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Geometrical Aspects of Statistical Learning Theory
Geometrical Aspects of Statistical Learning Theory Vom Fachbereich Informatik der Technischen Universit¨at Darmstadt genehmigte Dissertation zur Erlangung des akademischen Grades Doctor rerum naturalium (Dr. rer. nat.) vorgelegt von Dipl.-Phys. Matthias Hein aus Esslingen am Neckar Prufungskommission:¨ Vorsitzender: Prof. Dr. B. Schiele Erstreferent: Prof. Dr. T. Hofmann Korreferent : Prof. Dr. B. Sch¨olkopf Tag der Einreichung: 30.9.2005 Tag der Disputation: 9.11.2005 Darmstadt, 2005 Hochschulkennziffer: D17 Abstract Geometry plays an important role in modern statistical learning theory, and many different aspects of geometry can be found in this fast developing field. This thesis addresses some of these aspects. A large part of this work will be concerned with so called manifold methods, which have recently attracted a lot of interest. The key point is that for a lot of real-world data sets it is natural to assume that the data lies on a low-dimensional submanifold of a potentially high-dimensional Euclidean space. We develop a rigorous and quite general framework for the estimation and ap- proximation of some geometric structures and other quantities of this submanifold, using certain corresponding structures on neighborhood graphs built from random samples of that submanifold. Another part of this thesis deals with the generalizati- on of the maximal margin principle to arbitrary metric spaces. This generalization follows quite naturally by changing the viewpoint on the well-known support vector machines (SVM). It can be shown that the SVM can be seen as an algorithm which applies the maximum margin principle to a subclass of metric spaces. The motivati- on to consider the generalization to arbitrary metric spaces arose by the observation that in practice the condition for the applicability of the SVM is rather difficult to check for a given metric. -
Metric Spaces in Pure and Applied Mathematics
Documenta Math. 121 Metric Spaces in Pure and Applied Mathematics A. Dress, K. T. Huber1, V. Moulton2 Received: September 25, 2001 Revised: November 5, 2001 Communicated by Ulf Rehmann Abstract. The close relationship between the theory of quadratic forms and distance analysis has been known for centuries, and the the- ory of metric spaces that formalizes distance analysis and was devel- oped over the last century, has obvious strong relations to quadratic- form theory. In contrast, the first paper that studied metric spaces as such – without trying to study their embeddability into any one of the standard metric spaces nor looking at them as mere ‘presentations’ of the underlying topological space – was, to our knowledge, written in the late sixties by John Isbell. In particular, Isbell showed that in the category whose objects are metric spaces and whose morphisms are non-expansive maps, a unique injective hull exists for every object, he provided an explicit construction of this hull, and he noted that, at least for finite spaces, it comes endowed with an intrinsic polytopal cell structure. In this paper, we discuss Isbell’s construction, we summarize the his- tory of — and some basic questions studied in — phylogenetic analysis, and we explain why and how these two topics are related to each other. Finally, we just mention in passing some intriguing analogies between, on the one hand, a certain stratification of the cone of all metrics de- fined on a finite set X that is based on the combinatorial properties of the polytopal cell structure of Isbell’s injective hulls and, on the other, various stratifications of the cone of positive semi-definite quadratic forms defined on Rn that were introduced by the Russian school in the context of reduction theory. -
Phd Thesis, Stanford University
DISSERTATION TOPOLOGICAL, GEOMETRIC, AND COMBINATORIAL ASPECTS OF METRIC THICKENINGS Submitted by Johnathan E. Bush Department of Mathematics In partial fulfillment of the requirements For the Degree of Doctor of Philosophy Colorado State University Fort Collins, Colorado Summer 2021 Doctoral Committee: Advisor: Henry Adams Amit Patel Chris Peterson Gloria Luong Copyright by Johnathan E. Bush 2021 All Rights Reserved ABSTRACT TOPOLOGICAL, GEOMETRIC, AND COMBINATORIAL ASPECTS OF METRIC THICKENINGS The geometric realization of a simplicial complex equipped with the 1-Wasserstein metric of optimal transport is called a simplicial metric thickening. We describe relationships between these metric thickenings and topics in applied topology, convex geometry, and combinatorial topology. We give a geometric proof of the homotopy types of certain metric thickenings of the circle by constructing deformation retractions to the boundaries of orbitopes. We use combina- torial arguments to establish a sharp lower bound on the diameter of Carathéodory subsets of the centrally-symmetric version of the trigonometric moment curve. Topological information about metric thickenings allows us to give new generalizations of the Borsuk–Ulam theorem and a selection of its corollaries. Finally, we prove a centrally-symmetric analog of a result of Gilbert and Smyth about gaps between zeros of homogeneous trigonometric polynomials. ii ACKNOWLEDGEMENTS Foremost, I want to thank Henry Adams for his guidance and support as my advisor. Henry taught me how to be a mathematician in theory and in practice, and I was exceedingly fortu- nate to receive my mentorship in research and professionalism through his consistent, careful, and honest feedback. I could always count on him to make time for me and to guide me to interesting problems. -
