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13
Graph laplacians and their convergence on random neighborhood graphs
 Journal of Machine Learning Research
, 2006
"... Given a sample from a probability measure with support on a submanifold in Euclidean space one can construct a neighborhood graph which can be seen as an approximation of the submanifold. The graph Laplacian of such a graph is used in several machine learning methods like semisupervised learning, d ..."
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Cited by 25 (6 self)
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Given a sample from a probability measure with support on a submanifold in Euclidean space one can construct a neighborhood graph which can be seen as an approximation of the submanifold. The graph Laplacian of such a graph is used in several machine learning methods like semisupervised learning, dimensionality reduction and clustering. In this paper we determine the pointwise limit of three different graph Laplacians used in the literature as the sample size increases and the neighborhood size approaches zero. We show that for a uniform measure on the submanifold all graph Laplacians have the same limit up to constants. However in the case of a nonuniform measure on the submanifold only the so called random walk graph Laplacian converges to the weighted LaplaceBeltrami operator.
Transient random walks on graphs and metric spaces with applications to hyperbolic surfaces
 Proc. London Math. Soc
, 1992
"... We introduce an (r, /?)net (0 < 2r < R) of a metric space M as a maximal graph whose vertices are elements in M of pairwise distance at least r such that any two vertices of distance at most R are adjacent. We show that, for a large class of metric spaces, including many Riemannian manifolds, the p ..."
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Cited by 12 (2 self)
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We introduce an (r, /?)net (0 < 2r < R) of a metric space M as a maximal graph whose vertices are elements in M of pairwise distance at least r such that any two vertices of distance at most R are adjacent. We show that, for a large class of metric spaces, including many Riemannian manifolds, the property of transience of a net and the property of the net carrying a nonconstant harmonic function of bounded energy is independent of the choice of the net. We give a new necessary and sufficient condition for a graph with bounded degrees and satisfying an isoperimetric inequality to have no nonconstant harmonic functions. For this purpose we develop equivalent analytic conditions for graphs satisfying an isoperimetric inequality. Some of these results have been discovered recently by others in more general settings, but our treatment here is specific and selfcontained. We use graph transience to prove that Scherk's surface is hyperbolic, a problem posed by Osserman in 1965. 1.
Noncommutative Riemann integration and NovikovShubin invariants for Open Manifolds
, 2001
"... Given a C ∗algebra A with a semicontinuous semifinite trace τ acting on the Hilbert space H, we define the family A R of bounded Riemann measurable elements w.r.t. τ as a suitable closure, à la Dedekind, of A, in analogy with one of the classical characterizations of Riemann measurable functions [2 ..."
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Cited by 6 (3 self)
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Given a C ∗algebra A with a semicontinuous semifinite trace τ acting on the Hilbert space H, we define the family A R of bounded Riemann measurable elements w.r.t. τ as a suitable closure, à la Dedekind, of A, in analogy with one of the classical characterizations of Riemann measurable functions [26], and show that A R is a C ∗algebra, and τ extends to a semicontinuous semifinite trace on A R. Then, unbounded Riemann measurable operators are defined as the closed operators on H which are affiliated to A ′′ and can be approximated in measure by operators in A R, in analogy with unbounded Riemann integration. Unbounded Riemann measurable operators form a τa.e. bimodule on A R, denoted by A R, and such bimodule contains the functional calculi of selfadjoint elements of A R under unbounded Riemann measurable functions. Besides, τ extends to a bimodule trace on A R.
Noncommutative Riemann integration and singular traces for C ∗  algebras
"... Given a C ∗algebra A with a semicontinuous semifinite trace τ acting on the Hilbert space H, we define the family A R of bounded Riemann measurable elements w.r.t. τ as a suitable closure, à la Dedekind, of A, in analogy with one of the classical characterizations of Riemann measurable functions [1 ..."
