DOCSLIB.ORG

How Persistent Homology Can Be Useful in Predicting Proteins Functions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
How Persistent Homology Can Be Useful in Predicting Proteins Functions University of Rabat (Morocco) Faculty of Medicine MASTER BIOTECHNOLOGIE MEDICALE´ OPTION "BIOINFORMATIQUE" Projet Fin Annee´ (PFA) How persistent homology can be useful in predicting proteins functions Author: Zakaria Lamine Jury: President : ..... Examinator : Advisor : My Ismail Mamouni, CRMEF Rabat Soutenu le .... Dedication I dedicate this modest work to : My parents, especially my mother for all the sacrifices she has made, for all her assistance and presence in my life, my father for all his sacrifice and support. Thanks I thank GOD for giving me the strength to do this work. I thank my family for the support and the sacrifice. I thank my professor My Ismail Mamouni for his support and for all the discussions that have always been fruitful it is an honour to be one of his students. Introduction Persistent homology is a method for computing topological features of a space at different spatial resolutions. More persistent features are detected over a wide range of length and are deemed more likely to represent true features of the underlying space, rather than artifacts of sampling, noise, or particular choice of parameters. To find the persistent homology of a space, the space must first be represented as a simplicial complex. A distance function on the underlying space corresponds to a filtration of the simplicial complex, that is a nested sequence of increasing subsets. In other words persistent homology characterizes the geometric features with persistent topological invariants by defining a scale parameter relevant to topological events. Through filtration and persistence, persistent homology can capture topological structures continuously over a range of spatial scales. Unlike commonly used computational homology which results in truly metric free or coordinate free representations, persistent homol- ogy is able to embed geometric information to topological invariants so that \birth" and \death" of isolated components, circles, rings, loops, pockets, voids and cavities at all geometric scales can be monitored by topo- logical measurements. The basic concept was introduced by Frosini and Landi, and in a general form by Robins, Edelsbrunner and Zomorodian and Carlsson, independently. Efficient computational algorithms have been pro- posed to track topological variations during the filtration process. Usually, the persistent diagram is visualized through barcodes, in which various horizontal line segments or bars are the homology generators lasted over filtration scales. It has been applied to a variety of domains, including image analysis, image retrieval, chaotic dynamics verification, sensor network, complex network, data analysis, computer vision, shape recognition and computational biology. Compared with computational topology, and/or computational homology, persistent homology inherently has an additional dimension, the filtration parameter ,which can be utilized to embed some crucial geometric or quantitative information into the topological invariants. In medicine persistent homology has lead to a number of interesting results which can be visualized in chapter 4 by application of this method to detect some interesting properties of proteins which is an example from a bunch of other results . Contents 1 persistent homology of proteins6 1.1 proteins(overview)............................................6 1.2 structure-function relationship in proteins...............................6 1.3 introduction to persistent homology in analysis of proteins......................7 1.3.1 Prediction of the optimal characteristic distance using persistent homology........ 12 2 Conclusion 15 5 Chapter 1 Simplicial Homology 1.1 Simplicial complex In algebraic topology, simplicial homology formalizes the idea of the number of holes for a given dimension in a simplicial complex. This generalizes the number of connected components (the case of dimension 0). Simplicial homology arose as a way to study topological spaces whose building blocks are p-simplices, the p-dimensional analogs of triangles. This includes a point (0-simplex), a line segment (1-simplex), a triangle (2-simplex) and a tetrahedron (3-simplex). By defnition, such a space is homeomorphic to a simplicial complex (more precisely, the geometric realization of an abstract simplicial complex). Such a homeomorphism is referred to as a triangulation of the given space. Many topological spaces of interest can be triangulated, including every smooth manifold. Simplicial homology is defined by a simple recipe for any abstract simplicial complex. It is a remarkable fact that simplicial homology only depends on the associated topological space. As a result, it gives a computable way to distinguish one space from another. Singular homology is a related theory which is more commonly used by mathematicians today. Singular homology is defined for all topological spaces, and it agrees with simplicial homology for spaces which can be triangulated. Nonetheless, because it is possible to compute the simplicial homology of a simplicial complex automatically and efficiently, simplicial homology has become important for application to real-life situations, such as image analysis, medical imaging, and data analysis in general. In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their p- dimensional counterparts. Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. Figure 1.1: Simplicial complex p Definition 1 (p-simplex). A p-dimensional simplex (or p-simplex) σ = [e0; e1; :::; ep] is the smallest convex m set in a Euclidean space R containing the p + 1 points e0; :::; ep: We usually specify that for an p-simplex, we have that the points are not contained in any hyperplane of dimension less than p, The standard p-simplex is: p p p+1 X ∆ = f(t0; :::; tp) 2 R : ti = 1 and ti ≥ 0 for all i = 0; :::; pg i=0 Affinely independent means the p vectors xi − x0 for i = 1 : : : p are linearly independent, i.e., they are in general position. 6 ENSIAS MASTER BIOTECHNOLOGIE MEDICALE´ PFA : Persistent Homology Fac. Medecine OPTION "BIOINFORMATIQUE" The convex hull is simply the solid polyhedron determined by the p+1 vertices. A 0-simplex is a vertex, 1-simplex an edge, 2-simplex a triangle, and 3-simplex a tetrahedron Figure 1.2: 0,1,2 simplex Definition 2. Given a simplex σp, any (p − 1)-dimensional sub-simplex is called a face. For example, a tetrahedron has four triangle faces corresponding to the four subsets S obtained by removing one vertex at a time from σ. These four triangle faces are 2-simplices themselves. It also has six edge faces and four singleton vertex faces. Our space of interest is properly arranged simplices: Definition 3 (Simplicial complex). A simplicial complex K is a finite set of simplices satisfying the following conditions: 1. For all simplices A 2 K with α a face of A, we have α 2 K. 2. A; B 2 K ) A; B are properly situated. The dimension of a complex is the maximum dimension of the simplices contained in it. The intuition of simplicial complex is that if a simplex is in K, all its faces need to be in K, too. In addition, the simplicies have to be glued together along whole faces or be separate. The figure on the left is a simplicial complex, while the one on the right is not: Figure 1.3: Left simplicial complex ,Right not a simplicial complex Simplicial complex plays the role of the yellow space in the rubber band example. We next introduce the discrete version of the rubber bands. 1.2 Simplical homology groups Definition 4 (p-chains). A p-chains is a formal sum Np X p c = ciσi i=1 pi where σi are p-simplicies in K and ci 2 Z. Page 7 A. Akkari & Z. Lamine ENSIAS MASTER BIOTECHNOLOGIE MEDICALE´ PFA : Persistent Homology Fac. Medecine OPTION "BIOINFORMATIQUE" For example, let K be a tetrahedron. By definition the four triangle faces (i.e., 2-simplicies) are in K, too. A 2- chain is a subset of these four triangles, e.g., all four triangle, the bottom triangle face only, or the empty set. There are 24 distinct 2-chains. Similarly, by definition all six edges of the tetrahedron are in K, too. Thus, there are 26 distinct 1-chains. Despite the name "chain" a p-chain does not have to be connected. Figure 1.4: 2- chain on the left and a 1-chain (the blue edges) on the right Definition 5 (p-chain group). The p-chain group Cp(K) of k is a free Abelian group generated by the oriented p-simplexes of K. Np M Cp(K) ' Z i=1 By de finition Cp(K) = 0 for p > n When adding two p-chains we get another p- chain with duplicate p-simplices cancel out. We have a separate chain group for each dimension p. Figure 1.5: 1-chain addition Definition 6 (Boundary operator). The boundary operator is a homomorphism @p : Cp(K) ! Cp−1(K) defined as level of generator as follows: The boundary of an oriented p-simplex p σ = [p0; p1; ··· ; pp] is a (p - 1)-chain de fined by: p p X i @pσ = (−1) [e0; e1; :::; e^i; :::; ep] i=0 where e^i is omitted The boundary of a tetrahedron is the set of four triangles faces; the boundary of a triangle is its three edges; the boundary of an edge is its two vertices. The boundary of a tetrahedron is the set of four triangles faces; the boundary of a triangle is its three edges; the boundary of an edge is its two vertices. The boundary of a p-chain is de fined by linearity. Definition 7 (Chain complex). The chain complex is a sequence of free Abelian groups and homomorphisms i @p @p−1 @1 @0 0 ,! Cp(K) −! Cp−1(K) −! ::: −! C0(K) −! 0 where ,! denotes the inclusion map.
