose an improved cross-validation procedure which uses nonparametric testing coupled with sequential analysis to determine the best parameter set on linearly increasing subsets of the data. By eliminating underperforming candidates quickly and keeping promising candidates as long as possible, the method speeds up the computation while preserving the capability of the full cross-validation. Theoretical considerations underline the statistical power of our procedure. The experimental evaluation shows that our method reduces the computation time by a factor of up to 120 compared to a full cross-validation with a negligible impact on the accuracy.

Supervised statistical classification is a vital tool for satellite image processing. It is useful not only when a discrete result, such as feature extraction or surface type, is required, but also for continuum retrievals by dividing the quantity of interest into discrete ranges. Because of the high resolution of modern satellite instruments and because of the requirement for real-time processing, any algorithm has to be fast to be useful. Here we describe an algorithm based on kernel estimation called Adaptive Gaussian Filtering that incorporates several innovations to produce superior efficiency as compared to three other popular methods: k-nearest-neighbour (KNN), Learning Vector Quantization (LVQ) and Support Vector Machines (SVM). This efficiency is gained with no compromises: accuracy is maintained, while estimates of the conditional probabilities are returned. These are useful not only to gauge the accuracy of an estimate in the absence of its true value, but also to re-calibrate a retrieved image and as a proxy for a discretized continuum variable. The algorithm is demonstrated and compared with the other three on a pair of synthetic test classes and to map the waterways of the Netherlands. Software may be found at: http://libagf.sourceforge.net.

Cascades of information-sharing are a primary mechanism by which content reaches its audience on social media, and an active line of research has studied how such cascades, which form as content is reshared from person to person, develop and subside. In this paper, we perform a large-scale analysis of cascades on Facebook over significantly longer time scales, and find that a more complex picture emerges, in which many large cascades recur, exhibiting multiple bursts of popularity with periods of quiescence in between. We characterize recurrence by measuring the time elapsed between bursts, their overlap and proximity in the social network, and the diversity in the demographics of individuals participating in each peak. We discover that content virality, as revealed by its initial popularity, is a main driver of recurrence, with the availability of multiple copies of that content helping to spark new bursts. Still, beyond a certain popularity of content, the rate of recurrence drops as cascades start exhausting the population of interested individuals. We reproduce these observed patterns in a simple model of content recurrence simulated on a real social network. Using only characteristics of a cascade's initial burst, we demonstrate strong performance in predicting whether it will recur in the future.

In this paper we analyze approximate methods for undertaking a principal components analysis (PCA) on large data sets. PCA is a classical dimension reduction method that involves the projection of the data onto the subspace spanned by the leading eigenvectors of the covariance matrix. This projection can be used either for exploratory purposes or as an input for further analysis, e.g. regression. If the data have billions of entries or more, the computational and storage requirements for saving and manipulating the design matrix in fast memory is prohibitive. Recently, the Nystr\"om and column-sampling methods have appeared in the numerical linear algebra community for the randomized approximation of the singular value decomposition of large matrices. However, their utility for statistical applications remains unclear. We compare these approximations theoretically by bounding the distance between the induced subspaces and the desired, but computationally infeasible, PCA subspace. Additionally we show empirically, through simulations and a real data example involving a corpus of emails, the trade-off of approximation accuracy and computational complexity.

Traditionally, practitioners initialize the {\tt k-means} algorithm with centers chosen uniformly at random. Randomized initialization with uneven weights ({\tt k-means++}) has recently been used to improve the performance over this strategy in cost and run-time. We consider the k-means problem with semi-supervised information, where some of the data are pre-labeled, and we seek to label the rest according to the minimum cost solution. By extending the {\tt k-means++} algorithm and analysis to account for the labels, we derive an improved theoretical bound on expected cost and observe improved performance in simulated and real data examples. This analysis provides theoretical justification for a roughly linear semi-supervised clustering algorithm.

Mechanisms of human color vision are characterized by two phenomenological aspects: the system is nonlinear and adaptive to changing environments. Conventional attempts to derive these features from statistics use separate arguments for each aspect. The few statistical approaches that do consider both phenomena simultaneously follow parametric formulations based on empirical models. Therefore, it may be argued that the behavior does not come directly from the color statistics but from the convenient functional form adopted. In addition, many times the whole statistical analysis is based on simplified databases that disregard relevant physical effects in the input signal, as for instance by assuming flat Lambertian surfaces. Here we address the simultaneous statistical explanation of (i) the nonlinear behavior of achromatic and chromatic mechanisms in a fixed adaptation state, and (ii) the change of such behavior. Both phenomena emerge directly from the samples through a single data-driven method: the Sequential Principal Curves Analysis (SPCA) with local metric. SPCA is a new manifold learning technique to derive a set of sensors adapted to the manifold using different optimality criteria. A new database of colorimetrically calibrated images of natural objects under these illuminants was collected. The results obtained by applying SPCA show that the psychophysical behavior on color discrimination thresholds, discount of the illuminant and corresponding pairs in asymmetric color matching, emerge directly from realistic data regularities assuming no a priori functional form. These results provide stronger evidence for the hypothesis of a statistically driven organization of color sensors. Moreover, the obtained results suggest that color perception at this low abstraction level may be guided by an error minimization strategy rather than by the information maximization principle.

