In this paper we study nonconvex penalization using Bernstein functions. Since the Bernstein function is concave and nonsmooth at the origin, it can induce a class of nonconvex functions for high-dimensional sparse estimation problems. We derive a threshold function based on the Bernstein penalty and give its mathematical properties in sparsity modeling. We show that a coordinate descent algorithm is especially appropriate for penalized regression problems with the Bernstein penalty. Additionally, we prove that the Bernstein function can be defined as the concave conjugate of a $\varphi$-divergence and develop a conjugate maximization algorithm for finding the sparse solution. Finally, we particularly exemplify a family of Bernstein nonconvex penalties based on a generalized Gamma measure and conduct empirical analysis for this family.

Gaussian process state-space models (GP-SSMs) are a very flexible family of models of nonlinear dynamical systems. They comprise a Bayesian nonparametric representation of the dynamics of the system and additional (hyper-)parameters governing the properties of this nonparametric representation. The Bayesian formalism enables systematic reasoning about the uncertainty in the system dynamics. We present an approach to maximum likelihood identification of the parameters in GP-SSMs, while retaining the full nonparametric description of the dynamics. The method is based on a stochastic approximation version of the EM algorithm that employs recent developments in particle Markov chain Monte Carlo for efficient identification.

We introduce a recursive algorithm for performing compressed sensing on streaming data. The approach consists of a) recursive encoding, where we sample the input stream via overlapping windowing and make use of the previous measurement in obtaining the next one, and b) recursive decoding, where the signal estimate from the previous window is utilized in order to achieve faster convergence in an iterative optimization scheme applied to decode the new one. To remove estimation bias, a two-step estimation procedure is proposed comprising support set detection and signal amplitude estimation. Estimation accuracy is enhanced by a non-linear voting method and averaging estimates over multiple windows. We analyze the computational complexity and estimation error, and show that the normalized error variance asymptotically goes to zero for sublinear sparsity. Our simulation results show speed up of an order of magnitude over traditional CS, while obtaining significantly lower reconstruction error under mild conditions on the signal magnitudes and the noise level.

State-space models are successfully used in many areas of science, engineering and economics to model time series and dynamical systems. We present a fully Bayesian approach to inference \emph{and learning} (i.e. state estimation and system identification) in nonlinear nonparametric state-space models. We place a Gaussian process prior over the state transition dynamics, resulting in a flexible model able to capture complex dynamical phenomena. To enable efficient inference, we marginalize over the transition dynamics function and infer directly the joint smoothing distribution using specially tailored Particle Markov Chain Monte Carlo samplers. Once a sample from the smoothing distribution is computed, the state transition predictive distribution can be formulated analytically. Our approach preserves the full nonparametric expressivity of the model and can make use of sparse Gaussian processes to greatly reduce computational complexity.

Neurons in cortical circuits exhibit coordinated spiking activity, and can produce correlated synchronous spikes during behavior and cognition. We recently developed a method for estimating the dynamics of correlated ensemble activity by combining a model of simultaneous neuronal interactions (e.g., a spin-glass model) with a state-space method (Shimazaki et al. 2012 PLoS Comput Biol 8 e1002385). This method allows us to estimate stimulus-evoked dynamics of neuronal interactions which is reproducible in repeated trials under identical experimental conditions. However, the method may not be suitable for detecting stimulus responses if the neuronal dynamics exhibits significant variability across trials. In addition, the previous model does not include effects of past spiking activity of the neurons on the current state of ensemble activity. In this study, we develop a parametric method for simultaneously estimating the stimulus and spike-history effects on the ensemble activity from single-trial data even if the neurons exhibit dynamics that is largely unrelated to these effects. For this goal, we model ensemble neuronal activity as a latent process and include the stimulus and spike-history effects as exogenous inputs to the latent process. We develop an expectation-maximization algorithm that simultaneously achieves estimation of the latent process, stimulus responses, and spike-history effects. The proposed method is useful to analyze an interaction of internal cortical states and sensory evoked activity.

