Topic
Concave function
About: Concave function is a research topic. Over the lifetime, 1415 publications have been published within this topic receiving 33278 citations.
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TL;DR: For planar convex bodies, it was shown in this paper that for convex body with convex inverse functions, the Bruns-Minkowski inequality is equivalent to the Minkowski mixed volume inequalities.
Abstract: For strictly increasing concave functions $${\varphi}$$
whose inverse functions are log-concave, the $${\varphi}$$
-Brunn–Minkowski inequality for planar convex bodies is established. It is shown that for convex bodies in $${\mathbb{R}^n}$$
the $${\varphi}$$
-Brunn–Minkowski is equivalent to the $${\varphi}$$
-Minkowski mixed volume inequalities.
7 citations
TL;DR: In this article, the Dyck matching was introduced, and the average cost of connecting a blue point in x to a red point in y is the concave function |x − y| p, where the optimal matching is trivially determined, where the optimization is non-trivial.
Abstract: We consider models of assignment for random N blue points and N red points on an interval of length 2N , in which the cost for connecting a blue point in x to a red point in y is the concave function |x − y| p , for 0 1, where the optimal matching is trivially determined, here the optimization is non-trivial. The purpose of this paper is to introduce a special configuration, that we call the Dyck matching, and to study its statistical properties. We compute exactly the average cost, in the asymptotic limit of large N , together with the first subleading correction. The scaling is remarkable: it is of order N for p 1 2 , and it is universal for a wide class of models. We conjecture that the average cost of the Dyck matching has the same scaling in N as the cost of the optimal matching, and we produce numerical data in support of this conjecture. We hope to produce a proof of this claim in future work.
7 citations
01 Nov 1998
TL;DR: Investigations into the use of concave/Schur-concave functions as regularizing sparsity measures and their application to the problem of obtaining sparse representations, x, of environmentally generated signals y, and theproblem of learning environmentally adapted overcomplete dictionaries are discussed.
Abstract: Given a very overcomplete m/spl times/n dictionary of representation vectors a/sub i/, A=[a/sub 1/,...,a/sub n/], n/spl Gt/m, an environmentally generated signal, y, can be succinctly represented within the dictionary by obtaining a sparse solution, x, to the linear inverse problem Ax/spl ap/y using various previously proposed methodologies. In particular, sparse solutions can be found by an appropriately regularized minimization of the error e=y-Ax. In this paper we briefly discuss our investigations into the use of concave/Schur-concave functions as regularizing sparsity measures, and their application to the problem of obtaining sparse representations, x, of environmentally generated signals y, and the problem of learning environmentally adapted overcomplete dictionaries.
7 citations
TL;DR: It is demonstrated that a special case of identical concave functions is solvable in O(m), and both results significantly improve the previous bounds for these problems.
7 citations
TL;DR: In this paper, the authors review the results related to the notion of stability of convex sets and consider some of their applications, and prove a corollary of the stability property which enables them to develop an approximation technique for concave functions on a wide class of concave sets.
Abstract: We briefly review the results related to the notion of stability of convex sets and consider some of their applications. We prove a corollary of the stability property which enables us to develop an approximation technique for concave functions on a wide class of convex sets. This technique yields necessary and sufficient conditions for the local continuity of concave functions. We describe some examples of their applications.
7 citations