The consistency of a nonlocal anisotropic Ginzburg-Landau type functional for data classification and clustering is studied. The Ginzburg-Landau objective functional combines a double well potential, that favours indicator valued function, and the @math -Laplacian, that enforces regularity. Under appropriate scaling between the two terms minimisers exhibit a phase transition on the order of @math where @math is the number of data points. We study the large data asymptotics, i.e. as @math , in the regime where @math . The mathematical tool used to address this question is @math -convergence. In particular, it is proved that the discrete model converges to a weighted anisotropic perimeter.
The @math topology, as introduced by Garc ' i a Trillos and Slep c ev @cite_54 , provides a notion of convergence upon which the @math -convergence framework can be applied. This method has now been applied in many works, see, for instance, @cite_54 @cite_53 @cite_48 @cite_52 @cite_16 @cite_44 @cite_37 @cite_25 @cite_41 . Further studies on this topology can be found in @cite_54 @cite_48 @cite_32 @cite_17 .
@cite_37 @cite_41 @cite_48 @cite_54 @cite_53 @cite_52 @cite_32 @cite_44 @cite_16 @cite_25 @cite_17
2784708638 1945101555 2094096036 2100904599
Combining concepts from topology and algorithms, this book delivers what its title promises: an introduction to the field of computational topology. Starting with motivating problems in both mathematics and computer science and building up from classic topics in geometric and algebraic topology, the third part of the text advances to persistent homology. This point of view is critically important in turning a mostly theoretical field of mathematics into one that is relevant to a multitude of disciplines in the sciences and engineering. The main approach is the discovery of topology through algorithms. The book is ideal for teaching a graduate or advanced undergraduate course in computational topology, as it develops all the background of both the mathematical and algorithmic aspects of the subject from first principles. Thus the text could serve equally well in a course taught in a mathematics department or computer science department. Let @math be a smooth scheme over an algebraically closed field @math of characteristic zero and @math a regular function, and write @math Crit @math , as a closed subscheme of @math . The motivic vanishing cycle @math is an element of the @math -equivariant motivic Grothendieck ring @math defined by Denef and Loeser math.AG 0006050 and Looijenga math.AG 0006220, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants, arXiv:0811.2435. We prove three main results: (a) @math depends only on the third-order thickenings @math of @math . (b) If @math is another smooth scheme, @math is regular, @math Crit @math , and @math is an embedding with @math and @math an isomorphism, then @math equals @math "twisted" by a motive associated to a principal @math -bundle defined using @math , where now we work in a quotient ring @math of @math . (c) If @math is an "oriented algebraic d-critical locus" in the sense of Joyce arXiv:1304.4508, there is a natural motive @math , such that if @math is locally modelled on Crit @math , then @math is locally modelled on @math . Using results from arXiv:1305.6302, these imply the existence of natural motives on moduli schemes of coherent sheaves on a Calabi-Yau 3-fold equipped with "orientation data", as required in Kontsevich and Soibelman's motivic Donaldson-Thomas theory arXiv:0811.2435, and on intersections of oriented Lagrangians in an algebraic symplectic manifold. This paper is an analogue for motives of results on perverse sheaves of vanishing cycles proved in arXiv:1211.3259. We extend this paper to Artin stacks in arXiv:1312.0090. We show that if X is a complete separable metric space and C is a countable family of Borel subsets of X with finite VC dimension, then, for every stationary ergodic process with values in X, the relative frequencies of sets C ∈ C converge uniformly to their limiting probabilities. Beyond ergodicity, no assumptions are imposed on the sampling process, and no regularity conditions are imposed on the elements of C. The result extends existing work of Vapnik and Chervonenkis, among others, who have studied uniform convergence for i.i.d. and strongly mixing processes. Our method of proof is new and direct: it does not rely on symmetrization techniques, probability inequalities or mixing conditions. The uniform convergence of relative frequencies for VC-major and VC-graph classes of functions under ergodic sampling is established as a corollary of the basic result for sets. Recently, Hundal has constructed a hyperplane H , a cone K, and a starting point y0 in ‘2 such that the sequence of alternating projections ((PK PH ) n y0)n∈N converges weakly to some point in H ∩ K, but not in norm. We show how this construction results in a counterexample to norm convergence for iterates of averaged projections; hence, we give an a:rmative answer to a question raised by Reich two decades ago. Furthermore, new counterexamples to norm convergence for iterates of <rmly nonexpansive maps (=a la Genel and Lindenstrauss) and for the proximal point algorithm (= al a G@ uler) are provided. We also present a counterexample, along with some weak and norm convergence results, for the new framework of string-averaging projection methods introduced by Extensions to Banach spaces and the situation for the Hilbert ball are discussed as well. ? 2003 Elsevier Ltd. All rights reserved. MSC: 47H09; 47J25; 90C25