The process of dynamic state estimation (filtering) based on point process observations is in general intractable. Numerical sampling techniques are often practically useful, but lead to limited conceptual insight about optimal encoding/decoding strategies, which are of significant relevance to Computational Neuroscience. We develop an analytically tractable Bayesian approximation to optimal filtering based on point process observations, which allows us to introduce distributional assumptions about sensory cell properties, that greatly facilitates the analysis of optimal encoding in situations deviating from common assumptions of uniform coding. The analytic framework leads to insights which are difficult to obtain from numerical algorithms, and is consistent with experiments about the distribution of tuning curve centers. Interestingly, we find that the information gained from the absence of spikes may be crucial to performance.

We propose an extensive analysis of the behavior of majority votes in binary classification. In particular, we introduce a risk bound for majority votes, called the C-bound, that takes into account the average quality of the voters and their average disagreement. We also propose an extensive PAC-Bayesian analysis that shows how the C-bound can be estimated from various observations contained in the training data. The analysis intends to be self-contained and can be used as introductory material to PAC-Bayesian statistical learning theory. It starts from a general PAC-Bayesian perspective and ends with uncommon PAC-Bayesian bounds. Some of these bounds contain no Kullback-Leibler divergence and others allow kernel functions to be used as voters (via the sample compression setting). Finally, out of the analysis, we propose the MinCq learning algorithm that basically minimizes the C-bound. MinCq reduces to a simple quadratic program. Aside from being theoretically grounded, MinCq achieves state-of-the-art performance, as shown in our extensive empirical comparison with both AdaBoost and the Support Vector Machine.

In an effort to better understand meaning from natural language texts, we explore methods aimed at organizing lexical objects into contexts. A number of these methods for organization fall into a family defined by word ordering. Unlike demographic or spatial partitions of data, these collocation models are of special importance for their universal applicability. While we are interested here in text and have framed our treatment appropriately, our work is potentially applicable to other areas of research (e.g., speech, genomics, and mobility patterns) where one has ordered categorical data, (e.g., sounds, genes, and locations). Our approach focuses on the phrase (whether word or larger) as the primary meaning-bearing lexical unit and object of study. To do so, we employ our previously developed framework for generating word-conserving phrase-frequency data. Upon training our model with the Wiktionary---an extensive, online, collaborative, and open-source dictionary that contains over 100,000 phrasal-definitions---we develop highly effective filters for the identification of meaningful, missing phrase-entries. With our predictions we then engage the editorial community of the Wiktionary and propose short lists of potential missing entries for definition, developing a breakthrough, lexical extraction technique, and expanding our knowledge of the defined English lexicon of phrases.

Paragraph Vectors has been recently proposed as an unsupervised method for learning distributed representations for pieces of texts. In their work, the authors showed that the method can learn an embedding of movie review texts which can be leveraged for sentiment analysis. That proof of concept, while encouraging, was rather narrow. Here we consider tasks other than sentiment analysis, provide a more thorough comparison of Paragraph Vectors to other document modelling algorithms such as Latent Dirichlet Allocation, and evaluate performance of the method as we vary the dimensionality of the learned representation. We benchmarked the models on two document similarity data sets, one from Wikipedia, one from arXiv. We observe that the Paragraph Vector method performs significantly better than other methods, and propose a simple improvement to enhance embedding quality. Somewhat surprisingly, we also show that much like word embeddings, vector operations on Paragraph Vectors can perform useful semantic results.

This paper is concerned with a lesser-studied problem in the context of model-based, uncertainty quantification (UQ), that of optimization/design/control under uncertainty. The solution of such problems is hindered not only by the usual difficulties encountered in UQ tasks (e.g. the high computational cost of each forward simulation, the large number of random variables) but also by the need to solve a nonlinear optimization problem involving large numbers of design variables and potentially constraints. We propose a framework that is suitable for a large class of such problems and is based on the idea of recasting them as probabilistic inference tasks. To that end, we propose a Variational Bayesian (VB) formulation and an iterative VB-Expectation-Maximization scheme that is also capable of identifying a low-dimensional set of directions in the design space, along which, the objective exhibits the largest sensitivity. We demonstrate the validity of the proposed approach in the context of two numerical examples involving $\mathcal{O}(10^3)$ random and design variables. In all cases considered the cost of the computations in terms of calls to the forward model was of the order $\mathcal{O}(10^2)$. The accuracy of the approximations provided is assessed by appropriate information-theoretic metrics.

