We study the problem of predicting the future, though only in the probabilistic sense of estimating a future state of a time-varying probability distribution. This is not only an interesting academic problem, but solving this extrapolation problem also has many practical application, e.g. for training classifiers that have to operate under time-varying conditions. Our main contribution is a method for predicting the next step of the time-varying distribution from a given sequence of sample sets from earlier time steps. For this we rely on two recent machine learning techniques: embedding probability distributions into a reproducing kernel Hilbert space, and learning operators by vector-valued regression. We illustrate the working principles and the practical usefulness of our method by experiments on synthetic and real data. We also highlight an exemplary application: training a classifier in a domain adaptation setting without having access to examples from the test time distribution at training time.

This paper proposed a new regression model called $l_1$-regularized outlier isolation and regression (LOIRE) and a fast algorithm based on block coordinate descent to solve this model. Besides, assuming outliers are gross errors following a Bernoulli process, this paper also presented a Bernoulli estimate model which, in theory, should be very accurate and robust due to its complete elimination of affections caused by outliers. Though this Bernoulli estimate is hard to solve, it could be approximately achieved through a process which takes LOIRE as an important intermediate step. As a result, the approximate Bernoulli estimate is a good combination of Bernoulli estimate's accuracy and LOIRE regression's efficiency with several simulations conducted to strongly verify this point. Moreover, LOIRE can be further extended to realize robust rank factorization which is powerful in recovering low-rank component from massive corruptions. Extensive experimental results showed that the proposed method outperforms state-of-the-art methods like RPCA and GoDec in the aspect of computation speed with a competitive performance.

Data association, the problem of reasoning over correspondence between targets and measurements, is a fundamental problem in tracking. This paper presents a graphical model formulation of data association and applies an approximate inference method, belief propagation (BP), to obtain estimates of marginal association probabilities. We prove that BP is guaranteed to converge, and bound the number of iterations necessary. Experiments reveal a favourable comparison to prior methods in terms of accuracy and computational complexity.

Model-based methods and deep neural networks have both been tremendously successful paradigms in machine learning. In model-based methods, problem domain knowledge can be built into the constraints of the model, typically at the expense of difficulties during inference. In contrast, deterministic deep neural networks are constructed in such a way that inference is straightforward, but their architectures are generic and it is unclear how to incorporate knowledge. This work aims to obtain the advantages of both approaches. To do so, we start with a model-based approach and an associated inference algorithm, and \emph{unfold} the inference iterations as layers in a deep network. Rather than optimizing the original model, we \emph{untie} the model parameters across layers, in order to create a more powerful network. The resulting architecture can be trained discriminatively to perform accurate inference within a fixed network size. We show how this framework allows us to interpret conventional networks as mean-field inference in Markov random fields, and to obtain new architectures by instead using belief propagation as the inference algorithm. We then show its application to a non-negative matrix factorization model that incorporates the problem-domain knowledge that sound sources are additive. Deep unfolding of this model yields a new kind of non-negative deep neural network, that can be trained using a multiplicative backpropagation-style update algorithm. We present speech enhancement experiments showing that our approach is competitive with conventional neural networks despite using far fewer parameters.

Binary classification is a common statistical learning problem in which a model is estimated on a set of covariates for some outcome indicating the membership of one of two classes. In the literature, there exists a distinction between hard and soft classification. In soft classification, the conditional class probability is modeled as a function of the covariates. In contrast, hard classification methods only target the optimal prediction boundary. While hard and soft classification methods have been studied extensively, not much work has been done to compare the actual tasks of hard and soft classification. In this paper we propose a spectrum of statistical learning problems which span the hard and soft classification tasks based on fitting multiple decision rules to the data. By doing so, we reveal a novel collection of learning tasks of increasing complexity. We study the problems using the framework of large-margin classifiers and a class of piecewise linear convex surrogates, for which we derive statistical properties and a corresponding sub-gradient descent algorithm. We conclude by applying our approach to simulation settings and a magnetic resonance imaging (MRI) dataset from the Alzheimer's Disease Neuroimaging Initiative (ADNI) study.

