We present a practical, robust algorithm to locally minimize such functions, based on gradient sampling. Of particular interest is the case where f is not convex, and perhaps not even locally Lipschitz, but is a function whose gradient is easily computed where it is defined. Such functions arise in many applications, and very often minimizers are points at which f is not differentiable. Let f be a continuous function on R, and suppose f is continuously differentiable on an open dense subset. We evaluate the performance of the proposed method on simulatedĪnd publicly available data sets and ?nd that it compares favourably withĮxisting methods for projection pursuit and dimension reduction for Propose an approximation method using microclusters with provable approximationĮrror bounds. The computational cost associated with eachĮigen-problem is quadratic in the number of data. It also allows us to apply our methodology to the problem of ?nding This establishes a connection betweenĬonnectivity as measured by spectral graph theory and maximal Euclidean Vector normal to the maximum margin hyperplane through the data, as the scaling That the optimal projection based on spectral connectivity converges to the Which corresponds to a non-convex, non-smooth optimisation problem. Minimise the second eigenvalue of the Laplacian matrices of the projected data, This is achieved by ?nding projections which Maximising the separability of a binary partition of unlabeled data, as We study the problem of determining the optimal univariate subspace for Semi-supervised classification is evaluated on a number of benchmark datasets The performance of the proposed approach for unsupervised and Minimum density hyperplanes for unsupervised and semi-supervisedĬlassification. We propose a globally convergent algorithm for the estimation of Maximum margin clustering and semi-supervised support vector machineĬlassifiers. We are able to establish a link between the minimumĭensity and the maximum margin hyperplanes, thus linking this approach to The link between this approach and densityĬlustering is immediate. The minimumĭensity hyperplane minimises the integral of the empirical probability densityįunction along a hyperplane. Unlabelled data that avoids splitting high-density clusters. We propose a novel hyperplane classifier for Of low density, which is equivalent to assuming that high-density clusters areĪssociated with a single class. Semi-supervised classification, class boundaries are assumed to lie in regions Underlies a nonparametric approach known as density clustering. In unsupervised classification this cluster definition Statistical and machine learning approaches for the classification of High probability density (high-density clusters), is a central assumption in Associating distinct groups of objects (clusters) with contiguous regions of

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