The 2-class transduction problem, as formulated by Vapnik [1], involves finding a separating hyperplane for a labelled data set that is also maximally distant from a given set of unlabelled test points. In this form, the problem has exponential computational complexity in the size of the working set. So far it has been attacked by means of integer programming techniques [2] that do not scale to reasonable problem sizes, or by local search procedures [3]. In this paper we present a relaxation of this task based on semi- definite programming (SDP), resulting in a convex optimization problem that has polynomial complexity in the size of the data set. The results are very encouraging for mid sized data sets, however the cost is still too high for large scale problems, due to the high di- mensional search space.

The Minimax Probability Machine Classification (MPMC) framework [Lanckriet et al., 2002] builds classifiers by minimizing the maximum probability of misclassification, and gives direct estimates of the proba- bilistic accuracy bound Ω. The only assumptions that MPMC makes is that good estimates of means and covariance matrices of the classes exist. However, as with Support Vector Machines, MPMC is computationally expensive and requires extensive cross validation experiments to choose kernels and kernel parameters that give good performance. In this paper we address the computational cost of MPMC by proposing an algorithm that constructs nonlinear sparse MPMC (SMPMC) models by incremen- tally adding basis functions (i.e. Therefore the SMPMC algorithm simultaneously addresses the problem of kernel selection and feature selection (i.e.

Spectral methods for nonlinear dimensionality reduction (NLDR) impose a neighborhood graph on point data and compute eigenfunctions of a quadratic form generated from the graph. We introduce a more general and more robust formulation of NLDR based on the singular value de- composition (SVD). In this framework, most spectral NLDR principles can be recovered by taking a subset of the constraints in a quadratic form built from local nullspaces on the manifold. The minimax formulation also opens up an interesting class of methods in which the graph is "dec- orated" with information at the vertices, offering discrete or continuous maps, reduced computational complexity, and immunity to some solu- tion instabilities of eigenfunction approaches. Apropos, we show almost all NLDR methods based on eigenvalue decompositions (EVD) have a so- lution instability that increases faster than problem size.

We develop a family of upper and lower bounds on the worst-case ex- pected KL loss for estimating a discrete distribution on a finite number m of points, given N i.i.d. Our upper bounds are approximation- theoretic, similar to recent bounds for estimating discrete entropy; the lower bounds are Bayesian, based on averages of the KL loss under Dirichlet distributions. The upper bounds are convex in their parameters and thus can be minimized by descent methods to provide estimators with low worst-case error; the lower bounds are indexed by a one-dimensional parameter and are thus easily maximized. Asymptotic analysis of the bounds demonstrates the uniform KL-consistency of a wide class of es- timators as c N/m (no matter how slowly), and shows that no estimator is consistent for c bounded (in contrast to entropy estima- tion). Moreover, the bounds are asymptotically tight as c 0 or, and are shown numerically to be tight within a factor of two for all c.

This paper presents an application of Boosting for classifying labeled graphs, general structures for modeling a number of real-world data, such as chemical compounds, natural language texts, and bio sequences. The proposal consists of i) decision stumps that use subgraph as features, and ii) a Boosting algorithm in which subgraph-based decision stumps are used as weak learners. We also discuss the relation between our al- gorithm and SVMs with convolution kernels. Two experiments using natural language data and chemical compounds show that our method achieves comparable or even better performance than SVMs with convo- lution kernels as well as improves the testing efficiency. Most machine learning (ML) algorithms assume that given instances are represented in numerical vectors. However, much real-world data is not represented as numerical vectors, but as more complicated structures, such as sequences, trees, or graphs.

In this paper we consider the problem of finding sets of points that conform to a given underlying model from within a dense, noisy set of observations. This problem is motivated by the task of efficiently linking faint asteroid detections, but is applicable to a range of spatial queries. We survey current tree-based approaches, showing a trade-off exists between single tree and multiple tree algorithms. To this end, we present a new type of multiple tree algorithm that uses a variable number of trees to exploit the advantages of both approaches. We empirically show that this algorithm performs well using both simulated and astronomical data.

Sparse PCA seeks approximate sparse "eigenvectors" whose projections capture the maximal variance of data. As a cardinality-constrained and non-convex optimization problem, it is NP-hard and is encountered in a wide range of applied fields, from bio-informatics to finance. Recent progress has focused mainly on continuous approximation and convex relaxation of the hard cardinality constraint. In contrast, we consider an alternative discrete spectral formulation based on variational eigenvalue bounds and provide an effective greedy strategy as well as provably optimal solutions using branch-and-bound search. Moreover, the exact methodology used reveals a simple renormalization step that improves approximate solutions obtained by any continuous method.

We consider methods that try to find a good policy for a Markov decision process by choosing one from a given class. The policy is chosen based on its empirical performance in simulations. We are interested in conditions on the complexity of the policy class that ensure the success of such simulation based policy search methods. We show that under bounds on the amount of computation involved in computing policies, transition dynamics and rewards, uniform convergence of empirical estimates to true value functions occurs. Previously, such results were derived by assuming boundedness of pseudodimension and Lipschitz continuity.

Semi-supervised SVMs (S3 VM) attempt to learn low-density separators by maximizing the margin over labeled and unlabeled examples. The associated optimization problem is non-convex. To examine the full potential of S3 VMs modulo local minima problems in current implementations, we apply branch and bound techniques for obtaining exact, global ly optimal solutions. Empirical evidence suggests that the globally optimal solution can return excellent generalization performance in situations where other implementations fail completely. While our current implementation is only applicable to small datasets, we discuss variants that can potentially lead to practically useful algorithms.

In general, the problem of computing a maximum a posteriori (MAP) assignment in a Markov random field (MRF) is computationally intractable. However, in certain subclasses of MRF, an optimal or close-to-optimal assignment can be found very efficiently using combinatorial optimization algorithms: certain MRFs with mutual exclusion constraints can be solved using bipartite matching, and MRFs with regular potentials can be solved using minimum cut methods. However, these solutions do not apply to the many MRFs that contain such tractable components as sub-networks, but also other non-complying potentials. In this paper, we present a new method, called C O M P O S E, for exploiting combinatorial optimization for sub-networks within the context of a max-product belief propagation algorithm. C O M P O S E uses combinatorial optimization for computing exact maxmarginals for an entire sub-network; these can then be used for inference in the context of the network as a whole.