Injective Hulls of Certain Discrete Metric Spaces and Groups
Injective hulls of certain discrete metric spaces and groups Urs Lang∗ July 29, 2011; revised, June 28, 2012 Abstract Injective metric spaces, or absolute 1-Lipschitz retracts, share a number of properties with CAT(0) spaces. In the 1960es, J. R. Isbell showed that every metric space X has an injective hull E(X). Here it is proved that if X is the vertex set of a connected locally finite graph with a uniform stability property of intervals, then E(X) is a locally finite polyhedral complex with n finitely many isometry types of n-cells, isometric to polytopes in l∞, for each n. This applies to a class of finitely generated groups Γ, including all word hyperbolic groups and abelian groups, among others. Then Γ acts properly on E(Γ) by cellular isometries, and the first barycentric subdivision of E(Γ) is a model for the classifying space EΓ for proper actions. If Γ is hyperbolic, E(Γ) is finite dimensional and the action is cocompact. In particular, every hyperbolic group acts properly and cocompactly on a space of non-positive curvature in a weak (but non-coarse) sense. 1 Introduction A metric space Y is called injective if for every metric space B and every 1- Lipschitz map f : A → Y defined on a set A ⊂ B there exists a 1-Lipschitz extension f : B → Y of f. The terminology is in accordance with the notion of an injective object in category theory. Basic examples of injective metric spaces are the real line, all complete R-trees, and l∞(I) for an arbitrary index set I. -
Helly Meets Garside and Artin
Invent. math. (2021) 225:395–426 https://doi.org/10.1007/s00222-021-01030-8 Helly meets Garside and Artin Jingyin Huang1 · Damian Osajda2,3 Received: 7 June 2019 / Accepted: 7 January 2021 / Published online: 15 February 2021 © The Author(s) 2021 Abstract A graph is Helly if every family of pairwise intersecting combina- torial balls has a nonempty intersection. We show that weak Garside groups of finite type and FC-type Artin groups are Helly, that is, they act geometrically on Helly graphs. In particular, such groups act geometrically on spaces with a convex geodesic bicombing, equipping them with a nonpositive-curvature-like structure. That structure has many properties of a CAT(0) structure and, addi- tionally, it has a combinatorial flavor implying biautomaticity. As immediate consequences we obtain new results for FC-type Artin groups (in particular braid groups and spherical Artin groups) and weak Garside groups, including e.g. fundamental groups of the complements of complexified finite simplicial arrangements of hyperplanes, braid groups of well-generated complex reflec- tion groups, and one-relator groups with non-trivial center. Among the results are: biautomaticity, existence of EZ and Tits boundaries, the Farrell–Jones conjecture, the coarse Baum–Connes conjecture, and a description of higher B Damian Osajda [email protected] Jingyin Huang [email protected] 1 Department of Mathematics, The Ohio State University, 100 Math Tower, 231 W 18th Ave, Columbus, OH 43210, USA 2 Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50–384 Wrocław, Poland 3 Institute of Mathematics, Polish Academy of Sciences, Sniadeckich´ 8, 00-656 Warsaw, Poland 123 396 J. -
Linearly Rigid Metric Spaces and the Embedding Problem
Linearly rigid metric spaces and the embedding problem. J. Melleray,∗ F. V. Petrov,† A. M. Vershik‡ 06.03.08 Dedicated to 95-th anniversary of L.V.Kantorovich Abstract We consider the problem of isometric embedding of the metric spaces to the Banach spaces; and introduce and study the remarkable class of so called linearly rigid metric spaces: these are the spaces that admit a unique, up to isometry, linearly dense isometric embedding into a Banach space. The first nontrivial example of such a space was given by R. Holmes; he proved that the universal Urysohn space has this property. We give a criterion of linear rigidity of a metric space, which allows us to give a simple proof of the linear rigidity of the Urysohn space and some other metric spaces. The various properties of linearly rigid spaces and related spaces are considered. Introduction The goal of this paper is to describe the class of complete separable metric arXiv:math/0611049v4 [math.FA] 11 Apr 2008 (=Polish) spaces which have the following property: there is a unique (up to isometry) isometric embedding of this metric space (X, ρ) in a Banach space such that the affine span of the image of X is dense (in which case we say that X is linearly dense). We call such metric spaces linearly rigid spaces. ∗University of Illinois at Urbana-Champaign. E-mail: [email protected]. †St. Petersburg Department of Steklov Institute of Mathematics. E-mail: [email protected]. Supported by the grant NSh.4329.2006.1 ‡St. Petersburg Department of Steklov Institute of Mathematics. -
