Transductive-Inductive Cluster Approximation Via Multivariate Chebyshev Inequality

Approximating adequate number of clusters in multidimensional data is an open area of research, given a level of compromise made on the quality of acceptable results. The manuscript addresses the issue by formulating a transductive inductive learning algorithm which uses multivariate Chebyshev inequality. Considering clustering problem in imaging, theoretical proofs for a particular level of compromise are derived to show the convergence of the reconstruction error to a finite value with increasing (a) number of unseen examples and (b) the number of clusters, respectively. Upper bounds for these error rates are also proved. Non-parametric estimates of these error from a random sample of sequences empirically point to a stable number of clusters. Lastly, the generalization of algorithm can be applied to multidimensional data sets from different fields.Comment: 16 pages, 5 figure

Point in polygon problem via homotopy and Hopf's degree Theorem

The current work revisits the point-in-polygon problem by providing a novel solution that explicitly employs the properties of epigraphs and hypographs. Using concepts of epigraphs and hypographs, this manuscript provides a new definition of inaccessibility and inside, to accurately specify the meaning of inclusion of a point within or without a polygon. Via Poincaré's ideas on homotopy and Hopf's Degree Theorem from topology, a relationship between inaccessibility and inside is established and it is shown that consistent results are obtained for peculiar cases of both non-intersectingand self-intersecting polygons while investigating the point inclusion test w.r.t. a polygon. Through illustrative examples, the novel method addresses the issues of • ambiguous solutions given by the Cross Over for both non-intersecting and self-intersecting polygons and • a point being labeled as multi-ply inside a self-intersecting polygon by the Winding Number Rule, by providing an unambiguous and singular result for both kinds of polygons. The proposed solution bridgesthe gap between Cross Over and Winding Number Rule for complex cases