Despite many empirical successes of spectral clustering methods(cid:173) algorithms that cluster points using eigenvectors of matrices de(cid:173) rived from the data- there are several unresolved issues. First, there are a wide variety of algorithms that use the eigenvectors in slightly different ways. Second, many of these algorithms have no proof that they will actually compute a reasonable clustering. In this paper, we present a simple spectral clustering algorithm that can be implemented using a few lines of Matlab. Using tools from matrix perturbation theory, we analyze the algorithm, and give conditions under which it can be expected to do well.

We propose a new particle filter that incorporates a model of costs when generating particles. The approach is motivated by the observation that the costs of accidentally not tracking hypotheses might be significant in some areas of state space, and next to irrelevant in others. By incorporat- ing a cost model into particle filtering, states that are more critical to the system performance are more likely to be tracked. Automatic calculation of the cost model is implemented using an MDP value function calcula- tion that estimates the value of tracking a particular state. Experiments in two mobile robot domains illustrate the appropriateness of the approach.

We compare discriminative and generative learning as typified by logistic regression and naive Bayes. We show, contrary to a widely(cid:173) held belief that discriminative classifiers are almost always to be preferred, that there can often be two distinct regimes of per(cid:173) formance as the training set size is increased, one in which each algorithm does better. This stems from the observation- which is borne out in repeated experiments- that while discriminative learning has lower asymptotic error, a generative classifier may also approach its (higher) asymptotic error much faster.

A novel approach for comparing sequences of observations using an explicit-expansion kernel is demonstrated. The kernel is derived using the assumption of the independence of the sequence of observations and a mean-squared error training criterion. The use of an explicit expan- sion kernel reduces classifier model size and computation dramatically, resulting in model sizes and computation one-hundred times smaller in our application. The explicit expansion also preserves the computational advantages of an earlier architecture based on mean-squared error train- ing. Training using standard support vector machine methodology gives accuracy that significantly exceeds the performance of state-of-the-art mean-squared error training for a speaker recognition task.

Singularities are ubiquitous in the parameter space of hierarchical models such as multilayer perceptrons. At singularities, the Fisher information matrix degenerates, and the Cramer-Rao paradigm does no more hold, implying that the classical model selection the(cid:173) ory such as AIC and MDL cannot be applied. It is important to study the relation between the generalization error and the training error at singularities. The present paper demonstrates a method of analyzing these errors both for the maximum likelihood estima(cid:173) tor and the Bayesian predictive distribution in terms of Gaussian random fields, by using simple models.

Using methods of Statistical Physics, we investigate the rOle of model complexity in learning with support vector machines (SVMs). We show the advantages of using SVMs with kernels of infinite complexity on noisy target rules, which, in contrast to common theoretical beliefs, are found to achieve optimal general(cid:173) ization error although the training error does not converge to the generalization error. Moreover, we find a universal asymptotics of the learning curves which only depend on the target rule but not on the SVM kernel.

A mixed-signal paradigm is presented for high-resolution parallel inner- product computation in very high dimensions, suitable for efficient im- plementation of kernels in image processing. At the core of the externally digital architecture is a high-density, low-power analog array performing binary-binary partial matrix-vector multiplication. Full digital resolution is maintained even with low-resolution analog-to-digital conversion, ow- ing to random statistics in the analog summation of binary products. A random modulation scheme produces near-Bernoulli statistics even for highly correlated inputs. The approach is validated with real image data, and with experimental results from a CID/DRAM analog array prototype in 0.5

We present three ways of combining linear programming with the kernel trick to find value function approximations for reinforcement learning. One formulation is based on SVM regression; the second is based on the Bellman equation; and the third seeks only to ensure that good moves have an advantage over bad moves. All formu(cid:173) lations attempt to minimize the number of support vectors while fitting the data. Experiments in a difficult, synthetic maze problem show that all three formulations give excellent performance, but the advantage formulation is much easier to train. Unlike policy gradi(cid:173) ent methods, the kernel methods described here can easily'adjust the complexity of the function approximator to fit the complexity of the value function.

We propose a framework based on a parametric quadratic program(cid:173) ming (QP) technique to solve the support vector machine (SVM) training problem. This framework, can be specialized to obtain two SVM optimization methods. The first solves the fixed bias prob(cid:173) lem, while the second starts with an optimal solution for a fixed bias problem and adjusts the bias until the optimal value is found. The later method can be applied in conjunction with any other ex(cid:173) isting technique which obtains a fixed bias solution. Moreover, the second method can also be used independently to solve the com(cid:173) plete SVM training problem.

We propose a novel clustering method that is an extension of ideas inher- ent to scale-space clustering and support-vector clustering. Like the lat- ter, it associates every data point with a vector in Hilbert space, and like the former it puts emphasis on their total sum, that is equal to the scale- space probability function. The novelty of our approach is the study of an operator in Hilbert space, represented by the Schr odinger equation of which the probability function is a solution. This Schr odinger equation contains a potential function that can be derived analytically from the probability function. The method has one variable parameter, the scale of its Gaussian kernel.