Given a probability measure P and a reference measure µ, one is often interested in the minimum µ-measure set with P-measure at least α. Minimum volume sets of this type summarize the regions of greatest probability mass of P, and are useful for detecting anomalies and constructing confidence regions. This paper addresses the problem of estimating minimum volume sets based on independent samples distributed according to P. Other than these samples, no other information is available regarding P, but the reference measure µ is assumed to be known. We introduce rules for estimating minimum volume sets that parallel the empirical risk minimization and structural risk minimization principles in classification. As in classification, we show that the performances of our estimators are controlled by the rate of uniform convergence of empirical to true probabilities over the class from which the estimator is drawn. Thus we obtain finite sample size performance bounds in terms of VC dimension and related quantities. We also demonstrate strong universal consistency and an oracle inequality. Estimators based on histograms and dyadic partitions illustrate the proposed rules.

Reinforcement learning by direct policy gradient estimation is attractive in theory but in practice leads to notoriously ill-behaved optimization problems. We improve its robustness and speed of convergence with stochastic meta-descent, a gain vector adaptation method that employs fast Hessian-vector products. In our experiments the resulting algorithms outperform previously employed online stochastic, offline conjugate, and natural policy gradient methods.

We introduce a technique for dimensionality estimation based on the notion of quantization dimension, which connects the asymptotic optimal quantization error for a probability distribution on a manifold to its intrinsic dimension. The definition of quantization dimension yields a family of estimation algorithms, whose limiting case is equivalent to a recent method based on packing numbers. Using the formalism of high-rate vector quantization, we address issues of statistical consistency and analyze the behavior of our scheme in the presence of noise.

We consider criteria for variational representations of non-Gaussian latent variables, and derive variational EM algorithms in general form. We establish a general equivalence among convex bounding methods, evidence based methods, and ensemble learning/Variational Bayes methods, which has previously been demonstrated only for particular cases.

In this paper, we provide a general theorem that establishes a correspondence between surrogate loss functions in classification and the family of f-divergences. Moreover, we provide constructive procedures for determining the f-divergence induced by a given surrogate loss, and conversely for finding all surrogate loss functions that realize a given f-divergence. Next we introduce the notion of universal equivalence among loss functions and corresponding f-divergences, and provide necessary and sufficient conditions for universal equivalence to hold. These ideas have applications to classification problems that also involve a component of experiment design; in particular, we leverage our results to prove consistency of a procedure for learning a classifier under decentralization requirements.

We present a nonlinear, simple, yet effective, feature subset selection method for regression and use it in analyzing cortical neural activity. Our algorithm involves a feature-weighted version of the k-nearest-neighbor algorithm. It is able to capture complex dependency of the target function on its input and makes use of the leave-one-out error as a natural regularization. We explain the characteristics of our algorithm on synthetic problems and use it in the context of predicting hand velocity from spikes recorded in motor cortex of a behaving monkey. By applying feature selection we are able to improve prediction quality and suggest a novel way of exploring neural data.

We show that Queyranne's algorithm for minimizing symmetric submodular functions can be used for clustering with a variety of different objective functions. Two specific criteria that we consider in this paper are the single linkage and the minimum description length criteria. The first criterion tries to maximize the minimum distance between elements of different clusters, and is inherently "discriminative". It is known that optimal clusterings into k clusters for any given k in polynomial time for this criterion can be computed. The second criterion seeks to minimize the description length of the clusters given a probabilistic generative model. We show that the optimal partitioning into 2 clusters, and approximate partitioning (guaranteed to be within a factor of 2 of the the optimal) for more clusters can be computed. To the best of our knowledge, this is the first time that a tractable algorithm for finding the optimal clustering with respect to the MDL criterion for 2 clusters has been given. Besides the optimality result for the MDL criterion, the chief contribution of this paper is to show that the same algorithm can be used to optimize a broad class of criteria, and hence can be used for many application specific criterion for which efficient algorithm are not known.

This paper presents a diffusion based probabilistic interpretation of spectral clustering and dimensionality reduction algorithms that use the eigenvectors of the normalized graph Laplacian. Given the pairwise adjacency matrix of all points, we define a diffusion distance between any two data points and show that the low dimensional representation of the data by the first few eigenvectors of the corresponding Markov matrix is optimal under a certain mean squared error criterion.

Nested sampling is a new Monte Carlo method by Skilling [1] intended for general Bayesian computation. Nested sampling provides a robust alternative to annealing-based methods for computing normalizing constants. It can also generate estimates of other quantities such as posterior expectations. The key technical requirement is an ability to draw samples uniformly from the prior subject to a constraint on the likelihood. We provide a demonstration with the Potts model, an undirected graphical model.

In this paper we propose a new receiver for digital communications. We focus on the application of Gaussian Processes (GPs) to the multiuser detection (MUD) in code division multiple access (CDMA) systems to solve the near-far problem. Hence, we aim to reduce the interference from other users sharing the same frequency band. While usual approaches minimize the mean square error (MMSE) to linearly retrieve the user of interest, we exploit the same criteria but in the design of a nonlinear MUD. Since the optimal solution is known to be nonlinear, the performance of this novel method clearly improves that of the MMSE detectors. Furthermore, the GP based MUD achieves excellent interference suppression even for short training sequences. We also include some experiments to illustrate that other nonlinear detectors such as those based on Support Vector Machines (SVMs) exhibit a worse performance.