In this era of large-scale data, distributed systems built on top of clusters of commodity hardware provide cheap and reliable storage and scalable processing of massive data. Here, we review recent work on developing and implementing randomized matrix algorithms in large-scale parallel and distributed environments. Randomized algorithms for matrix problems have received a great deal of attention in recent years, thus far typically either in theory or in machine learning applications or with implementations on a single machine. Our main focus is on the underlying theory and practical implementation of random projection and random sampling algorithms for very large very overdetermined (i.e., overconstrained) $\ell_1$ and $\ell_2$ regression problems. Randomization can be used in one of two related ways: either to construct sub-sampled problems that can be solved, exactly or approximately, with traditional numerical methods; or to construct preconditioned versions of the original full problem that are easier to solve with traditional iterative algorithms. Theoretical results demonstrate that in near input-sparsity time and with only a few passes through the data one can obtain very strong relative-error approximate solutions, with high probability. Empirical results highlight the importance of various trade-offs (e.g., between the time to construct an embedding and the conditioning quality of the embedding, between the relative importance of computation versus communication, etc.) and demonstrate that $\ell_1$ and $\ell_2$ regression problems can be solved to low, medium, or high precision in existing distributed systems on up to terabyte-sized data.

We define a Hidden Markov Model (HMM) in which each hidden state has time-dependent $\textit{activity levels}$ that drive transitions and emissions, and show how to estimate its parameters. Our construction is motivated by the problem of inferring human mobility on sub-daily time scales from, for example, mobile phone records.

We present the first fully variational Bayesian inference scheme for continuous Gaussian-process-modulated Poisson processes. Such point processes are used in a variety of domains, including neuroscience, geo-statistics and astronomy, but their use is hindered by the computational cost of existing inference schemes. Our scheme: requires no discretisation of the domain; scales linearly in the number of observed events; and is many orders of magnitude faster than previous sampling based approaches. The resulting algorithm is shown to outperform standard methods on synthetic examples, coal mining disaster data and in the prediction of Malaria incidences in Kenya.

Linear regression is arguably the most prominent among statistical inference methods, popular both for its simplicity as well as its broad applicability. On par with data-intensive applications, the sheer size of linear regression problems creates an ever growing demand for quick and cost efficient solvers. Fortunately, a significant percentage of the data accrued can be omitted while maintaining a certain quality of statistical inference with an affordable computational budget. The present paper introduces means of identifying and omitting "less informative" observations in an online and data-adaptive fashion, built on principles of stochastic approximation and data censoring. First- and second-order stochastic approximation maximum likelihood-based algorithms for censored observations are developed for estimating the regression coefficients. Online algorithms are also put forth to reduce the overall complexity by adaptively performing censoring along with estimation. The novel algorithms entail simple closed-form updates, and have provable (non)asymptotic convergence guarantees. Furthermore, specific rules are investigated for tuning to desired censoring patterns and levels of dimensionality reduction. Simulated tests on real and synthetic datasets corroborate the efficacy of the proposed data-adaptive methods compared to data-agnostic random projection-based alternatives.

This paper presents a practical, and theoretically well-founded, approach to improve the speed of kernel manifold learning algorithms relying on spectral decomposition. Utilizing recent insights in kernel smoothing and learning with integral operators, we propose Reduced Set KPCA (RSKPCA), which also suggests an easy-to-implement method to remove or replace samples with minimal effect on the empirical operator. A simple data point selection procedure is given to generate a substitute density for the data, with accuracy that is governed by a user-tunable parameter . The effect of the approximation on the quality of the KPCA solution, in terms of spectral and operator errors, can be shown directly in terms of the density estimate error and as a function of the parameter . We show in experiments that RSKPCA can improve both training and evaluation time of KPCA by up to an order of magnitude, and compares favorably to the widely-used Nystrom and density-weighted Nystrom methods.