DOCSLIB.ORG

Embedding Smooth Diffeomorphisms in Flows

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Embedding Smooth Diffeomorphisms in Flows View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector J. Differential Equations 248 (2010) 1603–1616 Contents lists available at ScienceDirect Journal of Differential Equations www.elsevier.com/locate/jde Embedding smooth diffeomorphisms in flows Xiang Zhang 1 Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, People’s Republic of China article info abstract Article history: In this paper we study the problem on embedding germs of Received 24 May 2009 smooth diffeomorphisms in flows in higher dimensional spaces. Revised 10 August 2009 First we prove the existence of embedding vector fields for a local Available online 30 September 2009 diffeomorphism with its nonlinear term a resonant polynomial. Then using this result and the normal form theory, we obtain MSC: k ∈ N ∪{∞ } 34A34 a class of local C diffeomorphisms for k , ω which 34C41 admit embedding vector fields with some smoothness. Finally we 37G05 prove that for any k ∈ N ∪{∞} under the coefficient topology the 58D05 subset of local Ck diffeomorphisms having an embedding vector field with some smoothness is dense in the set of all local Ck Keywords: diffeomorphisms. Local diffeomorphism © 2009 Elsevier Inc. All rights reserved. Embedding flow Smoothness 1. Introduction and statement of the main results Let F(x) be a Ck smooth diffeomorphism on a smooth manifold M in Rn with k ∈ N ∪{∞, ω}, where N is the set of natural numbers and Cω denotes the class of analytic functions. A vector field X defined on the manifold M is called an embedding vector field of F(x) if F(x) is the Poincaré map of the vector field X . If the vector field X is autonomous, the corresponding flow denoted by {φt , t ∈ R} is called an embedding flow of F(x),i.e.wehaveφ1 = F. We note that when we consider the relation between diffeomorphisms and the Poincaré map of a vector field, it appears naturally the embedding problem. The embedding problem in the 1-dimensional case has been intensively studied (see for instance, [8,15–17,6,13]). In the higher dimensional case, Palis [19] in 1974 pointed out that in the Baire sense there are few diffeomorphisms which can admit embedding flows. For the existence of periodic embedding vector field, Arnold [2, p. 200] in 1988 mentioned that for ∞ n × n real matrices A and B with A = eB,ifF(x) = Ax + o(|x|) is a C diffeomorphism in (Rn, 0),then E-mail address: [email protected]. 1 The author is partially supported by NNSF of China grants 10671123 and 10831003, and Shanghai Pujiang Program. 0022-0396/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jde.2009.09.013 1604 X. Zhang / J. Differential Equations 248 (2010) 1603–1616 ∞ F(x) admits a C periodic embedding vector field Bx + r(x,t) of period 1 in (Rn, 0). In 1992 Kuksin and Pöschel [14] solved an embedding problem for a local analytic symplectic diffeomorphism to be embedded in a periodic analytic Hamiltonian vector field. The problem on the existence of the embedding flow of a given diffeomorphism becomes much more involved. Arnold [2, p. 200] mentioned that it is usually not possible to imbed a given map- ping in the phase flow of an autonomous system. Li et al. [18] in 2002 proved that for a germ of ∞ C diffeomorphisms in (Rn, 0) with the linear part A hyperbolic, if A has a real logarithm B with ∞ the eigenvalues weakly non-resonant, then the germ can be embedded in a germ of C autonomous ∞ vector fields having the linear part B. Recently this result was extended to the C local diffeomor- phisms in Banach spaces [24] by using the spectral theory of linear operators, the extended Floquet’s ∞ theory and the C conjugacy in Banach spaces. As we know there are few results on the existence of embedding flows for both finite smooth and analytic diffeomorphisms (see [23]). This paper will tackle this problem. The method of this paper is different from those of [18,24]. It is well known that if X (x) is an embedding vector field of a diffeomorphism F(x) on a manifold M and φt (x) is the corresponding embedding flow, then X (φt (x)) = D(φt (x))X (x) for x ∈ M and t ∈ R, where Dψ(x) denotes the Jacobian matrix of a differentiable map ψ at x.Consequently,we have X F(x) = D F(x) X (x). (1.1) It is called the embedding equation. Conversely a vector field X (x) satisfying equation (1.1) may not be an embedding vector field of the given diffeomorphism F(x). This fact will be verified via the remark following Example 1.6. Even though, by solving the embedding equation (1.1) we can obtain some candidate vector fields which may be used for solving the embedding problem. Another way to solve the embedding problem is to use the conjugacy: let G and H be two conju- gate diffeomorphisms on a manifold, i.e. there exists a diffeomorphism J such that J ◦ G = H ◦ J,ifX is an embedding vector field of G then J∗X is an embedding vector field of H, where J∗ denotes the tan- gent map induced by J. So, to solve the embedding problem for a given diffeomorphism we can try to − find a conjugate diffeomorphism which has an embedding flow. Since F is conjugate to G := B 1F ◦ B for any invertible n × n matrix B, in the following we can assume without loss of generality that the matrix A of the linear part of F is in the lower triangular Jordan normal form. In this paper we consider the germ F(x) of local Ck diffeomorphisms with k 1in(Rn, 0).Set F(x) = Ax + f(x) with A a real square matrix, f(0) = 0 and Df(0) = 0. In order for X (x) = Bx + v(x) to be an embedding vector field of F(x), the linear part of X should be an embedding vector field of Ax. That is to say, B must be a real logarithm of A,i.e.B = log A.SoifX (x) is an embedding vector field of F(x), v(x) should satisfy the equation v Ax + f(x) − Av(x) = Df(x)v(x) + Df(x)Bx − Bf(x). (1.2) = ∈ Cn Let the n-tuple λ (λ1,...,λn) be the eigenvalues of A. The eigenvalues λare resonant = m ∈{ } = ∈ Zn | |= n if λi λ for some i 1,...,n and m (m1,...,mn) + with the length m j=1 m j > 1, m = m1 · · mn Z ={ ∈ Z; } where λ λ1 ... λn , and + z z 0 . The eigenvalues λ satisfy finite resonant conditions n m if the set {m ∈ Z+; λi = λ for some i ∈{1,...,n} and |m| > 1} is finite. For the vector field f(x) = m m ( f1(x),..., fn(x)), a monomial amx of fi(x) is resonant if λi = λ , where am is the coefficient of the monomial depending on m. The vector field f(x) is resonant if its monomials are all resonant. The fixed point O of F(x) is hyperbolic if A is hyperbolic,i.e.A has no eigenvalues on the unit circle of the complex plane. We say that λ belong to the Poincaré domain if the n eigenvalues have modulus either all larger than 1 or all less than 1. If it is not the case, we say that λ belong to the Siegel domain.Forλ belonging to the Siegel domain, if there exist c, τ > 0 such that m c n 0 = λi − λ for i = 1,...,n, m ∈ Z+ and |m| 2, |m|τ X. Zhang / J. Differential Equations 248 (2010) 1603–1616 1605 we say that λ satisfy the Siegel condition.If ln ω − k < ∞, 2k k0 m n k where ωk = inf{|λi − λ |=0, 1 i n, m ∈ Z+, 2 |m| 2 },wecallλ satisfy the Brjuno diophan- tine condition. It is known that the Brjuno diophantine condition is weaker than the Siegel condition (see e.g. [21]). Our first result is on the existence of embedding flows for a polynomial map with its nonlinear terms all resonant. Its proof will be given in Section 2. Theorem 1.1. Assume that F(x) = Ax + f(x) with f(x) a resonant vector-valued polynomial without linear part and that A is hyperbolic and has a real logarithm B. If the eigenvalues λ of A satisfy finite resonant conditions, then F(x) has a unique embedding vector field X (x) = Bx + v(x) with v(x) a resonant polynomial vector field. This result will be the key point to prove the other ones of this paper. In Theorem 1.1 we have an additional assumption on the finite resonance of the eigenvalues λ. We should say that if the eigenvalues λ belong to the Poincaré domain, this assumption will be satisfied automatically. We remark that if the eigenvalues of A satisfy infinitely many resonant conditions, we do not know whether Theorem 1.1 holds. Because even though F(x) has its nonlinear part consisting of finitely many resonant monomials, the candidate embedding vector field probably has its nonlinear part formed by infinitely many number of resonant monomials. If it is the case, we do not know whether the infinite series is convergent. On the other hand, if the eigenvalues of A satisfy infinitely many res- onant conditions, our method cannot be used to prove the existence of the embedding vector field.
