An Optimal Generalization of the Colorful Carathéodory Theorem

International audienceThe Colorful Carathéodory theorem by Bárány (1982) states that given d + 1 sets of points in R d , the convex hull of each containing the origin, there exists a simplex (called a 'rainbow simplex') with at most one point from each point set, which also contains the origin. Equivalently, either there is a hyperplane separating one of these d + 1 sets of points from the origin, or there exists a rainbow simplex containing the origin. One of our results is the following extension of the Colorful Carathéodory theorem: given + 1 sets of points in R d and a convex object C, then either one set can be separated from C by a constant (depending only on d) number of hyperplanes, or there is a rainbow simplex intersecting C

AN L1 CRITERION FOR DICTIONARY LEARNING BY SUBSPACE IDENTIFICATION

Future and Emerging Technologies (FET) programme within the Seventh Framework Programme for Research of the European Commission, under FET-Open grant number: 225913 (project SMALL).EPSRC Leadership Fellowship (EP/G007177/1