Non-convex optimization problems often arise from probabilistic modeling, such as estimation of posterior distributions. Non-convexity makes the problems intractable, and poses various obstacles for us to design efficient algorithms. In this work, we attack non-convexity by first introducing the concept of \emph{probable convexity} for analyzing convexity of real functions in practice. We then use the new concept to analyze an inference problem in the \emph{Correlated Topic Model} (CTM) and related nonconjugate models. Contrary to the existing belief of intractability, we show that this inference problem is concave under certain conditions. One consequence of our analyses is a novel algorithm for learning CTM which is significantly more scalable and qualitative than existing methods. Finally, we highlight that stochastic gradient algorithms might be a practical choice to resolve efficiently non-convex problems. This finding might find beneficial in many contexts which are beyond probabilistic modeling.

Kernel approximation using randomized feature maps has recently gained a lot of interest. In this work, we identify that previous approaches for polynomial kernel approximation create maps that are rank deficient, and therefore do not utilize the capacity of the projected feature space effectively. To address this challenge, we propose compact random feature maps (CRAFTMaps) to approximate polynomial kernels more concisely and accurately. We prove the error bounds of CRAFTMaps demonstrating their superior kernel reconstruction performance compared to the previous approximation schemes. We show how structured random matrices can be used to efficiently generate CRAFTMaps, and present a single-pass algorithm using CRAFTMaps to learn non-linear multi-class classifiers. We present experiments on multiple standard data-sets with performance competitive with state-of-the-art results.

We present a model for the joint estimation of disparity and motion. The model is based on learning about the interrelations between images from multiple cameras, multiple frames in a video, or the combination of both. We show that learning depth and motion cues, as well as their combinations, from data is possible within a single type of architecture and a single type of learning algorithm, by using biologically inspired "complex cell" like units, which encode correlations between the pixels across image pairs. Our experimental results show that the learning of depth and motion makes it possible to achieve state-of-the-art performance in 3-D activity analysis, and to outperform existing hand-engineered 3-D motion features by a very large margin.

Fitting high-dimensional data involves a delicate tradeoff between faithful representation and the use of sparse models. Too often, sparsity assumptions on the fitted model are too restrictive to provide a faithful representation of the observed data. In this paper, we present a novel framework incorporating sparsity in different domains.We decompose the observed covariance matrix into a sparse Gaussian Markov model (with a sparse precision matrix) and a sparse independence model (with a sparse covariance matrix). Our framework incorporates sparse covariance and sparse precision estimation as special cases and thus introduces a richer class of high-dimensional models. We characterize sufficient conditions for identifiability of the two models, \viz Markov and independence models. We propose an efficient decomposition method based on a modification of the popular $\ell_1$-penalized maximum-likelihood estimator ($\ell_1$-MLE). We establish that our estimator is consistent in both the domains, i.e., it successfully recovers the supports of both Markov and independence models, when the number of samples $n$ scales as $n = \Omega(d^2 \log p)$, where $p$ is the number of variables and $d$ is the maximum node degree in the Markov model. Our experiments validate these results and also demonstrate that our models have better inference accuracy under simple algorithms such as loopy belief propagation.

Removing or filtering outliers and mislabeled instances prior to training a learning algorithm has been shown to increase classification accuracy. A popular approach for handling outliers and mislabeled instances is to remove any instance that is misclassified by a learning algorithm. However, an examination of which learning algorithms to use for filtering as well as their effects on multiple learning algorithms over a large set of data sets has not been done. Previous work has generally been limited due to the large computational requirements to run such an experiment, and, thus, the examination has generally been limited to learning algorithms that are computationally inexpensive and using a small number of data sets. In this paper, we examine 9 learning algorithms as filtering algorithms as well as examining the effects of filtering in the 9 chosen learning algorithms on a set of 54 data sets. In addition to using each learning algorithm individually as a filter, we also use the set of learning algorithms as an ensemble filter and use an adaptive algorithm that selects a subset of the learning algorithms for filtering for a specific task and learning algorithm. We find that for most cases, using an ensemble of learning algorithms for filtering produces the greatest increase in classification accuracy. We also compare filtering with a majority voting ensemble. The voting ensemble significantly outperforms filtering unless there are high amounts of noise present in the data set. Additionally, we find that a majority voting ensemble is robust to noise as filtering with a voting ensemble does not increase the classification accuracy of the voting ensemble.