Minimal Universal Metric Spaces
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 42, 2017, 1019–1064 MINIMAL UNIVERSAL METRIC SPACES Victoriia Bilet, Oleksiy Dovgoshey, Mehmet Küçükaslan and Evgenii Petrov Institute of Applied Mathematics and Mechaniks of NASU, Function Theory Department Dobrovolskogo str. 1, Slovyansk, 84100, Ukraine; [email protected] Institute of Applied Mathematics and Mechaniks of NASU, Function Theory Department Dobrovolskogo str. 1, Slovyansk, 84100, Ukraine; [email protected] Mersin University, Faculty of Art and Sciences, Department of Mathematics Mersin 33342, Turkey; [email protected] Institute of Applied Mathematics and Mechaniks of NASU, Function Theory Department Dobrovolskogo str. 1, Slovyansk, 84100, Ukraine; [email protected] Abstract. Let M be a class of metric spaces. A metric space Y is minimal M-universal if every X ∈ M can be isometrically embedded in Y but there are no proper subsets of Y satisfying this property. We find conditions under which, for given metric space X, there is a class M of metric spaces such that X is minimal M-universal. We generalize the notion of minimal M-universal metric space to notion of minimal M-universal class of metric spaces and prove the uniqueness, up to an isomorphism, for these classes. The necessary and sufficient conditions under which the disjoint union of the metric spaces belonging to a class M is minimal M-universal are found. Examples of minimal universal metric spaces are constructed for the classes of the three-point metric spaces and n-dimensional normed spaces. Moreover minimal universal metric spaces are found for some subclasses of the class of metric spaces X which possesses the following property. -
Spaces with Convex Geodesic Bicombings
Research Collection Doctoral Thesis Spaces with convex geodesic bicombings Author(s): Descombes, Dominic Publication Date: 2015 Permanent Link: https://doi.org/10.3929/ethz-a-010584573 Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library Diss. ETH No. 23109 Spaces with Convex Geodesic Bicombings A thesis submitted to attain the degree of Doctor of Sciences of ETH Zurich presented by Dominic Descombes Master of Science ETH in Mathematics citizen of Lignières, NE and citizen of Italy accepted on the recommendation of Prof. Dr. Urs Lang, examiner Prof. Dr. Alexander Lytchak, co-examiner 2015 Life; full of loneliness, and misery, and suffering, and unhappiness — and it's all over much too quickly. – Woody Allen Abstract In the geometry of CAT(0) or Busemann spaces every pair of geodesics, call them α and β, have convex distance; meaning d ◦ (α, β) is a convex function I → R provided the geodesics are parametrized proportional to arc length on the same interval I ⊂ R. Therefore, geodesics ought to be unique and thus even many normed spaces do not belong to these classes. We investigate spaces with non-unique geodesics where there exists a suitable selection of geodesics exposing the said (or a similar) convexity property; this structure will be called a bicombing. A rich class of such spaces arises naturally through the construction of the injective hull for arbitrary metric spaces or more generally as 1- Lipschitz retracts of normed spaces. -
Mathematics 595 (CAP/TRA) Fall 2005 2 Metric and Topological Spaces
Mathematics 595 (CAP/TRA) Fall 2005 2 Metric and topological spaces 2.1 Metric spaces The notion of a metric abstracts the intuitive concept of \distance". It allows for the development of many of the standard tools of analysis: continuity, convergence, compactness, etc. Spaces equipped with both a linear (algebraic) structure and a metric (analytic) structure will be considered in the next section. They provide suitable environments for the development of a rich theory of differential calculus akin to the Euclidean theory. Definition 2.1.1. A metric on a space X is a function d : X X [0; ) which is symmetric, vanishes at (x; y) if and only if x = y, and satisfies×the triangle! 1inequality d(x; y) d(x; z) + d(z; y) for all x; y; z X. ≤ 2 The pair (X; d) is called a metric space. Notice that we require metrics to be finite-valued. Some authors consider allow infinite-valued functions, i.e., maps d : X X [0; ]. A space equipped with × ! 1 such a function naturally decomposes into a family of metric spaces (in the sense of Definition 2.1.1), which are the equivalence classes for the equivalence relation x y if and only if d(x; y) < . ∼A pseudo-metric is a function1 d : X X [0; ) which satisfies all of the conditions in Definition 2.1.1 except that ×the condition! 1\d vanishes at (x; y) if and only if x = y" is replaced by \d(x; x) = 0 for all x". In other words, a pseudo-metric is required to vanish along the diagonal (x; y) X X : x = y , but may also vanish for some pairs (x; y) with x = y. -
