Notes on maxima in non-convex regions extended version

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The Hidden Convexity of Spectral Clustering

In recent years, spectral clustering has become a standard method for data analysis used in a broad range of applications. In this paper we propose a new class of algorithms for multiway spectral clustering based on optimization of a certain "contrast function" over the unit sphere. These algorithms, partly inspired by certain Independent Component Analysis techniques, are simple, easy to implement and efficient. Geometrically, the proposed algorithms can be interpreted as hidden basis recovery by means of function optimization. We give a complete characterization of the contrast functions admissible for provable basis recovery. We show how these conditions can be interpreted as a "hidden convexity" of our optimization problem on the sphere; interestingly, we use efficient convex maximization rather than the more common convex minimization. We also show encouraging experimental results on real and simulated data.Comment: 22 page

On the Oß-hull of a planar point set

© 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/We study the Oß-hull of a planar point set, a generalization of the Orthogonal Convex Hull where the coordinate axes form an angle ß. Given a set P of n points in the plane, we show how to maintain the Oß-hull of P while ß runs from 0 to p in T(n log n) time and O(n) space. With the same complexity, we also find the values of ß that maximize the area and the perimeter of the Oß-hull and, furthermore, we find the value of ß achieving the best fitting of the point set P with a two-joint chain of alternate interior angle ß.Peer ReviewedPostprint (author's final draft

The Expected Number of Maximal Points of the Convolution of Two 2-D Distributions

The Maximal points in a set S are those that are not dominated by any other point in S. Such points arise in multiple application settings and are called by a variety of different names, e.g., maxima, Pareto optimums, skylines. Their ubiquity has inspired a large literature on the expected number of maxima in a set S of n points chosen IID from some distribution. Most such results assume that the underlying distribution is uniform over some spatial region and strongly use this uniformity in their analysis. This research was initially motivated by the question of how this expected number changes if the input distribution is perturbed by random noise. More specifically, let B_p denote the uniform distribution from the 2-dimensional unit ball in the metric L_p. Let delta B_q denote the 2-dimensional L_q-ball, of radius delta and B_p + delta B_q be the convolution of the two distributions, i.e., a point v in B_p is reported with an error chosen from delta B_q. The question is how the expected number of maxima change as a function of delta. Although the original motivation is for small delta, the problem is well defined for any delta and our analysis treats the general case. More specifically, we study, as a function of n,delta, the expected number of maximal points when the n points in S are chosen IID from distributions of the type B_p + delta B_q where p,q in {1,2,infty} for delta > 0 and also of the type B_infty + delta B_q where q in [1,infty) for delta > 0. For fixed p,q we show that this function changes "smoothly" as a function of delta but that this smooth behavior sometimes transitions unexpectedly between different growth behaviors