Spectral Geometry for Structural Pattern Recognition
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Spectral Geometry Spring 2016
Spectral Geometry Spring 2016 Michael Magee Lecture 5: The resolvent. Last time we proved that the Laplacian has a closure whose domain is the Sobolev space H2(M)= u L2(M): u L2(M), ∆u L2(M) . { 2 kr k2 2 } We view the Laplacian as an unbounded densely defined self-adjoint operator on L2 with domain H2. We also stated the spectral theorem. We’ll now ask what is the nature of the pro- jection valued measure associated to the Laplacian (its ‘spectral decomposition’). Until otherwise specified, suppose M is a complete Riemannian manifold. Definition 1. The resolvent set of the Laplacian is the set of λ C such that λI ∆isa bijection of H2 onto L2 with a boundedR inverse (with respect to L2 norms)2 − 1 2 2 R(λ) (λI ∆)− : L (M) H (M). ⌘ − ! The operator R(λ) is called the resolvent of the Laplacian at λ. We write spec(∆) = C −R and call it the spectrum. Note that the projection valued measure of ∆is supported in R 0. Otherwise one can find a function in H2(M)where ∆ , is negative, but since can be≥ approximated in Sobolev norm by smooth functions, thish will contradicti ∆ , = g( , )Vol 0. (1) h i ˆ r r ≥ Note then the spectral theorem implies that the resolvent set contains C R 0 (use the Borel functional calculus). − ≥ Theorem 2 (Stone’s formula [1, VII.13]). Let P be the projection valued measure on R associated to the Laplacan by the spectral theorem. The following formula relates the resolvent to P b 1 1 lim(2⇡i)− [R(λ + i✏) R(λ i✏)] dλ = P[a,b] + P(a,b) . -
Three Conjectures in Extremal Spectral Graph Theory
Three conjectures in extremal spectral graph theory Michael Tait and Josh Tobin June 6, 2016 Abstract We prove three conjectures regarding the maximization of spectral invariants over certain families of graphs. Our most difficult result is that the join of P2 and Pn−2 is the unique graph of maximum spectral radius over all planar graphs. This was conjectured by Boots and Royle in 1991 and independently by Cao and Vince in 1993. Similarly, we prove a conjecture of Cvetkovi´cand Rowlinson from 1990 stating that the unique outerplanar graph of maximum spectral radius is the join of a vertex and Pn−1. Finally, we prove a conjecture of Aouchiche et al from 2008 stating that a pineapple graph is the unique connected graph maximizing the spectral radius minus the average degree. To prove our theorems, we use the leading eigenvector of a purported extremal graph to deduce structural properties about that graph. Using this setup, we give short proofs of several old results: Mantel's Theorem, Stanley's edge bound and extensions, the K}ovari-S´os-Tur´anTheorem applied to ex (n; K2;t), and a partial solution to an old problem of Erd}oson making a triangle-free graph bipartite. 1 Introduction Questions in extremal graph theory ask to maximize or minimize a graph invariant over a fixed family of graphs. Perhaps the most well-studied problems in this area are Tur´an-type problems, which ask to maximize the number of edges in a graph which does not contain fixed forbidden subgraphs. Over a century old, a quintessential example of this kind of result is Mantel's theorem, which states that Kdn=2e;bn=2c is the unique graph maximizing the number of edges over all triangle-free graphs. -
Manifold Learning
MANIFOLD LEARNING Kavé Salamatian- LISTIC Learning Machine Learning: develop algorithms to automatically extract ”patterns” or ”regularities” from data (generalization) Typical tasks Clustering: Find groups of similar points Dimensionality reduction: Project points in a lower dimensional space while preserving structure Semi-supervised: Given labelled and unlabelled points, build a labelling function Supervised: Given labelled points, build a labelling function All these tasks are not well-defined Clustering Identify the two groups Dimensionality Reduction 4096 Objects in R ? But only 3 parameters: 2 angles and 1 for illumination. They span a 3- dimensional sub-manifold of R4096! Semi-Supervised Learning Assign labels to black points Supervised learning Build a function that predicts the label of all points in the space Formal definitions Clustering: Given build a function Dimensionality reduction: Given build a function Semi-supervised: Given and with m << n, build a function Supervised: Given , build Geometric learning Consider and exploit the geometric structure of the data Clustering: two ”close” points should be in the same cluster Dimensionality reduction: ”closeness” should be preserved Semi-supervised/supervised: two ”close” points should have the same label Examples: k-means clustering, k-nearest neighbors. Manifold ? A manifold is a topological space that is locally Euclidian Around every point, there is a neighborhood that is topologically isomorph to unit ball Regularization Adopt a functional viewpoint -
Contemporary Spectral Graph Theory
CONTEMPORARY SPECTRAL GRAPH THEORY GUEST EDITORS Maurizio Brunetti, University of Naples Federico II, Italy, [email protected] Raffaella Mulas, The Alan Turing Institute, UK, [email protected] Lucas Rusnak, Texas State University, USA, [email protected] Zoran Stanić, University of Belgrade, Serbia, [email protected] DESCRIPTION Spectral graph theory is a striking theory that studies the properties of a graph in relationship to the characteristic polynomial, eigenvalues and eigenvectors of matrices associated with the graph, such as the adjacency matrix, the Laplacian matrix, the signless Laplacian matrix or the distance matrix. This theory emerged in the 1950s and 1960s. Over the decades it has considered not only simple undirected graphs but also their generalizations such as multigraphs, directed graphs, weighted graphs or hypergraphs. In the recent years, spectra of signed graphs and gain graphs have attracted a great deal of attention. Besides graph theoretic research, another major source is the research in a wide branch of applications in chemistry, physics, biology, electrical engineering, social sciences, computer sciences, information theory and other disciplines. This thematic special issue is devoted to the publication of original contributions focusing on the spectra of graphs or their generalizations, along with the most recent applications. Contributions to the Special Issue may address (but are not limited) to the following aspects: f relations between spectra of graphs (or their generalizations) and their structural properties in the most general shape, f bounds for particular eigenvalues, f spectra of particular types of graph, f relations with block designs, f particular eigenvalues of graphs, f cospectrality of graphs, f graph products and their eigenvalues, f graphs with a comparatively small number of eigenvalues, f graphs with particular spectral properties, f applications, f characteristic polynomials. -
Learning on Hypergraphs: Spectral Theory and Clustering
Learning on Hypergraphs: Spectral Theory and Clustering Pan Li, Olgica Milenkovic Coordinated Science Laboratory University of Illinois at Urbana-Champaign March 12, 2019 Learning on Graphs Graphs are indispensable mathematical data models capturing pairwise interactions: k-nn network social network publication network Important learning on graphs problems: clustering (community detection), semi-supervised/active learning, representation learning (graph embedding) etc. Beyond Pairwise Relations A graph models pairwise relations. Recent work has shown that high-order relations can be significantly more informative: Examples include: Understanding the organization of networks (Benson, Gleich and Leskovec'16) Determining the topological connectivity between data points (Zhou, Huang, Sch}olkopf'07). Graphs with high-order relations can be modeled as hypergraphs (formally defined later). Meta-graphs, meta-paths in heterogeneous information networks. Algorithmic methods for analyzing high-order relations and learning problems are still under development. Beyond Pairwise Relations Functional units in social and biological networks. High-order network motifs: Motif (Benson’16) Microfauna Pelagic fishes Crabs & Benthic fishes Macroinvertebrates Algorithmic methods for analyzing high-order relations and learning problems are still under development. Beyond Pairwise Relations Functional units in social and biological networks. Meta-graphs, meta-paths in heterogeneous information networks. (Zhou, Yu, Han'11) Beyond Pairwise Relations Functional units in social and biological networks. Meta-graphs, meta-paths in heterogeneous information networks. Algorithmic methods for analyzing high-order relations and learning problems are still under development. Review of Graph Clustering: Notation and Terminology Graph Clustering Task: Cluster the vertices that are \densely" connected by edges. Graph Partitioning and Conductance A (weighted) graph G = (V ; E; w): for e 2 E, we is the weight. -