Load more

Recommended publications
  • Planar Embeddings of Minc's Continuum and Generalizations
    PLANAR EMBEDDINGS OF MINC’S CONTINUUM AND GENERALIZATIONS ANA ANUSIˇ C´ Abstract. We show that if f : I → I is piecewise monotone, post-critically finite, x X I,f and locally eventually onto, then for every point ∈ =←− lim( ) there exists a planar embedding of X such that x is accessible. In particular, every point x in Minc’s continuum XM from [11, Question 19 p. 335] can be embedded accessibly. All constructed embeddings are thin, i.e., can be covered by an arbitrary small chain of open sets which are connected in the plane. 1. Introduction The main motivation for this study is the following long-standing open problem: Problem (Nadler and Quinn 1972 [20, p. 229] and [21]). Let X be a chainable contin- uum, and x ∈ X. Is there a planar embedding of X such that x is accessible? The importance of this problem is illustrated by the fact that it appears at three independent places in the collection of open problems in Continuum Theory published in 2018 [10, see Question 1, Question 49, and Question 51]. We will give a positive answer to the Nadler-Quinn problem for every point in a wide class of chainable continua, which includes←− lim(I, f) for a simplicial locally eventually onto map f, and in particular continuum XM introduced by Piotr Minc in [11, Question 19 p. 335]. Continuum XM was suspected to have a point which is inaccessible in every planar embedding of XM . A continuum is a non-empty, compact, connected, metric space, and it is chainable if arXiv:2010.02969v1 [math.GN] 6 Oct 2020 it can be represented as an inverse limit with bonding maps fi : I → I, i ∈ N, which can be assumed to be onto and piecewise linear.
    [Show full text]
  • Neural Subgraph Matching
    Neural Subgraph Matching NEURAL SUBGRAPH MATCHING Rex Ying, Andrew Wang, Jiaxuan You, Chengtao Wen, Arquimedes Canedo, Jure Leskovec Stanford University and Siemens Corporate Technology ABSTRACT Subgraph matching is the problem of determining the presence of a given query graph in a large target graph. Despite being an NP-complete problem, the subgraph matching problem is crucial in domains ranging from network science and database systems to biochemistry and cognitive science. However, existing techniques based on combinatorial matching and integer programming cannot handle matching problems with both large target and query graphs. Here we propose NeuroMatch, an accurate, efficient, and robust neural approach to subgraph matching. NeuroMatch decomposes query and target graphs into small subgraphs and embeds them using graph neural networks. Trained to capture geometric constraints corresponding to subgraph relations, NeuroMatch then efficiently performs subgraph matching directly in the embedding space. Experiments demonstrate that NeuroMatch is 100x faster than existing combinatorial approaches and 18% more accurate than existing approximate subgraph matching methods. 1.I NTRODUCTION Given a query graph, the problem of subgraph isomorphism matching is to determine if a query graph is isomorphic to a subgraph of a large target graph. If the graphs include node and edge features, both the topology as well as the features should be matched. Subgraph matching is a crucial problem in many biology, social network and knowledge graph applications (Gentner, 1983; Raymond et al., 2002; Yang & Sze, 2007; Dai et al., 2019). For example, in social networks and biomedical network science, researchers investigate important subgraphs by counting them in a given network (Alon et al., 2008).
