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27
Laplacian Eigenmaps for Dimensionality Reduction and Data Representation
 Neural Computation
, 2003
"... Abstract One of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data. We consider the problem of constructing a representation for data lying on a low dimensional manifold embedded in a high dimensional space. Drawing on the corr ..."
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Cited by 734 (15 self)
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Abstract One of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data. We consider the problem of constructing a representation for data lying on a low dimensional manifold embedded in a high dimensional space. Drawing on the correspondence between the graph Laplacian, the Laplace Beltrami operator on the manifold, and the connections to the heat equation, we propose a geometrically motivated algorithm for representing the high dimensional data. The algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering. Some potential applications and illustrative examples are discussed. 1 Introduction In many areas of artificial intelligence, information retrieval and data mining, one is often confronted with intrinsically low dimensional data lying in a very high dimensional space. Consider, for example, gray scale images of an object taken under fixed lighting conditions with a moving camera. Each such image would typically be represented by a brightness value at each pixel. If there were n 2
Nonembeddability theorems via Fourier analysis
"... Various new nonembeddability results (mainly into L1) are proved via Fourier analysis. In particular, it is shown that the Edit Distance on {0, 1}d has L1 distortion (log d) 12o(1). We also give new lower bounds on the L1 distortion of flat tori, quotients of the discrete hypercube under group ac ..."
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Cited by 43 (9 self)
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Various new nonembeddability results (mainly into L1) are proved via Fourier analysis. In particular, it is shown that the Edit Distance on {0, 1}d has L1 distortion (log d) 12o(1). We also give new lower bounds on the L1 distortion of flat tori, quotients of the discrete hypercube under group actions, and the transportation cost (Earthmover) metric.
A new class of transport distances between measures
 Calc. Var. Partial Differential Equations
"... Abstract We introduce a new class of distances between nonnegative Radon measures in Rd. They are modeled on the dynamical characterization of the KantorovichRubinsteinWasserstein distances proposed by BENAMOUBRENIER [7] and provide a wide family inSobolev distances. From the point of view of o ..."
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Cited by 14 (2 self)
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Abstract We introduce a new class of distances between nonnegative Radon measures in Rd. They are modeled on the dynamical characterization of the KantorovichRubinsteinWasserstein distances proposed by BENAMOUBRENIER [7] and provide a wide family inSobolev distances. From the point of view of optimal transport theory, these distances minimize a dynamical cost to move a given initial distribution of mass to a final configuration. An important difference with the classical setting in mass transport theory is that the cost not only depends on the velocity of the moving particles but also on the densities of the intermediate configurations with respect to a given reference measure γ. We study the topological and geometric properties of these new distances, comparing them with the notion of weak convergence of measures and the well established KantorovichRubinsteinWasserstein theory. An example of possible applications to the geometric theory of gradient flows is also given. terpolating between the Wasserstein and the homogeneous W −1,p γ
Physical measures at the boundary of hyperbolic maps
, 2004
"... ABSTRACT. We consider diffeomorphisms of a compact manifold with a dominated splitting which is hyperbolic except for a ”small ” subset of points (Hausdorff dimension smaller than one, e.g. a denumerable subset) and prove the existence of physical measures and their stochastic stability. The physica ..."
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Cited by 4 (2 self)
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ABSTRACT. We consider diffeomorphisms of a compact manifold with a dominated splitting which is hyperbolic except for a ”small ” subset of points (Hausdorff dimension smaller than one, e.g. a denumerable subset) and prove the existence of physical measures and their stochastic stability. The physical measures are obtained as zeronoise limits which are shown to satisfy the Entropy Formula. 1.
GEOMETRIC ANALYSIS
, 2005
"... This was a talk I gave in the occasion of the seventieth anniversary of the Chinese Mathematical Society. I dedicated it in memory of my teacher S. S. Chern who passed away half a year ago. During my graduate study, I was rather free in picking research topics. I [538] worked on fundamental groups o ..."
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Cited by 3 (0 self)
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This was a talk I gave in the occasion of the seventieth anniversary of the Chinese Mathematical Society. I dedicated it in memory of my teacher S. S. Chern who passed away half a year ago. During my graduate study, I was rather free in picking research topics. I [538] worked on fundamental groups of manifolds with nonpositive curvature. But in the second year of my study, I started to look into differential equations on manifolds. While Chern did not express much opinions on this part of my research, he started to appreciate it a few years later. In fact, after Chern gave a course on Calabi’s works on affine geometry in 1972 in Berkeley, Cheng told me these inspiring lectures. By 1973, Cheng and I started to work on some problems mentioned in his lectures. We did not realize that great geometers Pogorelov, Calabi and Nirenberg were also working on them. We were excited that we solved some of the conjectures of Calabi on improper affine spheres. But soon we found out that Pogorelov [398] published it right before us by different arguments. Nevertheless our ideas are useful to handle other problems in
The golden age of immersion theory in topology: 1959–1973. A mathematical survey from a historical perspective
 Bull. Amer. Math. Soc. (N.S
"... Abstract. We review the history of modern immersiontheoretic topology during the period 1959–1973, beginning with the work of S. Smale followed by the important contributions from the Leningrad school of topology, including the work of M. Gromov. We discuss the development of the major geometrical ..."
