Dimensionality reduction (DR) is often used as a preprocessing step in classification, but usually one first fixes the DR mapping, possibly using label information, and then learns a classifier (a filter approach). Best performance would be obtained by optimizing the classification error jointly over DR mapping and classifier (a wrapper approach), but this is a difficult nonconvex problem, particularly with nonlinear DR. Using the method of auxiliary coordinates, we give a simple, efficient algorithm to train a combination of nonlinear DR and a classifier, and apply it to a RBF mapping with a linear SVM. This alternates steps where we train the RBF mapping and a linear SVM as usual regression and classification, respectively, with a closed-form step that coordinates both. The resulting nonlinear low-dimensional classifier achieves classification errors competitive with the state-of-the-art but is fast at training and testing, and allows the user to trade off runtime for classification accuracy easily. We then study the role of nonlinear DR in linear classification, and the interplay between the DR mapping, the number of latent dimensions and the number of classes. When trained jointly, the DR mapping takes an extreme role in eliminating variation: it tends to collapse classes in latent space, erasing all manifold structure, and lay out class centroids so they are linearly separable with maximum margin.

With the rapidly growing scales of statistical problems, subset based communication-free parallel MCMC methods are a promising future for large scale Bayesian analysis. In this article, we propose a new Weierstrass sampler for parallel MCMC based on independent subsets. The new sampler approximates the full data posterior samples via combining the posterior draws from independent subset MCMC chains, and thus enjoys a higher computational efficiency. We show that the approximation error for the Weierstrass sampler is bounded by some tuning parameters and provide suggestions for choice of the values. Simulation study shows the Weierstrass sampler is very competitive compared to other methods for combining MCMC chains generated for subsets, including averaging and kernel smoothing.

We present methods for online linear optimization that take advantage of benign (as opposed to worst-case) sequences. Specifically if the sequence encountered by the learner is described well by a known "predictable process", the algorithms presented enjoy tighter bounds as compared to the typical worst case bounds. Additionally, the methods achieve the usual worst-case regret bounds if the sequence is not benign. Our approach can be seen as a way of adding prior knowledge about the sequence within the paradigm of online learning. The setting is shown to encompass partial and side information. Variance and path-length bounds can be seen as particular examples of online learning with simple predictable sequences. We further extend our methods and results to include competing with a set of possible predictable processes (models), that is "learning" the predictable process itself concurrently with using it to obtain better regret guarantees. We show that such model selection is possible under various assumptions on the available feedback. Our results suggest a promising direction of further research with potential applications to stock market and time series prediction.

We present a framework for learning human user models from joint-action demonstrations that enables the robot to compute a robust policy for a collaborative task with a human. The learning takes place completely automatically, without any human intervention. First, we describe the clustering of demonstrated action sequences into different human types using an unsupervised learning algorithm. These demonstrated sequences are also used by the robot to learn a reward function that is representative for each type, through the employment of an inverse reinforcement learning algorithm. The learned model is then used as part of a Mixed Observability Markov Decision Process formulation, wherein the human type is a partially observable variable. With this framework, we can infer, either offline or online, the human type of a new user that was not included in the training set, and can compute a policy for the robot that will be aligned to the preference of this new user and will be robust to deviations of the human actions from prior demonstrations. Finally we validate the approach using data collected in human subject experiments, and conduct proof-of-concept demonstrations in which a person performs a collaborative task with a small industrial robot.

In the last few years, several interesting variants of kernel-based spectral methods have arisen in the applied mathematics literature. These ideas appeared in connection with new types of data, where pairs of objects or measurements of interest have a relationship that is "blurred" by the action of a nuisance parameter. More specifically, we can find this type of data in a wide range of problems, for instance in the class averaging algorithm for the cryo-electron microscope (cryo-EM) problem [62, 71], in a modern light source imaging technique known as ptychography [45], in graph realization problems [24, 25], in vectored PageRank [20], in multi-channels image processing [5], etc... Before we give further details about the cryo-EM problem, let us present the main building blocks of the methods we will study. They depend on the following three components: 1. an undirected graph G (V, E) which describes all observations.