    [Show full text]
  • Some Planar Embeddings of Chainable Continua Can Be
    Some planar embeddings of chainable continua can be expressed as inverse limit spaces by Susan Pamela Schwartz A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Montana State University © Copyright by Susan Pamela Schwartz (1992) Abstract: It is well known that chainable continua can be expressed as inverse limit spaces and that chainable continua are embeddable in the plane. We give necessary and sufficient conditions for the planar embeddings of chainable continua to be realized as inverse limit spaces. As an example, we consider the Knaster continuum. It has been shown that this continuum can be embedded in the plane in such a manner that any given composant is accessible. We give inverse limit expressions for embeddings of the Knaster continuum in which the accessible composant is specified. We then show that there are uncountably many non-equivalent inverse limit embeddings of this continuum. SOME PLANAR EMBEDDINGS OF CHAIN ABLE OONTINUA CAN BE EXPRESSED AS INVERSE LIMIT SPACES by Susan Pamela Schwartz A thesis submitted in partial fulfillment of the requirements for the degree of . Doctor of Philosophy in Mathematics MONTANA STATE UNIVERSITY Bozeman, Montana February 1992 D 3 l% ii APPROVAL of a thesis submitted by Susan Pamela Schwartz This thesis has been read by each member of the thesis committee and has been found to be satisfactory regarding content, English usage, format, citations, bibliographic style, and consistency, and is ready for submission to the College of Graduate Studies. g / / f / f z Date Chairperson, Graduate committee Approved for the Major Department ___ 2 -J2 0 / 9 Date Head, Major Department Approved for the College of Graduate Studies Date Graduate Dean iii STATEMENT OF PERMISSION TO USE .
    [Show full text]
  • Cauchy Graph Embedding
    Cauchy Graph Embedding Dijun Luo [email protected] Chris Ding [email protected] Feiping Nie [email protected] Heng Huang [email protected] The University of Texas at Arlington, 701 S. Nedderman Drive, Arlington, TX 76019 Abstract classify unsupervised embedding approaches into two cat- Laplacian embedding provides a low- egories. Approaches in the first category are to embed data dimensional representation for the nodes of into a linear space via linear transformations, such as prin- a graph where the edge weights denote pair- ciple component analysis (PCA) (Jolliffe, 2002) and mul- wise similarity among the node objects. It is tidimensional scaling (MDS) (Cox & Cox, 2001). Both commonly assumed that the Laplacian embed- PCA and MDS are eigenvector methods and can model lin- ding results preserve the local topology of the ear variabilities in high-dimensional data. They have been original data on the low-dimensional projected long known and widely used in many machine learning ap- subspaces, i.e., for any pair of graph nodes plications. with large similarity, they should be embedded However, the underlying structure of real data is often closely in the embedded space. However, in highly nonlinear and hence cannot be accurately approx- this paper, we will show that the Laplacian imated by linear manifolds. The second category ap- embedding often cannot preserve local topology proaches embed data in a nonlinear manner based on differ- well as we expected. To enhance the local topol- ent purposes. Recently several promising nonlinear meth- ogy preserving property in graph embedding, ods have been proposed, including IsoMAP (Tenenbaum we propose a novel Cauchy graph embedding et al., 2000), Local Linear Embedding (LLE) (Roweis & which preserves the similarity relationships of Saul, 2000), Local Tangent Space Alignment (Zhang & the original data in the embedded space via a Zha, 2004), Laplacian Embedding/Eigenmap (Hall, 1971; new objective.
    [Show full text]
  • Strong Inverse Limit Reflection
    Strong Inverse Limit Reflection Scott Cramer March 4, 2016 Abstract We show that the axiom Strong Inverse Limit Reflection holds in L(Vλ+1) assuming the large cardinal axiom I0. This reflection theorem both extends results of [4], [5], and [3], and has structural implications for L(Vλ+1), as described in [3]. Furthermore, these results together highlight an analogy between Strong Inverse Limit Reflection and the Axiom of Determinacy insofar as both act as fundamental regularity properties. The study of L(Vλ+1) was initiated by H. Woodin in order to prove properties of L(R) under large cardinal assumptions. In particular he showed that L(R) satisfies the Axiom of Determinacy (AD) if there exists a non-trivial elementary embedding j : L(Vλ+1) ! L(Vλ+1) with crit (j) < λ (an axiom called I0). We investigate an axiom called Strong Inverse Limit Reflection for L(Vλ+1) which is in some sense analogous to AD for L(R). Our main result is to show that if I0 holds at λ then Strong Inverse Limit Reflection holds in L(Vλ+1). Strong Inverse Limit Reflection is a strong form of a reflection property for inverse limits. Axioms of this form generally assert the existence of a collection of embeddings reflecting a certain amount of L(Vλ+1), together with a largeness assumption on the collection. There are potentially many different types of axioms of this form which could be considered, but we concentrate on a particular form which, by results in [3], has certain structural consequences for L(Vλ+1), such as a version of the perfect set property.
