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11
Semisupervised learning on Riemannian manifolds
 Machine Learning
, 2004
"... We consider the general problem of utilizing both labeled and unlabeled data to improve classification accuracy. Under the assumption that the data lie on a submanifold in a high dimensional space, we develop an algorithmic framework to classify a partially labeled data set in a principled manner. T ..."
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Cited by 157 (8 self)
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We consider the general problem of utilizing both labeled and unlabeled data to improve classification accuracy. Under the assumption that the data lie on a submanifold in a high dimensional space, we develop an algorithmic framework to classify a partially labeled data set in a principled manner. The central idea of our approach is that classification functions are naturally defined only on the submanifold in question rather than the total ambient space. Using the LaplaceBeltrami operator one produces a basis (the Laplacian Eigenmaps) for a Hilbert space of square integrable functions on the submanifold. To recover such a basis, only unlabeled examples are required. Once such a basis is obtained, training can be performed using the labeled data set. Our algorithm models the manifold using the adjacency graph for the data and approximates the LaplaceBeltrami operator by the graph Laplacian. We provide details of the algorithm, its theoretical justification, and several practical applications for image, speech, and text classification. 1.
Weighted norm inequalities, offdiagonal estimates and elliptic operators, Part II: Offdiagonal estimates on spaces of homogeneous type
, 2005
"... Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincar ..."
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Cited by 23 (6 self)
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Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincaré inequalities. 1. Introduction and
Spectral Gap, Logarithmic Sobolev Constant, and Geometric Bunds
 Surveys in Diff. Geom., Vol. IX, 219–240, Int
, 2004
"... We survey recent works on the connection between spectral gap and logarithmic Sobolev constants, and exponential integrability of Lipschitz functions. In particular, tools from measure concentration are used to describe bounds on the diameter of a (compact) Riemannian manifold and of Markov ch ..."
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Cited by 14 (0 self)
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We survey recent works on the connection between spectral gap and logarithmic Sobolev constants, and exponential integrability of Lipschitz functions. In particular, tools from measure concentration are used to describe bounds on the diameter of a (compact) Riemannian manifold and of Markov chains in terms of the first eigenvalue of the Laplacian and the logarithmic Sobolev constant. We examine similarly dimension free isoperimetric bounds using these parameters.
The ZerointheSpectrum Question
"... Abstract. This is an expository article on the question of whether zero lies in the spectrum of the LaplaceBeltrami operator acting on differential forms on a manifold. 1. ..."
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Cited by 4 (1 self)
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Abstract. This is an expository article on the question of whether zero lies in the spectrum of the LaplaceBeltrami operator acting on differential forms on a manifold. 1.
On Improved Sobolev Embedding Theorems
"... .  We present a direct proof of some recent improved Sobolev inequalities put forward by A. Cohen, R. DeVore, P. Petrushev and H. Xu [CDVPX] in their wavelet analysis of the space BV (R 2 ). The argument, relying on pseudoPoincar'e inequalities, allows us to consider several extensions to m ..."
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Cited by 4 (0 self)
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.  We present a direct proof of some recent improved Sobolev inequalities put forward by A. Cohen, R. DeVore, P. Petrushev and H. Xu [CDVPX] in their wavelet analysis of the space BV (R 2 ). The argument, relying on pseudoPoincar'e inequalities, allows us to consider several extensions to manifolds and graphs. The classical Sobolev inequality indicates that for every function f on R n vanishing at infinity in some mild sense, kfk q C krfk 1 (1) where q = n n\Gamma1 and C ? 0 only depends on n. The Sobolev inequality (1) is invariant under the ax + b (a ? 0, b 2 R n ) group action, but is not under the WeylHeisenberg group action. Namely, if f(x) = f! (x) = e i! \Deltax '(x) where ' is in the Schwartz class, then krfk 1 = j!jk'k 1 +O(1) when j!j !1. In particular, (1) is not adapted to such modulated functions. In their study of the space BV (R 2 ), A. Cohen et al. [CDVPX] (see also [CMO]) improved the Sobolev inequality (1) into kfk q C krfk 1=q 1 kfk 1\Gamma(1=q) B (2) where B = B \Gamma(n\Gamma1) 1;1 is the homogeneous Besov space of indices (\Gamma(n \Gamma 1); 1;1). This improved Sobolev inequality is easily seen to be sharper than (1). Furthermore, if f = f! as above, then kfk B = j!j \Gamma(n\Gamma1) k'k 1 +O(j!j \Gamman ) so that (2) amounts in this case to the trivial bound kfk q k'k 1=q 1 k'k 1\Gamma(1=q) 1 . The proof of (2) in [CDVPX] and [CMO] is based on wavelet decompositions together with weak` 1 type estimates and interpolation results. The purpose of this note is to propose a direct semigroup argument without any use of wavelet decomposition. In particular, the approach we suggest emphasizes the use of pseudoPoincar'e inequalities (cf. [SC]) for families of operators (heat kernels for example) and ...