CS 598: Spectral Graph Theory. Lecture 10
CS 598: Spectral Graph Theory. Lecture 13 Expander Codes Alexandra Kolla Today Graph approximations, part 2. Bipartite expanders as approximations of the bipartite complete graph Quasi-random properties of bipartite expanders, expander mixing lemma Building expander codes Encoding, minimum distance, decoding Expander Graph Refresher We defined expander graphs to be d- regular graphs whose adjacency matrix eigenvalues satisfy |훼푖| ≤ 휖푑 for i>1, and some small 휖. We saw that such graphs are very good approximations of the complete graph. Expanders as Approximations of the Complete Graph Refresher Let G be a d-regular graph whose adjacency eigenvalues satisfy |훼푖| ≤ 휖푑. As its Laplacian eigenvalues satisfy 휆푖 = 푑 − 훼푖, all non-zero eigenvalues are between (1 − ϵ)d and 1 + ϵ 푑. 푇 푇 Let H=(d/n)Kn, 푠표 푥 퐿퐻푥 = 푑푥 푥 We showed that G is an ϵ − approximation of H 푇 푇 푇 1 − ϵ 푥 퐿퐻푥 ≼ 푥 퐿퐺푥 ≼ 1 + ϵ 푥 퐿퐻푥 Bipartite Expander Graphs Define them as d-regular good approximations of (some multiple of) the complete bipartite 푑 graph H= 퐾 : 푛 푛,푛 푇 푇 푇 1 − ϵ 푥 퐿퐻푥 ≼ 푥 퐿퐺푥 ≼ 1 + ϵ 푥 퐿퐻푥 ⇒ 1 − ϵ 퐻 ≼ 퐺 ≼ 1 + ϵ 퐻 G 퐾푛,푛 U V U V The Spectrum of Bipartite Expanders 푑 Eigenvalues of Laplacian of H= 퐾 are 푛 푛,푛 휆1 = 0, 휆2푛 = 2푑, 휆푖 = 푑, 0 < 푖 < 2푛 푇 푇 푇 1 − ϵ 푥 퐿퐻푥 ≼ 푥 퐿퐺푥 ≼ 1 + ϵ 푥 퐿퐻푥 Says we want d-regular graph G that has evals 휆1 = 0, 휆2푛 = 2푑, 휆푖 ∈ ( 1 − 휖 푑, 1 + 휖 푑), 0 < 푖 < 2푛 Construction of Bipartite Expanders Definition (Double Cover). -
Notes on Elementary Spectral Graph Theory Applications to Graph Clustering Using Normalized Cuts
University of Pennsylvania ScholarlyCommons Technical Reports (CIS) Department of Computer & Information Science 1-1-2013 Notes on Elementary Spectral Graph Theory Applications to Graph Clustering Using Normalized Cuts Jean H. Gallier University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/cis_reports Part of the Computer Engineering Commons, and the Computer Sciences Commons Recommended Citation Jean H. Gallier, "Notes on Elementary Spectral Graph Theory Applications to Graph Clustering Using Normalized Cuts", . January 2013. University of Pennsylvania Department of Computer and Information Science Technical Report No. MS-CIS-13-09. This paper is posted at ScholarlyCommons. https://repository.upenn.edu/cis_reports/986 For more information, please contact [email protected]. Notes on Elementary Spectral Graph Theory Applications to Graph Clustering Using Normalized Cuts Abstract These are notes on the method of normalized graph cuts and its applications to graph clustering. I provide a fairly thorough treatment of this deeply original method due to Shi and Malik, including complete proofs. I include the necessary background on graphs and graph Laplacians. I then explain in detail how the eigenvectors of the graph Laplacian can be used to draw a graph. This is an attractive application of graph Laplacians. The main thrust of this paper is the method of normalized cuts. I give a detailed account for K = 2 clusters, and also for K > 2 clusters, based on the work of Yu and Shi. Three points that do not appear to have been clearly articulated before are elaborated: 1. The solutions of the main optimization problem should be viewed as tuples in the K-fold cartesian product of projective space RP^{N-1}. -
Noncommutative Geometry, the Spectral Standpoint Arxiv
