We interpret nonnegative matrix factorization geometrically, as the problem of finding a simplicial cone which contains a cloud of data points and which is contained in the positive orthant. We show that under certain conditions, basically requiring that some of the data are spread across the faces of the positive orthant, there is a unique such simplicial cone. We give examples of synthetic image articulation databases which obey these conditions; these require separated support and factorial sampling. For such databases there is a generative model in terms of'parts' and NMF correctly identifies the'parts'. We show that our theoretical results are predictive of the performance of published NMF code, by running the published algorithms on one of our synthetic image articulation databases.

Many problems in information processing involve some form of dimensionality reduction. In this paper, we introduce Locality Preserving Projections (LPP). These are linear projective maps that arise by solving a variational problem that optimally preserves the neighborhood structure of the data set. LPP should be seen as an alternative to Principal Component Analysis (PCA) - a classical linear technique that projects the data along the directions of maximal variance. When the high dimensional data lies on a low dimensional manifold embedded in the ambient space, the Locality Preserving Projections are obtained by finding the optimal linear approximations to the eigenfunctions of the Laplace Beltrami operator on the manifold.

Predictive state representations (PSRs) use predictions of a set of tests to represent the state of controlled dynamical systems. One reason why this representation is exciting as an alternative to partially observable Markov decision processes (POMDPs) is that PSR models of dynamical systems may be much more compact than POMDP models. Empirical work on PSRs to date has focused on linear PSRs, which have not allowed for compression relative to POMDPs. We introduce a new notion of tests which allows us to define a new type of PSR that is nonlinear in general and allows for exponential compression in some deterministic dynamical systems. These new tests, called e-tests, are related to the tests used by Rivest and Schapire [1] in their work with the diversity representation, but our PSR avoids some of the pitfalls of their representation--in particular, its potential to be exponentially larger than the equivalent POMDP.

Despite the popularity of connectionist models in cognitive science, their performance can often be difficult to evaluate. Inspired by the geometric approach to statistical model selection, we introduce a conceptually similar method to examine the global behavior of a connectionist model, by counting the number and types of response patterns it can simulate. The Markov Chain Monte Carlo-based algorithm that we constructed Þnds these patterns efficiently. We demonstrate the approach using two localist network models of speech perception.

We propose a novel method of dimensionality reduction for supervised learning. Given a regression or classification problem in which we wish to predict a variable Y from an explanatory vector X, we treat the problem of dimensionality reduction as that of finding a low-dimensional "effective subspace" of X which retains the statistical relationship between X and Y. We show that this problem can be formulated in terms of conditional independence. To turn this formulation into an optimization problem, we characterize the notion of conditional independence using covariance operators on reproducing kernel Hilbert spaces; this allows us to derive a contrast function for estimation of the effective subspace. Unlike many conventional methods, the proposed method requires neither assumptions on the marginal distribution of X, nor a parametric model of the conditional distribution of Y.

The problem of extracting the relevant aspects of data was addressed through the information bottleneck (IB) method, by (soft) clustering one variable while preserving information about another - relevance - variable. An interesting question addressed in the current work is the extension of these ideas to obtain continuous representations that preserve relevant information, rather than discrete clusters. We give a formal definition of the general continuous IB problem and obtain an analytic solution for the optimal representation for the important case of multivariate Gaussian variables.

A recent area of significant progress in speaker recognition is the use of high level features--idiolect, phonetic relations, prosody, discourse structure, etc. A speaker not only has a distinctive acoustic sound but uses language in a characteristic manner. Large corpora of speech data available in recent years allow experimentation with long term statistics of phone patterns, word patterns, etc. of an individual. We propose the use of support vector machines and term frequency analysis of phone sequences to model a given speaker. To this end, we explore techniques for text categorization applied to the problem. We derive a new kernel based upon a linearization of likelihood ratio scoring. We introduce a new phone-based SVM speaker recognition approach that halves the error rate of conventional phone-based approaches.

We present test results from spike-timing correlation learning experiments carried out with silicon neurons with STDP (Spike Timing Dependent Plasticity) synapses. The weight change scheme of the STDP synapses can be set to either weight-independent or weight-dependent mode. We present results that characterise the learning window implemented for both modes of operation. When presented with spike trains with different types of synchronisation the neurons develop bimodal weight distributions. We also show that a 2-layered network of silicon spiking neurons with STDP synapses can perform hierarchical synchrony detection.

Loopy belief propagation (BP) has been successfully used in a number of difficult graphical models to find the most probable configuration of the hidden variables. In applications ranging from protein folding to image analysis one would like to find not just the best configuration but rather the top M. While this problem has been solved using the junction tree formalism, in many real world problems the clique size in the junction tree is prohibitively large. In this work we address the problem of finding the M best configurations when exact inference is impossible. We start by developing a new exact inference algorithm for calculating the best configurations that uses only max-marginals. For approximate inference, we replace the max-marginals with the beliefs calculated using max-product BP and generalized BP. We show empirically that the algorithm can accurately and rapidly approximate the M best configurations in graphs with hundreds of variables.

We describe a new approximation algorithm for solving partially observable MDPs. Our bounded policy iteration approach searches through the space of bounded-size, stochastic finite state controllers, combining several advantages of gradient ascent (efficiency, search through restricted controller space) and policy iteration (less vulnerability to local optima).