Most signal processing problems involve the challenging task of multidimensional probability density function (PDF) estimation. In this work, we propose a solution to this problem by using a family of Rotation-based Iterative Gaussianization (RBIG) transforms. The general framework consists of the sequential application of a univariate marginal Gaussianization transform followed by an orthonormal transform. The proposed procedure looks for differentiable transforms to a known PDF so that the unknown PDF can be estimated at any point of the original domain. In particular, we aim at a zero mean unit covariance Gaussian for convenience. RBIG is formally similar to classical iterative Projection Pursuit (PP) algorithms. However, we show that, unlike in PP methods, the particular class of rotations used has no special qualitative relevance in this context, since looking for interestingness is not a critical issue for PDF estimation. The key difference is that our approach focuses on the univariate part (marginal Gaussianization) of the problem rather than on the multivariate part (rotation). This difference implies that one may select the most convenient rotation suited to each practical application. The differentiability, invertibility and convergence of RBIG are theoretically and experimentally analyzed. Relation to other methods, such as Radial Gaussianization (RG), one-class support vector domain description (SVDD), and deep neural networks (DNN) is also pointed out. The practical performance of RBIG is successfully illustrated in a number of multidimensional problems such as image synthesis, classification, denoising, and multi-information estimation.

This paper presents a new framework for manifold learning based on a sequence of principal polynomials that capture the possibly nonlinear nature of the data. The proposed Principal Polynomial Analysis (PPA) generalizes PCA by modeling the directions of maximal variance by means of curves, instead of straight lines. Contrarily to previous approaches, PPA reduces to performing simple univariate regressions, which makes it computationally feasible and robust. Moreover, PPA shows a number of interesting analytical properties. First, PPA is a volume-preserving map, which in turn guarantees the existence of the inverse. Second, such an inverse can be obtained in closed form. Invertibility is an important advantage over other learning methods, because it permits to understand the identified features in the input domain where the data has physical meaning. Moreover, it allows to evaluate the performance of dimensionality reduction in sensible (input-domain) units. Volume preservation also allows an easy computation of information theoretic quantities, such as the reduction in multi-information after the transform. Third, the analytical nature of PPA leads to a clear geometrical interpretation of the manifold: it allows the computation of Frenet-Serret frames (local features) and of generalized curvatures at any point of the space. And fourth, the analytical Jacobian allows the computation of the metric induced by the data, thus generalizing the Mahalanobis distance. These properties are demonstrated theoretically and illustrated experimentally. The performance of PPA is evaluated in dimensionality and redundancy reduction, in both synthetic and real datasets from the UCI repository.

A successful class of image denoising methods is based on Bayesian approaches working in wavelet representations. However, analytical estimates can be obtained only for particular combinations of analytical models of signal and noise, thus precluding its straightforward extension to deal with other arbitrary noise sources. In this paper, we propose an alternative non-explicit way to take into account the relations among natural image wavelet coefficients for denoising: we use support vector regression (SVR) in the wavelet domain to enforce these relations in the estimated signal. Since relations among the coefficients are specific to the signal, the regularization property of SVR is exploited to remove the noise, which does not share this feature. The specific signal relations are encoded in an anisotropic kernel obtained from mutual information measures computed on a representative image database. Training considers minimizing the Kullback-Leibler divergence (KLD) between the estimated and actual probability functions of signal and noise in order to enforce similarity. Due to its non-parametric nature, the method can eventually cope with different noise sources without the need of an explicit re-formulation, as it is strictly necessary under parametric Bayesian formalisms. Results under several noise levels and noise sources show that: (1) the proposed method outperforms conventional wavelet methods that assume coefficient independence, (2) it is similar to state-of-the-art methods that do explicitly include these relations when the noise source is Gaussian, and (3) it gives better numerical and visual performance when more complex, realistic noise sources are considered. Therefore, the proposed machine learning approach can be seen as a more flexible (model-free) alternative to the explicit description of wavelet coefficient relations for image denoising.