We address the problem of learning a symmetric positive definite matrix. The central issue is to design parameter updates that preserve positive definiteness. Our updates are motivated with the von Neumann divergence. Rather than treating the most general case, we focus on two key applications that exemplify our methods: Online learning with a simple square loss and finding a symmetric positive definite matrix subject to symmetric linear constraints. The updates generalize the Exponentiated Gradient (EG) update and AdaBoost, respectively: the parameter is now a symmetric positive definite matrix of trace one instead of a probability vector (which in this context is a diagonal positive definite matrix with trace one). The generalized updates use matrix logarithms and exponentials to preserve positive definiteness. Most importantly, we show how the analysis of each algorithm generalizes to the non-diagonal case. We apply both new algorithms, called the Matrix Exponentiated Gradient (MEG) update and DefiniteBoost, to learn a kernel matrix from distance measurements.

This paper explores the computational consequences of simultaneous intrinsic and synaptic plasticity in individual model neurons. It proposes a new intrinsic plasticity mechanism for a continuous activation model neuron based on low order moments of the neuron's firing rate distribution. The goal of the intrinsic plasticity mechanism is to enforce a sparse distribution of the neuron's activity level. In conjunction with Hebbian learning at the neuron's synapses, the neuron is shown to discover sparse directions in the input.

We seek to both detect and segment objects in images. To exploit both local image data as well as contextual information, we introduce Boosted Random Fields (BRFs), which uses Boosting to learn the graph structure and local evidence of a conditional random field (CRF). The graph structure is learned by assembling graph fragments in an additive model. The connections between individual pixels are not very informative, but by using dense graphs, we can pool information from large regions of the image; dense models also support efficient inference. We show how contextual information from other objects can improve detection performance, both in terms of accuracy and speed, by using a computational cascade. We apply our system to detect stuff and things in office and street scenes.

Simple lexicographic decision heuristics that consider cues one at a time in a particular order and stop searching for cues as soon as a decision can be made have been shown to be both accurate and frugal in their use of information. But much of the simplicity and success of these heuristics comes from using an appropriate cue order. For instance, the Take The Best heuristic uses validity order for cues, which requires considerable computation, potentially undermining the computational advantages of the simple decision mechanism. But many cue orders can achieve good decision performance, and studies of sequential search for data records have proposed a number of simple ordering rules that may be of use in constructing appropriate decision cue orders as well. Here we consider a range of simple cue ordering mechanisms, including tallying, swapping, and move-to-front rules, and show that they can find cue orders that lead to reasonable accuracy and considerable frugality when used with lexicographic decision heuristics.

We propose the hierarchical Dirichlet process (HDP), a nonparametric Bayesian model for clustering problems involving multiple groups of data. Each group of data is modeled with a mixture, with the number of components being open-ended and inferred automatically by the model. Further, components can be shared across groups, allowing dependencies across groups to be modeled effectively as well as conferring generalization to new groups. Such grouped clustering problems occur often in practice, e.g. in the problem of topic discovery in document corpora. We report experimental results on three text corpora showing the effective and superior performance of the HDP over previous models.

We introduce a generalization of temporal-difference (TD) learning to networks of interrelated predictions. Rather than relating a single prediction to itself at a later time, as in conventional TD methods, a TD network relates each prediction in a set of predictions to other predictions in the set at a later time. TD networks can represent and apply TD learning to a much wider class of predictions than has previously been possible. Using a random-walk example, we show that these networks can be used to learn to predict by a fixed interval, which is not possible with conventional TD methods. Secondly, we show that if the interpredictive relationships are made conditional on action, then the usual learning-efficiency advantage of TD methods over Monte Carlo (supervised learning) methods becomes particularly pronounced.

We describe a three-dimensional geometric hand model suitable for visual tracking applications. The kinematic constraints implied by the model's joints have a probabilistic structure which is well described by a graphical model. Inference in this model is complicated by the hand's many degrees of freedom, as well as multimodal likelihoods caused by ambiguous image measurements. We use nonparametric belief propagation (NBP) to develop a tracking algorithm which exploits the graph's structure to control complexity, while avoiding costly discretization. While kinematic constraints naturally have a local structure, self-occlusions created by the imaging process lead to complex interpendencies in color and edge-based likelihood functions. However, we show that local structure may be recovered by introducing binary hidden variables describing the occlusion state of each pixel. We augment the NBP algorithm to infer these occlusion variables in a distributed fashion, and then analytically marginalize over them to produce hand position estimates which properly account for occlusion events. We provide simulations showing that NBP may be used to refine inaccurate model initializations, as well as track hand motion through extended image sequences.

It has been demonstrated that basic aspects of human visual motion perception are qualitatively consistent with a Bayesian estimation framework, where the prior probability distribution on velocity favors slow speeds. Here, we present a refined probabilistic model that can account for the typical trial-to-trial variabilities observed in psychophysical speed perception experiments. We also show that data from such experiments can be used to constrain both the likelihood and prior functions of the model. Specifically, we measured matching speeds and thresholds in a two-alternative forced choice speed discrimination task. Parametric fits to the data reveal that the likelihood function is well approximated by a LogNormal distribution with a characteristic contrast-dependent variance, and that the prior distribution on velocity exhibits significantly heavier tails than a Gaussian, and approximately follows a power-law function.

Go is an ancient oriental game whose complexity has defeated attempts to automate it. We suggest using probability in a Bayesian sense to model the uncertainty arising from the vast complexity of the game tree. We present a simple conditional Markov random field model for predicting the pointwise territory outcome of a game. The topology of the model reflects the spatial structure of the Go board. We describe a version of the Swendsen-Wang process for sampling from the model during learning and apply loopy belief propagation for rapid inference and prediction. The model is trained on several hundred records of professional games. Our experimental results indicate that the model successfully learns to predict territory despite its simplicity.

We establish learning rates to the Bayes risk for support vector machines (SVMs) with hinge loss. In particular, for SVMs with Gaussian RBF kernels we propose a geometric condition for distributions which can be used to determine approximation properties of these kernels. Finally, we compare our methods with a recent paper of G. Blanchard et al..