In this tutorial we give an overview of the basic ideas underlying Support Vector (SV) machines for function estimation. Furthermore, we include a summary of currently used algorithms for training SV machines, covering both the quadratic (or convex) programming part and advanced methods for dealing with large datasets. Finally, we mention some modifications and extensions that have been applied to the standard SV algorithm, and discuss the aspect of regularization from a SV perspective.

We had previously shown that regularization principles lead to approximation schemes which are equivalent to networks with one layer of hidden units, called Regularization Networks. In particular, standard smoothness functionals lead to a subclass of regularization networks, the well known Radial Basis Functions approximation schemes. This paper shows that regularization networks encompass a much broader range of approximation schemes, including many of the popular general additive models and some of the neural networks. In particular, we introduce new classes of smoothness functionals that lead to different classes of basis functions. Additive splines as well as some tensor product splines can be obtained from appropriate classes of smoothness functionals. Furthermore, the same generalization that extends Radial Basis Functions (RBF) to Hyper Basis Functions (HBF) also leads from additive models to ridge approximation models, containing as special cases Breiman's hinge functions, som...

. We explore the effect of dimensionality on the "nearest neighbor " problem. We show that under a broad set of conditions (much broader than independent and identically distributed dimensions), as dimensionality increases, the distance to the nearest data point approaches the distance to the farthest data point. To provide a practical perspective, we present empirical results on both real and synthetic data sets that demonstrate that this effect can occur for as few as 10-15 dimensions. These results should not be interpreted to mean that high-dimensional indexing is never meaningful; we illustrate this point by identifying some high-dimensional workloads for which this effect does not occur. However, our results do emphasize that the methodology used almost universally in the database literature to evaluate high-dimensional indexing techniques is flawed, and should be modified. In particular, most such techniques proposed in the literature are not evaluated versus simple...

Exploratory projection pursuit is concerned with finding relatively highly revealing lower dimensional projections of high dimensional data. The intent is to discover views of the multivariate data set that exhibit nonlinear effects-clustering, concentrations near nonlinear manifolds- that are not captured by the linear correlation structure. This paper presents a new algorithm for this purpose that has both statistical and computational advantages over previous methods. A connection to density estimation is established. Examples are presented and issues related to practical application are discussed.

The success of reinforcement learning in practical problems depends on the ability tocombine function approximation with temporal difference methods such as value iteration. Experiments in this area have produced mixed results; there have been both notable successes and notable disappointments. Theory has been scarce, mostly due to the difficulty of reasoning about function approximators that generalize beyond the observed data. We provide a proof of convergence for a wide class of temporal difference methods involving function approximators such as k-nearest-neighbor, and show experimentally that these methods can be useful. The proof is based on a view of function approximators as expansion or contraction mappings. In addition, we present a novel view of approximate value iteration: an approximate algorithm for one environment turns out to be an exact algorithm for a different environment.

Abstract. The classification problem is considered in which an output variable y assumes discrete values with respective probabilities that depend upon the simultaneous values of a set of input variables x ={x1,...,xn}.At issue is how error in the estimates of these probabilities affects classification error when the estimates are used in a classification rule. These effects are seen to be somewhat counter intuitive in both their strength and nature. In particular the bias and variance components of the estimation error combine to influence classification in a very different way than with squared error on the probabilities themselves. Certain types of (very high) bias can be canceled by low variance to produce accurate classification. This can dramatically mitigate the effect of the bias associated with some simple estimators like “naive ” Bayes, and the bias induced by the curse-of-dimensionality on nearest-neighbor procedures. This helps explain why such simple methods are often competitive with and sometimes superior to more sophisticated ones for classification, and why “bagging/aggregating ” classifiers can often improve accuracy. These results also suggest simple modifications to these procedures that can (sometimes dramatically) further improve their classification performance.

This paper is concerned with modeling planning problems involving uncertainty as discrete-time, finite-state stochastic automata. Solving planning problems is reduced to computing policies for Markov decision processes. Classical methods for solving Markov decision processes cannot cope with the size of the state spaces for typical problems encountered in practice. As an alternative, we investigate methods that decompose global planning problems into a number of local problems, solve the local problems separately, and then combine the local solutions to generate a global solution. We present algorithms that decompose planning problems into smaller problems given an arbitrary partition of the state space. The local problems are interpreted as Markov decision processes and solutions to the local problems are interpreted as policies restricted to the subsets of the state space defined by the partition. One algorithm relies on constructing and solving an abstract version of the original de...

In this paper, we present an objective function formulation of the BCM theory of visual cortical plasticity that permits us to demonstrate the connection between the unsupervised BCM learning procedure and various statistical methods, in particular, that of Projection Pursuit. This formulation provides a general method for stability analysis of the fixed points of the theory and enables us to analyze the behavior and the evolution of the network under various visual rearing conditions. It also allows comparison with many existing unsupervised methods. This model has been shown successful in various applications such as phoneme and 3D object recognition. We thus have the striking and possibly highly significant result that a biological neuron is performing a sophisticated statistical procedure.

Nearest neighbor classification assumes locally constant class conditional probabilities. This assumption becomes invalid in high dimensions with finite samples due to the curse of dimensionality. Severe bias can be introduced under these conditions when using the nearest neighbor rule. We propose a locally adaptive nearest neighbor classification method to try to minimize bias. We use a Chisquared distance analysis to compute a flexible metric for producing neighborhoods that are highly adaptive to query locations. Neighborhoods are elongated along less relevant feature dimensions and constricted along most influential ones. As a result, the class conditional probabilities tend to be smoother in the modified neighborhoods, whereby better classification performance can be achieved. The efficacy of our method is validated and compared against other techniques using a variety of simulated and real world data. 1 Introduction In a classification problem, we are given J classes and N tra...

Financial forecasting is an example of a signal processing problem which is challenging due to small sample sizes, high noise, non-stationarity, and non-linearity. Neural networks have been very successful in a number of signal processing applications. We discuss fundamental limitations and inherent difficulties when using neural networks for the processing of high noise, small sample size signals. We introduce a new intelligent signal processing method which addresses the difficulties. The method proposed uses conversion into a symbolic representation with a selforganizing map, and grammatical inference with recurrent neural networks. We apply the method to the prediction of daily foreign exchange rates, addressing difficulties with non-stationarity, overfitting, and unequal a priori class probabilities, and we find significant predictability in comprehensive experiments covering 5 different foreign exchange rates. The method correctly predicts the direction of change for th...