    [Show full text]
  • Lecture 9: the Whitney Embedding Theorem
    LECTURE 9: THE WHITNEY EMBEDDING THEOREM Historically, the word \manifold" (Mannigfaltigkeit in German) first appeared in Riemann's doctoral thesis in 1851. At the early times, manifolds are defined extrinsi- cally: they are the set of all possible values of some variables with certain constraints. Translated into modern language,\smooth manifolds" are objects that are (locally) de- fined by smooth equations and, according to last lecture, are embedded submanifolds in Euclidean spaces. In 1912 Weyl gave an intrinsic definition for smooth manifolds. A natural question is: what is the difference between the extrinsic definition and the intrinsic definition? Is there any \abstract" manifold that cannot be embedded into any Euclidian space? In 1930s, Whitney and others settled this foundational problem: the two ways of defining smooth manifolds are in fact the same. In fact, Whitney's result is much more stronger than this. He showed that not only one can embed any smooth manifold into some Euclidian space, but that the dimension of the Euclidian space can be chosen to be (as low as) twice the dimension of the manifold itself! Theorem 0.1 (The Whitney embedding theorem). Any smooth manifold M of di- mension m can be embedded into R2m+1. Remark. In 1944, by using completely different techniques (now known as the \Whitney trick"), Whitney was able to prove Theorem 0.2 (The Strong Whitney Embedding Theorem). Any smooth man- ifold M of dimension m ≥ 2 can be embedded into R2m (and can be immersed into R2m−1). We will not prove this stronger version in this course, but just mention that the Whitney trick was further developed in h-cobordism theory by Smale, using which he proved the Poincare conjecture in dimension ≥ 5 in 1961! Remark.
    [Show full text]
  • A Novel Approach to Embedding of Metric Spaces
    A Novel Approach to Embedding of Metric Spaces Thesis submitted for the degree of Doctor of Philosophy By Ofer Neiman Submitted to the Senate of the Hebrew University of Jerusalem April, 2009 This work was carried out under the supervision of: Prof. Yair Bartal 1 Abstract An embedding of one metric space (X, d) into another (Y, ρ) is an injective map f : X → Y . The central genre of problems in the area of metric embedding is finding such maps in which the distances between points do not change “too much”. Metric Embedding plays an important role in a vast range of application areas such as computer vision, computational biology, machine learning, networking, statistics, and mathematical psychology, to name a few. The mathematical theory of metric embedding is well studied in both pure and applied analysis and has more recently been a source of interest for computer scientists as well. Most of this work is focused on the development of bi-Lipschitz mappings between metric spaces. In this work we present new concepts in metric embeddings as well as new embedding methods for metric spaces. We focus on finite metric spaces, however some of the concepts and methods may be applicable in other settings as well. One of the main cornerstones in finite metric embedding theory is a celebrated theorem of Bourgain which states that every finite metric space on n points embeds in Euclidean space with O(log n) distortion. Bourgain’s result is best possible when considering the worst case distortion over all pairs of points in the metric space.