Correlation filters (CFs) are a class of classifiers that are attractive for object localization and tracking applications. Traditionally, CFs have been designed in the frequency domain using the discrete Fourier transform (DFT), where correlation is efficiently implemented. However, existing CF designs do not account for the fact that the multiplication of two DFTs in the frequency domain corresponds to a circular correlation in the time/spatial domain. Because this was previously unaccounted for, prior CF designs are not truly optimal, as their optimization criteria do not accurately quantify their optimization intention. In this paper, we introduce new zero-aliasing constraints that completely eliminate this aliasing problem by ensuring that the optimization criterion for a given CF corresponds to a linear correlation rather than a circular correlation. This means that previous CF designs can be significantly improved by this reformulation. We demonstrate the benefits of this new CF design approach with several important CFs. We present experimental results on diverse data sets and present solutions to the computational challenges associated with computing these CFs. Code for the CFs described in this paper and their respective zero-aliasing versions is available at http://vishnu.boddeti.net/projects/correlation-filters.html

Modeling dynamical systems with ordinary differential equations implies a mechanistic view of the process underlying the dynamics. However in many cases, this knowledge is not available. To overcome this issue, we introduce a general framework for nonparametric ODE models using penalized regression in Reproducing Kernel Hilbert Spaces (RKHS) based on operator-valued kernels. Moreover, we extend the scope of gradient matching approaches to nonparametric ODE. A smooth estimate of the solution ODE is built to provide an approximation of the derivative of the ODE solution which is in turn used to learn the nonparametric ODE model. This approach benefits from the flexibility of penalized regression in RKHS allowing for ridge or (structured) sparse regression as well. Very good results are shown on 3 different ODE systems.

Computational intensity and sequential nature of estimation techniques for Bayesian methods in statistics and machine learning, combined with their increasing applications for big data analytics, necessitate both the identification of potential opportunities to parallelize techniques such as MCMC sampling, and the development of general strategies for mapping such parallel algorithms to modern CPUs in order to elicit the performance up the compute-based and/or memory-based hardware limits. Two opportunities for Single-Instruction Multiple-Data (SIMD) parallelization of MCMC sampling for probabilistic graphical models are presented. In exchangeable models with many observations such as Bayesian Generalized Linear Models, child-node contributions to the conditional posterior of each node can be calculated concurrently. In undirected graphs with discrete nodes, concurrent sampling of conditionally-independent nodes can be transformed into a SIMD form. High-performance libraries with multi-threading and vectorization capabilities can be readily applied to such SIMD opportunities to gain decent speedup, while a series of high-level source-code and runtime modifications provide further performance boost by reducing parallelization overhead and increasing data locality for NUMA architectures. For big-data Bayesian GLM graphs, the end-result is a routine for evaluating the conditional posterior and its gradient vector that is 5 times faster than a naive implementation using (built-in) multi-threaded Intel MKL BLAS, and reaches within the striking distance of the memory-bandwidth-induced hardware limit. The proposed optimization strategies improve the scaling of performance with number of cores and width of vector units (applicable to many-core SIMD processors such as Intel Xeon Phi and GPUs), resulting in cost-effectiveness, energy efficiency, and higher speed on multi-core x86 processors.

In this article, we derive a new stepsize adaptation for the normalized least mean square algorithm (NLMS) by describing the task of linear acoustic echo cancellation from a Bayesian network perspective. Similar to the well-known Kalman filter equations, we model the acoustic wave propagation from the loudspeaker to the microphone by a latent state vector and define a linear observation equation (to model the relation between the state vector and the observation) as well as a linear process equation (to model the temporal progress of the state vector). Based on additional assumptions on the statistics of the random variables in observation and process equation, we apply the expectation-maximization (EM) algorithm to derive an NLMS-like filter adaptation. By exploiting the conditional independence rules for Bayesian networks, we reveal that the resulting EM-NLMS algorithm has a stepsize update equivalent to the optimal-stepsize calculation proposed by Yamamoto and Kitayama in 1982, which has been adopted in many textbooks. As main difference, the instantaneous stepsize value is estimated in the M step of the EM algorithm (instead of being approximated by artificially extending the acoustic echo path). The EM-NLMS algorithm is experimentally verified for synthesized scenarios with both, white noise and male speech as input signal.

We derive a convex optimization problem for the task of segmenting sequential data, which explicitly treats presence of outliers. We describe two algorithms for solving this problem, one exact and one a top-down novel approach, and we derive a consistency results for the case of two segments and no outliers. Robustness to outliers is evaluated on two real-world tasks related to speech segmentation. Our algorithms outperform baseline segmentation algorithms.