We introduce a new sparse estimator of the covariance matrix for high-dimensional models in which the variables have a known ordering. Our estimator, which is the solution to a convex optimization problem, is equivalently expressed as an estimator which tapers the sample covariance matrix by a Toeplitz, sparsely-banded, data-adaptive matrix. As a result of this adaptivity, the convex banding estimator enjoys theoretical optimality properties not attained by previous banding or tapered estimators. In particular, our convex banding estimator is minimax rate adaptive in Frobenius and operator norms, up to log factors, over commonly-studied classes of covariance matrices, and over more general classes. Furthermore, it correctly recovers the bandwidth when the true covariance is exactly banded. Our convex formulation admits a simple and efficient algorithm. Empirical studies demonstrate its practical effectiveness and illustrate that our exactly-banded estimator works well even when the true covariance matrix is only close to a banded matrix, confirming our theoretical results. Our method compares favorably with all existing methods, in terms of accuracy and speed. We illustrate the practical merits of the convex banding estimator by showing that it can be used to improve the performance of discriminant analysis for classifying sound recordings.

We consider the problem of learning soft assignments of $N$ items to $K$ categories given two sources of information: an item-category similarity matrix, which encourages items to be assigned to categories they are similar to (and to not be assigned to categories they are dissimilar to), and an item-item similarity matrix, which encourages similar items to have similar assignments. We propose a simple quadratic programming model that captures this intuition. We give necessary conditions for its solution to be unique, define an out-of-sample mapping, and derive a simple, effective training algorithm based on the alternating direction method of multipliers. The model predicts reasonable assignments from even a few similarity values, and can be seen as a generalization of semisupervised learning. It is particularly useful when items naturally belong to multiple categories, as for example when annotating documents with keywords or pictures with tags, with partially tagged items, or when the categories have complex interrelations (e.g. hierarchical) that are unknown.

Real-world data typically contain repeated and periodic patterns. This suggests that they can be effectively represented and compressed using only a few coefficients of an appropriate basis (e.g., Fourier, Wavelets, etc.). However, distance estimation when the data are represented using different sets of coefficients is still a largely unexplored area. This work studies the optimization problems related to obtaining the \emph{tightest} lower/upper bound on Euclidean distances when each data object is potentially compressed using a different set of orthonormal coefficients. Our technique leads to tighter distance estimates, which translates into more accurate search, learning and mining operations \textit{directly} in the compressed domain. We formulate the problem of estimating lower/upper distance bounds as an optimization problem. We establish the properties of optimal solutions, and leverage the theoretical analysis to develop a fast algorithm to obtain an \emph{exact} solution to the problem. The suggested solution provides the tightest estimation of the $L_2$-norm or the correlation. We show that typical data-analysis operations, such as k-NN search or k-Means clustering, can operate more accurately using the proposed compression and distance reconstruction technique. We compare it with many other prevalent compression and reconstruction techniques, including random projections and PCA-based techniques. We highlight a surprising result, namely that when the data are highly sparse in some basis, our technique may even outperform PCA-based compression. The contributions of this work are generic as our methodology is applicable to any sequential or high-dimensional data as well as to any orthogonal data transformation used for the underlying data compression scheme.

Conventional approaches of sampling signals follow the celebrated theorem of Nyquist and Shannon. Compressive sampling, introduced by Donoho, Romberg and Tao, is a new paradigm that goes against the conventional methods in data acquisition and provides a way of recovering signals using fewer samples than the traditional methods use. Here we suggest an alternative way of reconstructing the original signals in compressive sampling using EM algorithm. We first propose a naive approach which has certain computational difficulties and subsequently modify it to a new approach which performs better than the conventional methods of compressive sampling. The comparison of the different approaches and the performance of the new approach has been studied using simulated data.

The Collective Graphical Model (CGM) models a population of independent and identically distributed individuals when only collective statistics (i.e., counts of individuals) are observed. Exact inference in CGMs is intractable, and previous work has explored Markov Chain Monte Carlo (MCMC) and MAP approximations for learning and inference. This paper studies Gaussian approximations to the CGM. As the population grows large, we show that the CGM distribution converges to a multivariate Gaussian distribution (GCGM) that maintains the conditional independence properties of the original CGM. If the observations are exact marginals of the CGM or marginals that are corrupted by Gaussian noise, inference in the GCGM approximation can be computed efficiently in closed form. If the observations follow a different noise model (e.g., Poisson), then expectation propagation provides efficient and accurate approximate inference. The accuracy and speed of GCGM inference is compared to the MCMC and MAP methods on a simulated bird migration problem. The GCGM matches or exceeds the accuracy of the MAP method while being significantly faster.