ELLIPTIC OPERATORS ON INFINITE GRAPHS
, 2005
"... Abstract. We present some applications of ideas from partial differential equations and differential geometry to the study of difference equations on infinite graphs. All operators that we consider are examples of ”elliptic operators ” as defined by Y. Colin de Verdiere [4]. For such operators, we d ..."
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Abstract. We present some applications of ideas from partial differential equations and differential geometry to the study of difference equations on infinite graphs. All operators that we consider are examples of ”elliptic operators ” as defined by Y. Colin de Verdiere [4]. For such operators, we discuss analogs of inequalities of Cheeger and Harnack and of the maximum principle (in both elliptic and parabolic versions), and apply them to study spectral theory, the ground state and the heat semigroup associated to these operators. 1. Preliminaries We consider graphs (without loops or multiple connections) G = (V,E) where V is a set whose elements are called vertices and E, the set of edges, is a subset of the set of twoelement subsets of V. For an edge e = {x,y} ∈ E, we will denote by [x,y] the oriented edge from x to y and write E for the set of all oriented edges. We also write x ∼ y if {x,y} is an edge. All graphs considered will be connected. By a function on a graph we will mean a mapping f: V − → C. By an operator on a graph, we shall always mean an operator acting on functions and follow [4] in defining the notion of “selfadjoint, positive, elliptic operator.” Observe first that every operator L is given by a matrix (bx,y). We require our operators to be local, i.e. Thus bx,y = 0 if {x,y} is not an edge and x ̸ = y. Lf(x) = bx,xf(x) + ∑ bx,yf(y). The constant functions are annihilated by L if and only if ∑ y∼x bx,y = −bx,x for every x ∈ V. Every local operator L can be rewritten in the form (1) Lf(x) = W(x)f(x) + ∑ ax,y(f(x) − f(y)) where W(x) = bx,x + ∑ y∼x bx,y and ax,y = −bx,y. We will often write L = A + W, where A, given by the sum in the formula above, annihilates constant functions and W denotes the operator of multiplication by the function W(x). y∼x x∼y
Examining Committee:
"... MultiNet: An interactive program for analysing and visualizing complex networks by ..."
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MultiNet: An interactive program for analysing and visualizing complex networks by
Special Functions of Bounded . . .
"... In this paper we extend the theory of special functions of bounded variation (characterised by a total variation measure which is the sum of a “volume ” energy and of a “surface ” energy) to doubling metric measure spaces endowed with a Poincaré inequality. In this framework, which includes all Carn ..."
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In this paper we extend the theory of special functions of bounded variation (characterised by a total variation measure which is the sum of a “volume ” energy and of a “surface ” energy) to doubling metric measure spaces endowed with a Poincaré inequality. In this framework, which includes all Carnot–Carathéodory spaces, we use and improve previous results in [2], [4], [41] to show the basic compactness theorem of special functions of bounded variation. In a particular class of “local ” spaces, which includes all Carnot groups of step 2 and spaces induced by a continuous and strong A ∞ weight, we are able to show the lower semicontinuity of a Mumford–Shah type functional, extending previous results by Song and Yang [45], Citti, Manfredini and Sarti [17] in the Heisenberg group and by Franchi and Baldi [10] in weighted spaces.
Surveys in Differential Geometry IX, International Press
"... An excursion into geometric analysis ..."