Load more

Recommended publications
  • Christopher J. Tralie –
    B [email protected] Í www.ctralie.com Christopher J. Tralie ctralie Research Interests Geometric signal processing, applied topology, nonlinear time series analysis, music information retrieval, video processing, computer graphics Academic Positions 8/1/2019 - Assistant Professor of Mathematics And Computer Science, Ursinus College. Col- Present legeville, Pennsylvania. 1/2019 - Postdoctoral Fellow, Department of Chemical And Biomedical Engineering, Johns 6/2019 Hopkins University. Baltimore, Maryland. 4/2017 - Postdoctoral Associate, Department of Mathematics, Duke University. Durham, 12/2018 North Carolina. Education 2011 - 2017 Ph.D., Duke University, Durham, NC, Electrical and Computer Engineering with Cer- tificate in College Teaching. Advisers: Guillermo Sapiro, John Harer Dissertation Title: “Geometric Multimedia Time Series” 2011 - 2013 M.S., Duke University, Durham, NC, Electrical and Computer Engineering. 2007 - 2011 B.S.E., Princeton University, Princeton, NJ, Electrical Engineering with Certificate in Computer Science (Cum Laude). Honors and Awards 2018 Deezer Hacking Audio And Music Research (HAMR) Best Code Award, Paris, France 2016 Top 5% Teachers At Duke: Dean’s award for ranking among top 5% (university wide) in student evaluations for Quality of Course or Intellectual Stimulation, Duke University, Spring 2016 2015 Duke University Department of Electrical Engineering Best Poster Award 2015 Duke University Bass Family Teaching Fellowship 2011 National Science Foundation Graduate Fellowship 2011 G. David Forney Jr. Prize in Signals and Systems at Princeton University 2009 Summer Undergraduate Fellowship in Robotics at Duke University: Awarded through the National Science Foundation’s Research Experience for Undergraduates (REU) Program 2007 Lockheed Martin National Merit Scholar 2006 Pennsylvania Governor School for the Sciences, Carnegie Mellon University Publications Journal Publications Paul Bendich, Ellen Gasparovic, John Harer, and Christopher J.
    [Show full text]
  • Download Malware Onto Their Devices
    P a g e | 1 Application of Data Analytics to Cyber Forensic Data A Major Qualifying Project Report submitted to the Faculty of the Worcester Polytechnic Institute in partial fulfillment of the requirements for the Degree of Bachelor of Science By Yao Yuan Chow Date: October 13, 2016 Approved: Professor: George T. Heineman, Computer Science Advisor Professor: Randy C. Paffenroth, Mathematical Sciences Advisor Sponsored by: MITRE Corporation P a g e | 2 Table of Contents Table of Figures..................................................................................................................................4 Table of Tables...................................................................................................................................5 Abstract .............................................................................................................................................6 Acknowledgements............................................................................................................................7 Executive Summary............................................................................................................................8 1 Introduction ............................................................................................................................. 12 2 Background .............................................................................................................................. 15 2.1 Tactical Cyber Threat Intelligence ......................................................................................
