We study the problem of determining the optimal low dimensional projection for maximising the separability of a binary partition of an unlabelled dataset, as measured by spectral graph theory. This is achieved by finding projections which minimise the second eigenvalue of the graph Laplacian of the projected data, which corresponds to a non-convex, non-smooth optimisation problem. We show that the optimal univariate projection based on spectral connectivity converges to the vector normal to the maximum margin hyperplane through the data, as the scaling parameter is reduced to zero. This establishes a connection between connectivity as measured by spectral graph theory and maximal Euclidean separation. The computational cost associated with each eigen-problem is quadratic in the number of data. To mitigate this issue, we propose an approximation method using microclusters with provable approximation error bounds. Combining multiple binary partitions within a divisive hierarchical model allows us to construct clustering solutions admitting clusters with varying scales and lying within different subspaces. We evaluate the performance of the proposed method on a large collection of benchmark datasets and find that it compares favourably with existing methods for projection pursuit and dimension reduction for data clustering.
Principal component analysis and independent component analysis have been used in the context of clustering, however their objectives do not correspond exactly with those of the clustering task and the justification of their use is based more on common-sense reasoning. Nonetheless, these methods have shown good empirical performance on a number of applications . Some recent approaches to projection pursuit for clustering rely on the non-parametric statistical notion clusters, i.e., that clusters are regions of high density in a probability distribution from which the data are assumed to have arisen. @cite_9 proposed using as projection index the of the projected data. The dip is a measure of departure from unimodality, and so maximising the dip tends to projections which have mutlimodal marginal density, and therefore separate high density clusters. The authors establish that the dip is differentiable for any projection vector onto which the projected data are unique, and use a simple gradient ascent method to find local optima.
@cite_9
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Methods that find insightful low-dimensional projections are essential to effectively explore high-dimensional data. Principal Component Analysis is used pervasively to find low-dimensional projections, not only because it is straightforward to use, but it is also often effective, because the variance in data is often dominated by relevant structure. However, even if the projections highlight real structure in the data, not all structure is interesting to every user. If a user is already aware of, or not interested in the dominant structure, Principal Component Analysis is less effective for finding interesting components. We introduce a new method called Subjectively Interesting Component Analysis (SICA), designed to find data projections that are subjectively interesting, i.e, projections that truly surprise the end-user. It is rooted in information theory and employs an explicit model of a user's prior expectations about the data. The corresponding optimization problem is a simple eigenvalue problem, and the result is a trade-off between explained variance and novelty. We present five case studies on synthetic data, images, time-series, and spatial data, to illustrate how SICA enables users to find (subjectively) interesting projections. An integrated framework for density-based cluster analysis, outlier detection, and data visualization is introduced in this article. The main module consists of an algorithm to compute hierarchical estimates of the level sets of a density, following Hartigan’s classic model of density-contour clusters and trees. Such an algorithm generalizes and improves existing density-based clustering techniques with respect to different aspects. It provides as a result a complete clustering hierarchy composed of all possible density-based clusters following the nonparametric model adopted, for an infinite range of density thresholds. The resulting hierarchy can be easily processed so as to provide multiple ways for data visualization and exploration. It can also be further postprocessed so that: (i) a normalized score of “outlierness” can be assigned to each data object, which unifies both the global and local perspectives of outliers into a single definition; and (ii) a “flat” (i.e., nonhierarchical) clustering solution composed of clusters extracted from local cuts through the cluster tree (possibly corresponding to different density thresholds) can be obtained, either in an unsupervised or in a semisupervised way. In the unsupervised scenario, the algorithm corresponding to this postprocessing module provides a global, optimal solution to the formal problem of maximizing the overall stability of the extracted clusters. If partially labeled objects or instance-level constraints are provided by the user, the algorithm can solve the problem by considering both constraints violations satisfactions and cluster stability criteria. An asymptotic complexity analysis, both in terms of running time and memory space, is described. Experiments are reported that involve a variety of synthetic and real datasets, including comparisons with state-of-the-art, density-based clustering and (global and local) outlier detection methods. We present a practical approach to nonparametric cluster analysis of large data sets. The number of clusters and the cluster centres are automatically derived by mode seeking with the mean shift procedure on a reduced set of points randomly selected from the data. The cluster boundaries are delineated using a k-nearest neighbour technique. The proposed algorithm is stable and efficient, a 10,000 point data set being decomposed in only a few seconds. Complex clustering examples and applications are discussed, and convergence of the gradient ascent mean shift procedure is demonstrated for arbitrary distribution and cardinality of the data. A new formulation for multiway spectral clustering is proposed. This method corresponds to a weighted kernel principal component analysis (PCA) approach based on primal-dual least-squares support vector machine (LS-SVM) formulations. The formulation allows the extension to out-of-sample points. In this way, the proposed clustering model can be trained, validated, and tested. The clustering information is contained on the eigendecomposition of a modified similarity matrix derived from the data. This eigenvalue problem corresponds to the dual solution of a primal optimization problem formulated in a high-dimensional feature space. A model selection criterion called the balanced line fit (BLF) is also proposed. This criterion is based on the out-of-sample extension and exploits the structure of the eigenvectors and the corresponding projections when the clusters are well formed. The BLF criterion can be used to obtain clustering parameters in a learning framework. Experimental results with difficult toy problems and image segmentation show improved performance in terms of generalization to new samples and computation times.