Finite Metric Space Is of Euclidean Type
Finite Metric Spaces of Euclidean Type1 Peter J. Kahn Department of Mathematics Cornell University March, 2021 n Abstract: Finite subsets of R may be endowed with the Euclidean metric. Do all finite metric spaces arise in this way for some n? If two such finite metric spaces are abstractly isometric, must the isometry be the restriction of an isometry of the ambient Euclidean space? This note shows that the answer to the first question is no and the answer to the second is yes. These answers are reversed if the sup metric is used in place of the Euclidean metric. 1. Introduction Metric spaces typically arise in connection with the foundations of calculus or anal- ysis, providing the necessary general structure for discussing the concepts of conver- gence, continuity, compactness and the like. These spaces usually have the cardinality of the continuum. But metric spaces of finite cardinality also appear naturally in many mathematical contexts (e.g., graph theory, string metrics, coding theory) and have applications in the sciences (e.g., DNA analysis, network theory, phylogenetics). Any finite subset X of a metric space (Y; e) inherits a metric d = ejX ×X, allowing us to produce many examples of finite metric spaces (X; d) when (Y; e) is known. n n Often (Y; e) is taken to be (R ; d2), where R is the standard Euclidean space of dimension n and d2 is the usual Euclidean metric. In such a case we say that the induced finite metric space is of Euclidean type, and we also use this designation for any metric space isometric to it. -
Geometric Embeddings of Metric Spaces by Juha Heinonen Lectures in the Finnish Graduate School of Mathematics, University Of
Geometric embeddings of metric spaces By Juha Heinonen Lectures in the Finnish Graduate School of Mathematics, University of Jyv¨askyl¨a, January 2003. 1 Basic Concepts. 2 Gromov-Hausdorff convergence. 3 Some fundamental embeddings. 4 Strong A∞-deformations. 1 2 Preface These notes form a slightly expanded version of the lectures that I gave in the Finnish Graduate School of Mathematics at the University of Jyv¨askyl¨a in January 2003.The purpose of this mini-course was to introduce beginning graduate students to some easily accessible ques- tions of current interest in metric geometry.Besides additional remarks and references, the only topics that were not discussed in the course, but are included here, are the proof of the existence of a Urysohn uni- versal metric space and the proof of Semmes’s theorem 4.5. I thank Tero Kilpel¨ainen for inviting me to give these lectures, Pekka Koskela for offering to publish the lecture notes in the Jyv¨askyl¨a Math- ematics Department Reports series, and Juha Inkeroinen for typing a preliminary version of this mansucript from my hand written notes. I am grateful to Bruce Hanson who carefully read the entire manu- script and made useful suggestions.Finally, I thank the Mathematical Sciences Research Institute and NSF (Grant DMS 9970427) for their support. Berkeley, September 2003 Juha Heinonen 3 1. Basic Concepts Let X =(X, d)=(X, dX ) denote a metric space.Throughout these lectures, we will consider quite general metric spaces.However, the reader should not think of anything pathological here (like the discrete metric on some huge set). -
Arxiv:2002.06895V2 [Math.GR] 19 Feb 2021 3.3
HELLY GROUPS JER´ EMIE´ CHALOPIN, VICTOR CHEPOI, ANTHONY GENEVOIS, HIROSHI HIRAI, AND DAMIAN OSAJDA Abstract. Helly graphs are graphs in which every family of pairwise intersecting balls has a non-empty intersection. This is a classical and widely studied class of graphs. In this article we focus on groups acting geometrically on Helly graphs { Helly groups. We provide numerous examples of such groups: all (Gromov) hyperbolic, CAT(0) cubical, finitely presented graph- ical C(4)−T(4) small cancellation groups, and type-preserving uniform lattices in Euclidean buildings of type Cn are Helly; free products of Helly groups with amalgamation over finite subgroups, graph products of Helly groups, some diagram products of Helly groups, some right- angled graphs of Helly groups, and quotients of Helly groups by finite normal subgroups are Helly. We show many properties of Helly groups: biautomaticity, existence of finite dimensional models for classifying spaces for proper actions, contractibility of asymptotic cones, existence of EZ-boundaries, satisfiability of the Farrell-Jones conjecture and of the coarse Baum-Connes conjecture. This leads to new results for some classical families of groups (e.g. for FC-type Artin groups) and to a unified approach to results obtained earlier. Contents 1. Introduction 2 1.1. Motivations and main results 2 1.2. Discussion of consequences of main results 5 1.3. Organization of the article and further results 6 2. Preliminaries 7 2.1. Graphs 7 2.2. Complexes 10 2.3. CAT(0) spaces and Gromov hyperbolicity 11 2.4. Group actions 12 2.5. Hypergraphs (set families) 12 2.6.