    [Show full text]
  • Persistent Homology and the Topology of Motor Cortical Activity
    Persistent homology and the topology of motor cortical activity Julian Salazar and Emma West [email protected] emma [email protected] · Abstract. We introduce the underlying principles of topological data analysis via persistent homology. We then preview how these can be applied to analyze the activity-based topologies of various neuronal classes in the motor cortex, along with their evolution as induced by engagement in a physical task. Contents 1. Introduction1 1.1. Topological data analysis1 1.2. Motor cortical activity3 2. Mathematical background4 2.1. Simplicial complexes4 2.2. Homology groups6 2.3. Persistent homology 10 2.4. Witness complexes 12 3. Data collection and classification 13 3.1. Experimental setup 13 3.2. Neuronal classifications 14 4. Topological analysis 16 4.1. Prior work 16 4.2. Activity and witness spaces 16 4.3. Persistence barcodes 17 5. Conclusions 18 Acknowledgements 20 References 20 1. Introduction 1.1. Topological data analysis Topology is the branch of mathematics which studies the qualitative geometry of a space. This includes information about connectivity and higher-dimensional structures such as holes and voids, attributes that are preserved under homeo- morphisms like continuous deformation. In recent years, the field of topological data analysis (TDA) has emerged as a way to employ topological tools such as homology to point clouds of discrete data. This technique is particularly useful in recovering qualitative information about a data set that might not be recognized using conventional analytic methods, as these methods often make assumptions of distance-dependence and linear separability. TDA generally returns a summary of 1 2 JULIAN SALAZAR AND EMMA WEST the overall data structure that is insensitive to coordinates and metric, via exam- ination of the relationships between functorial geometric constructions that arise from the data [Car09].
    [Show full text]
  • Persistence Homology of Networks: Methods and Applications
    Aktas et al. RESEARCH Persistence Homology of Networks: Methods and Applications Mehmet E Aktas1*, Esra Akbas2 and Ahmed El Fatmaoui1 *Correspondence: [email protected] Abstract 1Department of Mathematics and Statistics, University of Information networks are becoming increasingly popular to capture complex Central Oklahoma, Edmond, OK, relationships across various disciplines, such as social networks, citation networks, USA and biological networks. The primary challenge in this domain is measuring Full list of author information is available at the end of the article similarity or distance between networks based on topology. However, classical graph-theoretic measures are usually local and mainly based on differences between either node or edge measurements or correlations without considering the topology of networks such as the connected components or holes. In recent years, mathematical tools and deep learning based methods have become popular to extract the topological features of networks. Persistent homology (PH) is a mathematical tool in computational topology that measures the topological features of data that persist across multiple scales with applications ranging from biological networks to social networks. In this paper, we provide a conceptual review of key advancements in this area of using PH on complex network science. We give a brief mathematical background on PH, review different methods (i.e. filtrations) to define PH on networks and highlight different algorithms and applications where PH is used in solving network mining problems. In doing so, we develop a unified framework to describe these recent approaches and emphasize major conceptual distinctions. We conclude with directions for future work. We focus our review on recent approaches that get significant attention in the mathematics and data mining communities working on network data.
    [Show full text]
  • Multiradial (Multi)Filtrations and Persistent Homology
    MARTIN, JOSHUA M., M.A. Multiradial (Multi)Filtrations and Persistent Homology. (2016) Directed by Dr. Gregory Bell. 144 pp. Motivated by the problem of optimizing sensor network covers, we general- ize the persistent homology of simplicial complexes over a single radial parameter to the context of multiple radial parameters. The persistent homology of so-called multiradial (multi)filtrations is identified as a special case of multidimensional per- sistence. Specifically, we exhibit that the persistent homology of (multi)filtrations N corresponds to both generalized persistence modules of the form Z≥0 ! ModR and (multi)graded modules over a polynomial ring. The stability of persistence barcodes/diagrams of multiradial filtrations is derived, along with explicit bounds associated to perturbations in both radii and vertex position. A strengthening of the Vietoris-Rips lemma of [DSG07, p. 346] to the setting of multiple radial param- eters is obtained. We also use the categorical framework of [BdSS15] to show the persistent homology modules of multiradial (multi)filtrations are stable. MULTIRADIAL (MULTI)FILTRATIONS AND PERSISTENT HOMOLOGY by Joshua M. Martin A Thesis Submitted to the Faculty of The Graduate School at The University of North Carolina at Greensboro in Partial Fulfillment of the Requirements for the Degree Master of Arts Greensboro 2016 Approved by Committee Chair For my family ii APPROVAL PAGE This thesis written by Joshua M. Martin has been approved by the following committee of the Faculty of The Graduate School at The University of North Car- olina at Greensboro. Committee Chair Gregory Bell Committee Members Maya Chhetri Clifford Smyth Date of Acceptance by Committee Date of Final Oral Examination iii ACKNOWLEDGMENTS I would like to thank Austin Lawson, Cliff Smyth, Greg Bell, and James Rudzin- ski, whose input over the past two years made this thesis possible.