    [Show full text]
  • Isomorphism and Embedding Problems for Infinite Limits of Scale
    Isomorphism and Embedding Problems for Infinite Limits of Scale-Free Graphs Robert D. Kleinberg ∗ Jon M. Kleinberg y Abstract structure of finite PA graphs; in particular, we give a The study of random graphs has traditionally been characterization of the graphs H for which the expected dominated by the closely-related models (n; m), in number of subgraph embeddings of H in an n-node PA which a graph is sampled from the uniform distributionG graph remains bounded as n goes to infinity. on graphs with n vertices and m edges, and (n; p), in n G 1 Introduction which each of the 2 edges is sampled independently with probability p. Recen tly, however, there has been For decades, the study of random graphs has been dom- considerable interest in alternate random graph models inated by the closely-related models (n; m), in which designed to more closely approximate the properties of a graph is sampled from the uniformG distribution on complex real-world networks such as the Web graph, graphs with n vertices and m edges, and (n; p), in n G the Internet, and large social networks. Two of the most which each of the 2 edges is sampled independently well-studied of these are the closely related \preferential with probability p.The first was introduced by Erd}os attachment" and \copying" models, in which vertices and R´enyi in [16], the second by Gilbert in [19]. While arrive one-by-one in sequence and attach at random in these random graphs have remained a central object \rich-get-richer" fashion to d earlier vertices.
    [Show full text]
  • Inverse Limit Spaces of Interval Maps
    FACULTY OF SCIENCE DEPARTMENT OF MATHEMATICS Ana Anušic´ INVERSE LIMIT SPACES OF INTERVAL MAPS DOCTORAL THESIS Zagreb, 2018 PRIRODOSLOVNO - MATEMATICKIˇ FAKULTET MATEMATICKIˇ ODSJEK Ana Anušic´ INVERZNI LIMESI PRESLIKAVANJA NA INTERVALU DOKTORSKI RAD Zagreb, 2018. FACULTY OF SCIENCE DEPARTMENT OF MATHEMATICS Ana Anušic´ INVERSE LIMIT SPACES OF INTERVAL MAPS DOCTORAL THESIS Supervisors: Univ.-Prof. PhD Henk Bruin izv. prof. dr. sc. Sonja Štimac Zagreb, 2018 PRIRODOSLOVNO - MATEMATICKIˇ FAKULTET MATEMATICKIˇ ODSJEK Ana Anušic´ INVERZNI LIMESI PRESLIKAVANJA NA INTERVALU DOKTORSKI RAD Mentori: Univ.-Prof. PhD Henk Bruin izv. prof. dr. sc. Sonja Štimac Zagreb, 2018. Acknowledgements During my PhD studies I have met so many extraordinary people who became not only my future colleagues but my dear friends. They all deserve to be mentioned here and it is going to be really hard not to leave somebody out. I would like to express my deepest gratitude to my supervisors Sonja Štimac and Henk Bruin. Sonja, thank you for introducing me to the area, giving me a push into the community, and for the life lessons I am still to comprehend. Henk, thank you for openly sharing your knowledge, treating me like an equal from the very beginning, never locking your doors, and turtle keeping it simple. I am also deeply indebted to Jernej and Vesna Činč. Guys, thank you for being the best friends a person can have. Jernej, I also have to thank you for your patience during our collaboration. The completion of this thesis would not have been possible without Martina Stojić and Goran Erceg who shared their template with me, Mario Stipčić who helped me hand the thesis in, and the committee members who carefully read the first drafts and improved it with valuable comments.
    [Show full text]
  • Lecture 5: Submersions, Immersions and Embeddings
    LECTURE 5: SUBMERSIONS, IMMERSIONS AND EMBEDDINGS 1. Properties of the Differentials Recall that the tangent space of a smooth manifold M at p is the space of all 1 derivatives at p, i.e. all linear maps Xp : C (M) ! R so that the Leibnitz rule holds: Xp(fg) = g(p)Xp(f) + f(p)Xp(g): The differential (also known as the tangent map) of a smooth map f : M ! N at p 2 M is defined to be the linear map dfp : TpM ! Tf(p)N such that dfp(Xp)(g) = Xp(g ◦ f) 1 for all Xp 2 TpM and g 2 C (N). Remark. Two interesting special cases: • If γ :(−"; ") ! M is a curve such that γ(0) = p, then dγ0 maps the unit d d tangent vector dt at 0 2 R to the tangent vectorγ _ (0) = dγ0( dt ) of γ at p 2 M. • If f : M ! R is a smooth function, we can identify Tf(p)R with R by identifying d a dt with a (which is merely the \derivative $ vector" correspondence). Then for any Xp 2 TpM, dfp(Xp) 2 R. Note that the map dfp : TpM ! R is linear. ∗ In other words, dfp 2 Tp M, the dual space of TpM. We will call dfp a cotangent vector or a 1-form at p. Note that by taking g = Id 2 C1(R), we get Xp(f) = dfp(Xp): For the differential, we still have the chain rule for differentials: Theorem 1.1 (Chain rule). Suppose f : M ! N and g : N ! P are smooth maps, then d(g ◦ f)p = dgf(p) ◦ dfp.