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Cited by 4 (4 self)
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Given a C ∗algebra A with a semicontinuous semifinite trace τ acting on the Hilbert space H, we define the family A R of bounded Riemann measurable elements w.r.t. τ as a suitable closure, à la Dedekind, of A, in analogy with one of the classical characterizations of Riemann measurable functions [16], and show that A R is a C ∗algebra, and τ extends to a semicontinuous semifinite trace on A R. Then, unbounded Riemann measurable operators are defined as the closed operators on H which are affiliated to A ′′ and can be approximated in measure by operators in A R, in analogy with improper Riemann integration. Unbounded Riemann measurable operators form a τa.e. bimodule on A R, denoted by AR, and such bimodule contains the functional calculi of selfadjoint elements of A R under unbounded Riemann measurable functions. Besides, τ extends to a bimodule trace on AR. As type II1 singular traces for a semifinite von Neumann algebra M with a normal semifinite faithful (nonatomic) trace τ have been defined as traces on M − Mbimodules of unbounded τmeasurable operators [5], type II1 singular traces for a C ∗algebra A with a semicontinuous semifinite (nonatomic) trace τ are defined here as traces on A − Abimodules of unbounded Riemann measurable operators (in AR) for any faithful representation of A. An application of singular traces for C ∗algebras is contained in [6].
Discretization of bounded harmonic functions on Riemannian manifolds and entropy
"... We give conditions under which the space of bounded harmonic functions on a Riemannian manifold M is naturally isomorphic to the space of bounded harmonic functions of a Markov chain on a discrete net X ae M arising from a discretization procedure for the pair (M; X). If, further, M is a regular co ..."
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Cited by 4 (0 self)
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We give conditions under which the space of bounded harmonic functions on a Riemannian manifold M is naturally isomorphic to the space of bounded harmonic functions of a Markov chain on a discrete net X ae M arising from a discretization procedure for the pair (M; X). If, further, M is a regular covering manifold and the net is invariant with respect to the deck transformation group, then the entropy of the arising random walk on X equals the entropy of the Brownian motion on M times the average stopping time of the discretization procedure. 1980 Mathematics Subject Classification (1985 Revision): 31C12, 58G32, 60J50. 0. Introduction During the last few years a lot of papers devoted to the discrete potential theory has appeared. It turns out that this theory is to a large extent parallel to the potential theory on Riemannian manifolds (see, e.g., a survey [1]). Thus one can naturally ask about any direct relationships between the potential theory on Riemannian manifolds and on graph...
A Semicontinuous Trace for Almost Local Operators on an Open Manifold
, 2001
"... A semicontinuous semifinite trace is constructed on the C*algebra ..."
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Cited by 3 (3 self)
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A semicontinuous semifinite trace is constructed on the C*algebra
Transient Networks and Fattenings of the Line
"... . It is wellknown that a random walk on the onedimensional lattice Z 1 is recurrent. An interesting question is, how much can we "fatten" Z 1 before it becomes transient? We show that for 0 fi 1 a network which is a oneway infinite path with vertices in order v 0 ; v 1 ; v 2 ; : : : and ad ..."
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. It is wellknown that a random walk on the onedimensional lattice Z 1 is recurrent. An interesting question is, how much can we "fatten" Z 1 before it becomes transient? We show that for 0 fi 1 a network which is a oneway infinite path with vertices in order v 0 ; v 1 ; v 2 ; : : : and additional edges vnv n+bn 1\Gammafi c ; n = 1; 2; 3; : : : having corresponding conductances cn , is transient if P n0 c \Gamma1 n n \Gamma2(1\Gammafi) ! 1: As a consequence, we show that if we fatten Z 2 by adding edges (n; j)(n + bn 1\Gammafi c; j); n 1; j 0 with corresponding conductances cn , then this network is transient if P n0 c \Gamma 1 2 n n 2(1\Gammafi) ! 1: We discuss these "fattenings" and their relationships to isoperimetric inequalities and the classification of Riemann surfaces. In particular, it was recently shown that the Scherk surface is hyperbolic by showing that a certain net on the surface is transient. We give a simple proof that it's transience ...