    [Show full text]
  • Topological Methods for 3D Point Cloud Processing
    Topological Methods for 3D Point Cloud Processing A DISSERTATION SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY William Joseph Beksi IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy Nikolaos Papanikolopoulos, Adviser August, 2018 c William Joseph Beksi 2018 ALL RIGHTS RESERVED Acknowledgements First and foremost, I thank my adviser Nikolaos Papanikolopoulos. Nikos was the first person in my career to see my potential and give me the freedom to pursue my own research interests. My success is a direct result of his advice, encouragement, and support over the years. I am beholden to my labmates at the Center for Distributed Robotics for their questions, comments, and healthy criticism of my work. I'm especially grateful to Duc Fehr for letting me get involved in his research as a new member of the lab. I also thank all of my committee members for their feedback through out this entire process. The inspiration for this research is due to the Institute for Mathematics and its Applications (IMA) located at the University of Minnesota. Each year the IMA explores a theme in applied mathematics through a series of tutorials, workshops, and seminars. It was during the Scientific and Engineering Applications of Algebraic Topology (2013- 2014) that I began to conceive of the ideas for this thesis. This work would not be possible without the computational, storage, and software resources provided by the Minnesota Supercomputing Institute (MSI) and its excellent staff. I especially thank Jeffrey McDonald for his friendship and support. I acknowledge and thank the National Science Foundation (NSF) for supporting this research and the University of Minnesota Informatics Institute (UMII) for funding my fellowship.
    [Show full text]
  • Multi-Scale Persistent Homology
    LAWSON, AUSTIN, M.A. Multi-Scale Persistent Homology. (2016) Directed by Dr. Greg Bell. 62 pp. Data has shape and that shape is important. This is the anthem of Topological Data Analysis (TDA) as often stated by Gunnar Carlsson. In this paper, we take a common method of persistence involving the growing of balls of the same size and generalize this situation to one where the balls have dierent sizes. This helps us to better understand the outlier and coverage problems. We begin with a summary of classical persistence theory and stability. We then move on to generalizing the Rips and ech complexes as well as generalizing the Rips lemma. We transition into 3 notions of stability in terms of bottleneck distance. For the outlier problem, we show that it is possible to interpolate between persistence on a set with no noise and a set with noise. For the coverage problem, we present an algorithm which provides a cheap way of covering a compact domain. MULTI-SCALE PERSISTENT HOMOLOGY by Austin Lawson A Thesis Submitted to the Faculty of The Graduate School at The University of North Carolina at Greensboro in Partial Fulllment of the Requirements for the Degree Master of Arts Greensboro 2016 Approved by Committee Chair APPROVAL PAGE This thesis written by Austin Lawson has been approved by the following com- mittee of the Faculty of The Graduate School at The University of North Carolina at Greensboro. Committee Chair Greg Bell Committee Members Cli Smyth Maya Chhetri Date of Acceptance by Committee Date of Final Oral Examination ii ACKNOWLEDGMENTS I would like to take this time to express my sincere gratitude to my advisor, Dr.