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Abstract. We review the history of modern immersiontheoretic topology during the period 1959–1973, beginning with the work of S. Smale followed by the important contributions from the Leningrad school of topology, including the work of M. Gromov. We discuss the development of the major geometrical ideas in immersiontheoretic topology during this period. Historical remarks are included and technical concepts are introduced informally. 1. Brief overview In this article1 I briefly review selected contributions to immersiontheoretic topology during the early “golden ” period, from about 1959 to 1973, during which time the subject received its initial important developments from leading topologists in many countries. Modern immersiontheoretic topology began with the work of Stephen Smale ([51]), ([52]) on the classification of immersions of the sphere Sn into Euclidean space R q, n ≥ 2, q ≥ n + 1. During approximately the next 15 years the methods introduced by Smale were generalized in various ways to analyse and solve an astonishing variety of geometrical and topological problems. In particular, at Leningrad University during the late 1960s and early 1970s, M. Gromov and
FOLDFORMS FOR FOURFOLDS
"... Abstract. This paper explains an application of Gromov’s hprinciple to prove the existence, on any orientable 4manifold, of a folded symplectic form. That is a closed 2form which is symplectic except on a separating hypersurface where the form singularities are like the pullback of a symplectic f ..."
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Cited by 2 (0 self)
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Abstract. This paper explains an application of Gromov’s hprinciple to prove the existence, on any orientable 4manifold, of a folded symplectic form. That is a closed 2form which is symplectic except on a separating hypersurface where the form singularities are like the pullback of a symplectic form by a folding map. We use the hprinciple for folding maps (a theorem of Eliashberg) and the hprinciple for symplectic forms on open manifolds (a theorem of Gromov) to show that, for orientable evendimensional manifolds, the existence of a stable almost complex structure is necessary and sufficient to warrant the existence of a folded symplectic form. 1.
hPRINCIPLE AND RIGIDITY FOR C 1,α ISOMETRIC EMBEDDINGS
, 905
"... Abstract. In this paper we study the embedding of Riemannian manifolds in low codimension. The wellknown result of Nash and Kuiper [21, 20] says that any short embedding in codimension one can be uniformly approximated by C 1 isometric embeddings. This statement clearly cannot be true for C 2 embed ..."
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Cited by 2 (2 self)
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Abstract. In this paper we study the embedding of Riemannian manifolds in low codimension. The wellknown result of Nash and Kuiper [21, 20] says that any short embedding in codimension one can be uniformly approximated by C 1 isometric embeddings. This statement clearly cannot be true for C 2 embeddings in general, due to the classical rigidity in the Weyl problem. In fact Borisov extended the latter to embeddings of class C 1,α with α> 2/3 in [3, 5]. On the other hand he announced in [6] that the NashKuiper statement can be extended to local C 1,α embeddings with α < (1+n+n 2) −1, where n is the dimension of the manifold, provided the metric is analytic. Subsequently a proof of the 2dimensional case appeared in [7]. In this paper we provide analytic proofs of all these statements, for general dimension and general metric. 1.
OBSTRUCTIONS TO EMBEDDABILITY INTO HYPERQUADRICS AND EXPLICIT EXAMPLES
"... The purpose of this paper is to propose a method of constructing explicit examples of real submanifolds in C n that do not admit holomorphic embeddings into hyperquardics of larger dimension. It is always a natural question whether one can embedd a general manifold with a given structure ..."
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Cited by 1 (0 self)
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The purpose of this paper is to propose a method of constructing explicit examples of real submanifolds in C n that do not admit holomorphic embeddings into hyperquardics of larger dimension. It is always a natural question whether one can embedd a general manifold with a given structure
Relative isometric embeddings of Riemannian manifolds in R n , preprint available at www.math.gatech.edu/∼ghomi
"... Abstract. We prove the existence of C 1 isometric embeddings, and C ∞ approximate isometric embeddings, of Riemannian manifolds into Euclidean space with prescribed values in a neighborhood of a point. 1. ..."
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Cited by 1 (1 self)
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Abstract. We prove the existence of C 1 isometric embeddings, and C ∞ approximate isometric embeddings, of Riemannian manifolds into Euclidean space with prescribed values in a neighborhood of a point. 1.