    [Show full text]
  • Metric Manifold Learning: Preserving the Intrinsic Geometry
    Metric Learning and Manifolds Metric Manifold Learning: Preserving the Intrinsic Geometry Dominique Perrault-Joncas [email protected] Google, Inc. Seattle, WA 98103, USA Marina Meilă [email protected] Department of Statistics University of Washington Seattle, WA 98195-4322, USA Editor: Abstract A variety of algorithms exist for performing non-linear dimension reduction, but these algorithms do not preserve the original geometry of the data except in special cases. In general, in the low-dimensional representations obtained, distances are distorted, as well as angles, areas, etc. This paper proposes a generic method to estimate the distortion incurred at each point of an embedding, and subsequently to “correct” distances and other intrinsic geometric quantities back to their original values (up to sampling noise). Our approach is based on augmenting the output of an embedding algorithm with geometric information embodied in the Riemannian metric of the manifold. The Riemannian metric allows one to compute geometric quantities (such as angle, length, or volume) for any coordinate system or embedding of the manifold. In this work, we provide an algorithm for estimating the Riemannian metric from data, consider its consistency, and demonstrate the uses of our approach in a variety of examples. 1. Introduction When working with high-dimensional data, one is regularly confronted with the problem of tractabil- ity and interpretability of the data. An appealing approach to this problem is dimension reduction: finding a low-dimensional representation of the data that preserves all or most of the “important information”. One popular idea for data consisting of vectors in Rr is to assume the so-called man- ifold hypothesis, whereby the data lie on a low-dimensional smooth manifold embedded in the high dimensional space.
    [Show full text]
  • HYPERSPACES of TOPOLOGICAL VECTOR SPACES: THEIR EMBEDDING in TOPOLOGICAL VECTOR SPACES PREM Prakash and Murat R
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 61, Number 1, November 1976 HYPERSPACES OF TOPOLOGICAL VECTOR SPACES: THEIR EMBEDDING IN TOPOLOGICAL VECTOR SPACES PREM prakash and murat r. SERTEL Abstract. Let L be a real (Hausdorff) topological vector space. The space %[L] of nonempty compact subsets of L forms a (Hausdorff) topological semivector space with singleton origin when 3C[L] is given the uniform (equivalently, the finite) hyperspace topology determined by L. Then %[L] is locally compact iff L is so. Furthermore, 90.[Z,], the set of nonempty compact convex subsets of L, is the largest pointwise convex subset of %[L] and is a cancellative topological semivector space. For any nonempty compact and convex set X C L, the collection SCSlX] C CK2[L]is nonempty compact and convex. L is iseomorphically embeddable in 9©[L] and, in turn, there is a smallest vector space £ in which 3<S[L] is algebraically embeddable (as a cone). Furthermore, when L is locally convex, £ can be given a locally convex vector topology 9" such mat the algebraic embedding of 9©[L] in £ is an iseomorphism, and then £ is normable iff L is so; indeed, ?Tcan be so chosen that, when L is normed, the embedding of L in 9<2,[L]and that of CK2[L]in £ are both iseometries. 1. Preliminaries. R denotes the set of real numbers with the usual topology, and R+ = (A G 7?|A > 0). For any set X, [X] denotes the set of nonempty subsets of X. When A is a topological space, %[X] denotes the set of compact nonempty subsets of X.
    [Show full text]