Noncommutative Geometry, the spectral standpoint Alain Connes October 24, 2019 In memory of John Roe, and in recognition of his pioneering achievements in coarse geometry and index theory. Abstract We update our Year 2000 account of Noncommutative Geometry in [68]. There, we de- scribed the following basic features of the subject: I The natural “time evolution" that makes noncommutative spaces dynamical from their measure theory. I The new calculus which is based on operators in Hilbert space, the Dixmier trace and the Wodzicki residue. I The spectral geometric paradigm which extends the Riemannian paradigm to the non- commutative world and gives a new tool to understand the forces of nature as pure gravity on a more involved geometric structure mixture of continuum with the discrete. I The key examples such as duals of discrete groups, leaf spaces of foliations and de- formations of ordinary spaces, which showed, very early on, that the relevance of this new paradigm was going far beyond the framework of Riemannian geometry. Here, we shall report1 on the following highlights from among the many discoveries made since 2000: 1. The interplay of the geometry with the modular theory for noncommutative tori. 2. Great advances on the Baum-Connes conjecture, on coarse geometry and on higher in- dex theory. 3. The geometrization of the pseudo-differential calculi using smooth groupoids. 4. The development of Hopf cyclic cohomology. 5. The increasing role of topological cyclic homology in number theory, and of the lambda arXiv:1910.10407v1 [math.QA] 23 Oct 2019 operations in archimedean cohomology. 6. The understanding of the renormalization group as a motivic Galois group. -
Spectral Theory of Translation Surfaces : a Short Introduction Volume 28 (2009-2010), P
Institut Fourier — Université de Grenoble I Actes du séminaire de Théorie spectrale et géométrie Luc HILLAIRET Spectral theory of translation surfaces : A short introduction Volume 28 (2009-2010), p. 51-62. <http://tsg.cedram.org/item?id=TSG_2009-2010__28__51_0> © Institut Fourier, 2009-2010, tous droits réservés. L’accès aux articles du Séminaire de théorie spectrale et géométrie (http://tsg.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://tsg.cedram.org/legal/). cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Séminaire de théorie spectrale et géométrie Grenoble Volume 28 (2009-2010) 51-62 SPECTRAL THEORY OF TRANSLATION SURFACES : A SHORT INTRODUCTION Luc Hillairet Abstract. — We define translation surfaces and, on these, the Laplace opera- tor that is associated with the Euclidean (singular) metric. This Laplace operator is not essentially self-adjoint and we recall how self-adjoint extensions are chosen. There are essentially two geometrical self-adjoint extensions and we show that they actually share the same spectrum Résumé. — On définit les surfaces de translation et le Laplacien associé à la mé- trique euclidienne (avec singularités). Ce laplacien n’est pas essentiellement auto- adjoint et on rappelle la façon dont les extensions auto-adjointes sont caractérisées. Il y a deux choix naturels dont on montre que les spectres coïncident. 1. Introduction Spectral geometry aims at understanding how the geometry influences the spectrum of geometrically related operators such as the Laplace oper- ator. The more interesting the geometry is, the more interesting we expect the relations with the spectrum to be. -
Discrete Spectral Geometry
Discrete Spectral Geometry Discrete Spectral Geometry Amir Daneshgar Department of Mathematical Sciences Sharif University of Technology Hossein Hajiabolhassan is mathematically dense in this talk!! May 31, 2006 Discrete Spectral Geometry Outline Outline A viewpoint based on mappings A general non-symmetric discrete setup Weighted energy spaces Comparison and variational formulations Summary A couple of comparison theorems Graph homomorphisms A comparison theorem The isoperimetric spectrum ε-Uniformizers Symmetric spaces and representation theory Epilogue Discrete Spectral Geometry A viewpoint based on mappings Basic Objects I H is a geometric objects, we are going to analyze. Discrete Spectral Geometry A viewpoint based on mappings Basic Objects I There are usually some generic (i.e. well-known, typical, close at hand, important, ...) types of these objects. Discrete Spectral Geometry A viewpoint based on mappings Basic Objects (continuous case) I These are models and objects of Euclidean and non-Euclidean geometry. Discrete Spectral Geometry A viewpoint based on mappings Basic Objects (continuous case) I Two important generic objects are the