    [Show full text]
  • Ripser.Py: a Lean Persistent Homology Library for Python
    Ripser.py: A Lean Persistent Homology Library for Python Christopher Tralie1, Nathaniel Saul2, and Rann Bar-On1 1 Department of Mathematics, Duke University 2 Department of Mathematics and Statistics, DOI: 10.21105/joss.00925 Washington State University Software • Review Summary • Repository • Archive Topological data analysis (TDA) (Edelsbrunner & Harer, 2010), (Carlsson, 2009) is a Submitted: 17 August 2018 field focused on understanding the shape and structure of data by computing topological Published: 13 September 2018 descriptors that summarize features as connected components, loops, and voids. TDA has found wide applications across nonlinear time series analysis (Perea & Harer, 2015), com- License Authors of papers retain copyright puter vision (Perea & Carlsson, 2014), computational neuroscience (Giusti, Pastalkova, and release the work under a Cre- Curto, & Itskov, 2015), (Bendich, Marron, Miller, Pieloch, & Skwerer, 2016), compu- ative Commons Attribution 4.0 In- tational biology (Iyer-Pascuzzi et al., 2010), (Wu et al., 2017), and materials science ternational License (CC-BY). (Kramar, Goullet, Kondic, & Mischaikow, 2013), to name a few of the many areas of impact in recent years. Persistent homology (Edelsbrunner & Harer, 2010) is the main workhorse of TDA, and it computes a data structure known as the persistence diagram to summarize the space of stable topological features. The most commonly used scheme for generating persistence diagrams is the Vietoris Rips filtration (VR) since it is easily defined for any point cloud. In its naive implementation, VR is prohibitively slow, but recently a C++ library known as Ripser (Bauer, 2017) has been devised to aggregate all known computational speedups of the VR filtration into one concise implementation.
    [Show full text]
  • Homology and Persistent Homology Bootcamp Notes Summer@ICERM 2017
    Homology and Persistent Homology Bootcamp Notes Summer@ICERM 2017 Melissa McGuirl September 19, 2017 1 Preface This bootcamp is intended to be a review session for the material covered in Week 1 of Summer@ICERM 2017. My goal is to clarify any confusing topics covered in Week 1 and to present the material in a way that connects all of the material into a full picture. If you have any questions about the material please feel free to email me at melissa [email protected]. Most of the exercises have been taken directly from the references listed at the end. 2 Introduction to Topological Data Analysis Topological data analysis (TDA) lies in the intersection of data science and algebraic topol- ogy. In data science, one of the main goals is to extract important features from large datasets and then classify or cluster the data based on their features. There are several methods for doing this and the choice of method depends largely on the data being studied. In algebraic topology, one of the main goals is to classify topological spaces based on their topological features. There are two primary tools for classifying topological spaces: homo- topy and homology. While homotopy is easier to define, it is nearly impossible to compute the homotopy type of a given space. In contrast, homology is tricky to define but as we have seen it is \easy" to compute. For this reason, homology is the tool that connects topology to data science. Datasets often come in the form of point clouds lying in some metric space.
