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.

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.

This paper introduces a new unsupervised method for dimensionality reduction via regression (DRR). The algorithm belongs to the family of invertible transforms that generalize Principal Component Analysis (PCA) by using curvilinear instead of linear features. DRR identifies the nonlinear features through multivariate regression to ensure the reduction in redundancy between he PCA coefficients, the reduction of the variance of the scores, and the reduction in the reconstruction error. More importantly, unlike other nonlinear dimensionality reduction methods, the invertibility, volume-preservation, and straightforward out-of-sample extension, makes DRR interpretable and easy to apply. The properties of DRR enable learning a more broader class of data manifolds than the recently proposed Non-linear Principal Components Analysis (NLPCA) and Principal Polynomial Analysis (PPA). We illustrate the performance of the representation in reducing the dimensionality of remote sensing data. In particular, we tackle two common problems: processing very high dimensional spectral information such as in hyperspectral image sounding data, and dealing with spatial-spectral image patches of multispectral images. Both settings pose collinearity and ill-determination problems. Evaluation of the expressive power of the features is assessed in terms of truncation error, estimating atmospheric variables, and surface land cover classification error. Results show that DRR outperforms linear PCA and recently proposed invertible extensions based on neural networks (NLPCA) and univariate regressions (PPA).

In this work we propose a new deep learning tool called deep dictionary learning. Multi-level dictionaries are learnt in a greedy fashion, one layer at a time. This requires solving a simple (shallow) dictionary learning problem, the solution to this is well known. We apply the proposed technique on some benchmark deep learning datasets. We compare our results with other deep learning tools like stacked autoencoder and deep belief network; and state of the art supervised dictionary learning tools like discriminative KSVD and label consistent KSVD. Our method yields better results than all.

A key question in modern statistics is how to make fast and reliable inferences for complex, high-dimensional data. While there has been much interest in sparse techniques, current methods do not generalize well to data with nonlinear structure. In this work, we present an orthogonal series estimator for predictors that are complex aggregate objects, such as natural images, galaxy spectra, trajectories, and movies. Our series approach ties together ideas from kernel machine learning, and Fourier methods. We expand the unknown regression on the data in terms of the eigenfunctions of a kernel-based operator, and we take advantage of orthogonality of the basis with respect to the underlying data distribution, P, to speed up computations and tuning of parameters. If the kernel is appropriately chosen, then the eigenfunctions adapt to the intrinsic geometry and dimension of the data. We provide theoretical guarantees for a radial kernel with varying bandwidth, and we relate smoothness of the regression function with respect to P to sparsity in the eigenbasis. Finally, using simulated and real-world data, we systematically compare the performance of the spectral series approach with classical kernel smoothing, k-nearest neighbors regression, kernel ridge regression, and state-of-the-art manifold and local regression methods.

In today's world, we follow news which is distributed globally. Significant events are reported by different sources and in different languages. In this work, we address the problem of tracking of events in a large multilingual stream. Within a recently developed system Event Registry we examine two aspects of this problem: how to compare articles in different languages and how to link collections of articles in different languages which refer to the same event. Taking a multilingual stream and clusters of articles from each language, we compare different cross-lingual document similarity measures based on Wikipedia. This allows us to compute the similarity of any two articles regardless of language. Building on previous work, we show there are methods which scale well and can compute a meaningful similarity between articles from languages with little or no direct overlap in the training data. Using this capability, we then propose an approach to link clusters of articles across languages which represent the same event. We provide an extensive evaluation of the system as a whole, as well as an evaluation of the quality and robustness of the similarity measure and the linking algorithm.

We present a novel framework for kernel learning with sequential data of any kind, such as time series, sequences of graphs, or strings. Our approach is based on signature features which can be seen as an ordered variant of sample (cross-)moments; it allows to obtain a "sequentialized" version of any static kernel. The sequential kernels are efficiently computable for discrete sequences and are shown to approximate a continuous moment form in a sampling sense. A number of known kernels for sequences arise as "sequentializations" of suitable static kernels: string kernels may be obtained as a special case, and alignment kernels are closely related up to a modification that resolves their open non-definiteness issue. Our experiments indicate that our signature-based sequential kernel framework may be a promising approach to learning with sequential data, such as time series, that allows to avoid extensive manual pre-processing.

Diffusion magnetic resonance imaging (dMRI) and tractography provide means to study the anatomical structures within the white matter of the brain. When studying tractography data across subjects, it is usually necessary to align, i.e. to register, tractographies together. This registration step is most often performed by applying the transformation resulting from the registration of other volumetric images (T1, FA). In contrast with registration methods that "transform" tractographies, in this work, we try to find which streamline in one tractography correspond to which streamline in the other tractography, without any transformation. In other words, we try to find a "mapping" between the tractographies. We propose a graph-based solution for the tractography mapping problem and we explain similarities and differences with the related well-known graph matching problem. Specifically, we define a loss function based on the pairwise streamline distance and reformulate the mapping problem as combinatorial optimization of that loss function. We show preliminary promising results where we compare the proposed method, implemented with simulated annealing, against a standard registration techniques in a task of segmentation of the corticospinal tract.

We briefly review recent progress in techniques for modeling and analyzing hyperspectral images and movies, in particular for detecting plumes of both known and unknown chemicals. For detecting chemicals of known spectrum, we extend the technique of using a single subspace for modeling the background to a "mixture of subspaces" model to tackle more complicated background. Furthermore, we use partial least squares regression on a resampled training set to boost performance. For the detection of unknown chemicals we view the problem as an anomaly detection problem, and use novel estimators with low-sampled complexity for intrinsically low-dimensional data in high-dimensions that enable us to model the "normal" spectra and detect anomalies. We apply these algorithms to benchmark data sets made available by the Automated Target Detection program co-funded by NSF, DTRA and NGA, and compare, when applicable, to current state-of-the-art algorithms, with favorable results.

This paper proposes a decision support system to aid movie investment decisions at the early stage of movie productions. The system predicts the success of a movie based on its profitability by leveraging historical data from various sources. Using social network analysis and text mining techniques, the system automatically extracts several groups of features, including "who" are on the cast, "what" a movie is about, "when" a movie will be released, as well as "hybrid" features that match "who" with "what", and "when" with "what". Experiment results with movies during an 11-year period showed that the system outperforms benchmark methods by a large margin in predicting movie profitability. Novel features we proposed also made great contributions to the prediction. In addition to designing a decision support system with practical utilities, our analysis of key factors for movie profitability may also have implications for theoretical research on team performance and the success of creative work.