sphere and the upper half plane. Discrete Spectral Geometry A viewpoint based on mappings Basic Objects (discrete case) I Some important generic objects in this case are complete graphs, infinite k-regular tree and fractal-meshes in Rn. Discrete Spectral Geometry A viewpoint based on mappings Basic Objects (discrete case) I These are models and objects in network analysis and design, discrete state-spaces of algorithms and discrete geometry (e.g. in geometric group theory). Discrete Spectral Geometry A viewpoint based on mappings Comparison method I The basic idea here is to try to understand (or classify if we are lucky!) an object by comparing it with the generic ones. -
Pose Analysis Using Spectral Geometry
Vis Comput DOI 10.1007/s00371-013-0850-0 ORIGINAL ARTICLE Pose analysis using spectral geometry Jiaxi Hu · Jing Hua © Springer-Verlag Berlin Heidelberg 2013 Abstract We propose a novel method to analyze a set 1 Introduction of poses of 3D models that are represented with trian- gle meshes and unregistered. Different shapes of poses are Shape deformation is one of the major research topics in transformed from the 3D spatial domain to a geometry spec- computer graphics. Recently there have been extensive lit- trum domain that is defined by Laplace–Beltrami opera- eratures on this topic. There are two major categories of re- tor. During this space-spectrum transform, all near-isometric searches on this topic. One is surface-based and the other deformations, mesh triangulations and Euclidean transfor- one is skeleton-driven method. Shape interpolation [2, 9, 11] mations are filtered away. The different spatial poses from is a surface-based approach. The basic idea is that given two a 3D model are represented with near-isometric deforma- key frames of a shape, the intermediate deformation can be tions; therefore, they have similar behaviors in the spec- generated with interpolation and further deformations can be tral domain. Semantic parts of that model are then deter- predicted with extrapolation. It can also blend among sev- mined based on the computed geometric properties of all the eral shapes. Shape interpolation is convenient for animation mapped vertices in the geometry spectrum domain. Seman- generation but is not flexible enough for shape editing due tic skeleton can be automatically built with joints detected to the lack of local control. -
Spectral Graph Theory – from Practice to Theory
Spectral Graph Theory – From Practice to Theory Alexander Farrugia [email protected] Abstract Graph theory is the area of mathematics that studies networks, or graphs. It arose from the need to analyse many diverse network-like structures like road networks, molecules, the Internet, social networks and electrical networks. In spectral graph theory, which is a branch of graph theory, matrices are constructed from such graphs and analysed from the point of view of their so-called eigenvalues and eigenvectors. The first practical need for studying graph eigenvalues was in quantum chemistry in the thirties, forties and fifties, specifically to describe the Hückel molecular orbital theory for unsaturated conjugated hydrocarbons. This study led to the field which nowadays is called chemical graph theory. A few years later, during the late fifties and sixties, graph eigenvalues also proved to be important in physics, particularly in the solution of the membrane vibration problem via the discrete approximation of the membrane as a graph. This paper delves into the journey of how the practical needs of quantum chemistry and vibrating membranes compelled the creation of the more abstract spectral graph theory. Important, yet basic, mathematical results stemming from spectral graph theory shall be mentioned in this paper. Later, areas of study that make full use of these mathematical results, thus benefitting greatly from spectral graph theory, shall be described. These fields of study include the P versus NP problem in the field of computational complexity, Internet search, network centrality measures and control theory. Keywords: spectral graph theory, eigenvalue, eigenvector, graph. Introduction: Spectral Graph Theory Graph theory is the area of mathematics that deals with networks.