Results 1 
6 of
6
Towards a theoretical foundation for Laplacianbased manifold methods
, 2005
"... Abstract. In recent years manifold methods have attracted a considerable amount of attention in machine learning. However most algorithms in that class may be termed “manifoldmotivated ” as they lack any explicit theoretical guarantees. In this paper we take a step towards closing the gap between t ..."
Abstract

Cited by 158 (11 self)
 Add to MetaCart
(Show Context)
Abstract. In recent years manifold methods have attracted a considerable amount of attention in machine learning. However most algorithms in that class may be termed “manifoldmotivated ” as they lack any explicit theoretical guarantees. In this paper we take a step towards closing the gap between theory and practice for a class of Laplacianbased manifold methods. We show that under certain conditions the graph Laplacian of a point cloud converges to the LaplaceBeltrami operator on the underlying manifold. Theorem 1 contains the first result showing convergence of a random graph Laplacian to manifold Laplacian in the machine learning context. 1
Convergence of laplacian eigenmaps
 In NIPS
, 2006
"... Geometrically based methods for various tasks of machine learning have attracted considerable attention over the last few years. In this paper we show convergence of eigenvectors of the point cloud Laplacian to the eigenfunctions of the LaplaceBeltrami operator on the underlying manifold, thus esta ..."
Abstract

Cited by 46 (3 self)
 Add to MetaCart
(Show Context)
Geometrically based methods for various tasks of machine learning have attracted considerable attention over the last few years. In this paper we show convergence of eigenvectors of the point cloud Laplacian to the eigenfunctions of the LaplaceBeltrami operator on the underlying manifold, thus establishing the first convergence results for a spectral dimensionality reduction algorithm in the manifold setting. 1
On the relation between low density separation, spectral clustering and graph cuts
 Advances in Neural Information Processing Systems (NIPS) 19
, 2006
"... One of the intuitions underlying many graphbased methods for clustering and semisupervised learning, is that class or cluster boundaries pass through areas of low probability density. In this paper we provide some formal analysis of that notion for a probability distribution. We introduce a notion ..."
Abstract

Cited by 24 (2 self)
 Add to MetaCart
(Show Context)
One of the intuitions underlying many graphbased methods for clustering and semisupervised learning, is that class or cluster boundaries pass through areas of low probability density. In this paper we provide some formal analysis of that notion for a probability distribution. We introduce a notion of weighted boundary volume, which measures the length of the class/cluster boundary weighted by the density of the underlying probability distribution. We show that sizes of the cuts of certain commonly used data adjacency graphs converge to this continuous weighted volume of the boundary. keywords: Clustering, SemiSupervised Learning 1
Understanding the use of unlabelled data in predictive modelling
 Statistical Science
, 2006
"... The incorporation of unlabelled data in statistical machine learning methods for prediction, including regression and classification, has demonstrated the potential for improved accuracy in prediction in a number of recent examples. The statistical basis for this semisupervised analysis does not, h ..."
Abstract

Cited by 23 (10 self)
 Add to MetaCart
(Show Context)
The incorporation of unlabelled data in statistical machine learning methods for prediction, including regression and classification, has demonstrated the potential for improved accuracy in prediction in a number of recent examples. The statistical basis for this semisupervised analysis does not, however, appear to have been well delineated in the literature to date. Nor, perhaps, are statisticians as fully engaged in the vigourous research in this area of machine learning as might be desired. Much of the theoretical work in the literature has focused, for example, on geometric and structural properties of the unlabeled data in the context of particular algorithms, rather than probabilistic and statistical questions. This paper overviews the fundamental statistical foundations for predictive modelling and the general questions associated with unlabelled data, highlighting the relevance of venerable concepts of sampling design and prior specification. This theory, illustrated with a series of simple but central examples, shows precisely when, why and how unlabelled data matter.
Abstract Towards a Theoretical Foundation for LaplacianBased Manifold Methods
"... In recent years manifold methods have attracted a considerable amount of attention in machine learning. However most algorithms in that class may be termed “manifoldmotivated ” as they lack any explicit theoretical guarantees. In this paper we take a step towards closing the gap between theory and ..."
Abstract
 Add to MetaCart
(Show Context)
In recent years manifold methods have attracted a considerable amount of attention in machine learning. However most algorithms in that class may be termed “manifoldmotivated ” as they lack any explicit theoretical guarantees. In this paper we take a step towards closing the gap between theory and practice for a class of Laplacianbased manifold methods. These methods utilize the graph Laplacian associated to a data set for a variety of applications in semisupervised learning, clustering, data representation. We show that under certain conditions the graph Laplacian of a point cloud of data samples converges to the LaplaceBeltrami operator on the underlying manifold. Theorem 3.1 contains the first result showing convergence of a random graph Laplacian to the manifold Laplacian in the context of machine learning. Key words: LaplaceBeltrami operator, graph Laplacian, manifold methods 1
A Regularity conditions on p and S We make the following assumptions about p:
"... 1. p can be extended to a function p ′ that is L−Lipshitz and which is bounded above by pmax. 2. For 0 < t < t0, min(p(x), Kt(x, y)p(y)dy) ≥ pmin. Note that this is a property of both of the boundary ∂M and p. We note that since p ′ is L−Lipshitz over R d, so is � M Kt(x, z)p ′ (z)dz. We assu ..."
Abstract
 Add to MetaCart
1. p can be extended to a function p ′ that is L−Lipshitz and which is bounded above by pmax. 2. For 0 < t < t0, min(p(x), Kt(x, y)p(y)dy) ≥ pmin. Note that this is a property of both of the boundary ∂M and p. We note that since p ′ is L−Lipshitz over R d, so is � M Kt(x, z)p ′ (z)dz. We assume that S has condition number 1/τ. We also make the following assumption about S:The volume of the set of points whose distance to both S and ∂M is ≤ R, is O(R 2) as R → 0. This is reasonable, and is true if S ∩ ∂M is a manifold of codimension 2.