    [Show full text]
  • Javaplex Tutorial
    JAVAPLEX TUTORIAL HENRY ADAMS AND ANDREW TAUSZ Contents 1. Introduction 2 1.1. Javaplex 2 1.2. License 2 1.3. Installation for Matlab 2 1.4. Accompanying files 3 2. Math review 3 2.1. Simplicial complexes 3 2.2. Homology 3 2.3. Filtered simplicial complexes 3 2.4. Persistent homology 3 3. Explicit simplex streams 3 3.1. Computing homology 4 3.2. Computing persistent homology 6 4. Point cloud data 10 4.1. Euclidean metric spaces 10 4.2. Explicit metric spaces 11 5. Streams from point cloud data 13 5.1. Vietoris{Rips streams 13 5.2. Landmark selection 17 5.3. Witness streams 19 5.4. Lazy witness streams 21 6. Examples with real data 23 6.1. Range image patches 23 6.2. Optical image patches 26 6.3. Cyclo-octane molecule conformations 31 7. Remarks 33 7.1. Java heap size 33 7.2. Matlab functions with Javaplex commands 34 7.3. Displaying the simplices in a stream 34 7.4. Displaying the boundary matrix of a homology computation 34 7.5. Computing the bottleneck distance 34 8. Acknowledgements 34 Appendices 34 Appendix A. Dense core subsets 34 Appendix B. Exercise solutions 36 References 41 Date: July 18, 2019. 1 1. Introduction 1.1. Javaplex. Javaplex is a Java software package for computing the persistent homology of filtered sim- plicial complexes (or more generally, filtered chain complexes), with special emphasis on applications arising in topological data analysis [Tausz et al. 2014]. The main author is Andrew Tausz. Javaplex is a re- write of the JPlex package, which was written by Harlan Sexton and Mikael Vejdemo{Johansson.
    [Show full text]
  • Finding Hidden Order in Spin Models with Persistent Homology
    PHYSICAL REVIEW RESEARCH 2, 043308 (2020) Finding hidden order in spin models with persistent homology Bart Olsthoorn ,1,* Johan Hellsvik ,1,† and Alexander V. Balatsky 1,2,‡ 1Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden 2Department of Physics, University of Connecticut, Storrs, Connecticut 06269, USA (Received 10 September 2020; revised 31 October 2020; accepted 3 November 2020; published 2 December 2020) Persistent homology (PH) is a relatively new field in applied mathematics that studies the components and shapes of discrete data. In this paper, we demonstrate that PH can be used as a universal framework to identify phases of classical spins on a lattice. This demonstration includes hidden order such as spin-nematic ordering and spin liquids. By converting a small number of spin configurations to barcodes we obtain a descriptive picture of configuration space. Using dimensionality reduction to reduce the barcode space to color space leads to a visualization of the phase diagram. DOI: 10.1103/PhysRevResearch.2.043308 I. INTRODUCTION that both the easily identifiable phases, such as the ferromag- net, and more complicated and not easily visualized orders, In condensed-matter physics we often deal with materials such as spin nematic, do describe strong correlations between where collective degrees of freedom such as magnetization, spins. Of relevance for the present work are classical and density modulations, coherent condensate formation, etc., can quantum spin liquids [1–5], the spin-ice phase [6–8], and undergo order-to-disorder transitions as a function of external nematic ordering that involves breaking of spin rotational parameters.
    [Show full text]
  • Statistical Learning and Analysis on Homology- Based Features
    DEGREE PROJECT IN MATHEMATICS, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2020 Statistical Learning and Analysis on Homology- Based Features JENS AGERBERG KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES Statistical Learning and Analysis on Homology-Based Features JENS AGERBERG Degree Projects in Mathematical Statistics (30 ECTS credits) Master's Programme in Applied and Computational Mathematics KTH Royal Institute of Technology year 2020 Supervisor at RISE SICS: Ather Gattami Supervisors at KTH: Martina Scolamiero, Wojciech Chachólski Examiner at KTH: Martina Scolamiero TRITA-SCI-GRU 2020:056 MAT-E 2020:019 Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci Abstract Stable rank has recently been proposed as an invariant to encode the result of persistent homology, a method used in topological data analysis. In this thesis we develop methods for statistical analysis as well as machine learning methods based on stable rank. As stable rank may be viewed as a mapping to a Hilbert space, a kernel can be constructed from the inner product in this space. First, we investigate this kernel in the context of kernel learning methods such as support-vector machines. Next, using the theory of kernel embedding of probability distributions, we give a statistical treatment of the kernel by showing some of its properties and develop a two-sample hypothesis test based on the kernel. As an alternative approach, a mapping to Rd with learnable parameters can be conceived, serving as an input layer to a neural network. The developed methods are first evaluated on synthetic data.
    [Show full text]