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.

The classifiers are based on dyadic classification trees (DCTs), which involve adaptively pruned partitions of the feature space. A key aspect of DCTs is their spatial adaptivity, which enables local (rather than global) fitting of the decision boundary. Our risk analysis involves a spatial decomposition of the usual concentration inequalities, leading to a spatially adaptive, data-dependent pruning criterion. For any distribution on (X, Y) whose Bayes decision boundary behaves locally like a Lipschitz smooth function, we show that the DCT error converges to the Bayes error at a rate within a logarithmic factor of the minimax optimal rate.

We investigate improvements of AdaBoost that can exploit the fact that the weak hypotheses are one-sided, i.e. either all its positive (or negative) predictions are correct.

In order to understand AdaBoost's dynamics, especially its ability to maximize margins, we derive an associated simplified nonlinear iterated map and analyze its behavior in low-dimensional cases. We find stable cycles for these cases, which can explicitly be used to solve for Ada-Boost's output. By considering AdaBoost as a dynamical system, we are able to prove Rätsch and Warmuth's conjecture that AdaBoost may fail to converge to a maximal-margin combined classifier when given a'nonoptimal' weak learning algorithm.

We consider an online learning scenario in which the learner can make predictions on the basis of a fixed set of experts. We derive upper and lower relative loss bounds for a class of universal learning algorithms involving a switching dynamics over the choice of the experts. On the basis of the performance bounds we provide the optimal a priori discretization for learning the parameter that governs the switching dynamics. We demonstrate the new algorithm in the context of wireless networks.

This paper is concerned with transductive learning. Although transduction appears to be an easier task than induction, there have not been many provably useful algorithms and bounds for transduction. We present explicit error bounds for transduction and derive a general technique for devising bounds within this setting. The technique is applied to derive error bounds for compression schemes such as (transductive) SVMs and for transduction algorithms based on clustering.

The purpose of this paper is to investigate infinity-sample properties of risk minimization based multi-category classification methods. These methods can be considered as natural extensions to binary large margin classification. We establish conditions that guarantee the infinity-sample consistency of classifiers obtained in the risk minimization framework. Examples are provided for two specific forms of the general formulation, which extend a number of known methods. Using these examples, we show that some risk minimization formulations can also be used to obtain conditional probability estimates for the underlying problem. Such conditional probability information will be useful for statistical inferencing tasks beyond classification.

The decision functions constructed by support vector machines (SVM's) usually depend only on a subset of the training set--the so-called support vectors. We derive asymptotically sharp lower and upper bounds on the number of support vectors for several standard types of SVM's. In particular, we show for the Gaussian RBF kernel that the fraction of support vectors tends to twice the Bayes risk for the L1-SVM, to the probability of noise for the L2-SVM, and to 1 for the LS-SVM.

We demonstrate improvements over a previous chip by moving toward a significantly more versatile device. This includes a larger number of silicon neurons, more sophisticated neurons including voltage dependent charging and relative and absolute refractory periods, and enhanced programmability of neural networks. This chip builds on the basic results achieved on a previous chip and expands its versatility to get closer to a self-contained locomotion controller for walking robots. 1 Introduction Legged locomotion is a system level behavior that engages most senses and activates most muscles in the human body. Understanding of biological systems is exceedingly difficult and usually defies any unifying analysis. Walking behavior is no exception. Theories of walking are likely incomplete, often in ways that are invisible to the scientist studying these behavior in animal or human systems. Biological systems often fill in gaps and details. One way of exposing our incomplete understanding is through the process of synthesis. In this paper we report on continued progress in building the basic elements of a motor pattern generator sufficient to control a legged robot.

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.