Results 1  10
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100
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 ..."
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Cited by 103 (10 self)
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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
Discrete Laplace Operator on Meshed Surfaces
"... In recent years a considerable amount of work in graphics and geometric optimization used tools based on the LaplaceBeltrami operator on a surface. The applications of the Laplacian include mesh editing, surface smoothing, and shape interpolations among others. However, it has been shown [12, 23, 2 ..."
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Cited by 37 (11 self)
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In recent years a considerable amount of work in graphics and geometric optimization used tools based on the LaplaceBeltrami operator on a surface. The applications of the Laplacian include mesh editing, surface smoothing, and shape interpolations among others. However, it has been shown [12, 23, 25] that the popular cotangent approximation schemes do not provide convergent pointwise (or even L2) estimates, while many applications rely on pointwise estimation. Existence of such schemes has been an open question [12]. In this paper we propose the first algorithm for approximating the Laplace operator of a surface from a mesh with pointwise convergence guarantees applicable to arbitrary meshed surfaces. We show that for a sufficiently fine mesh over an arbitrary surface, our mesh Laplacian is close to the LaplaceBeltrami operator on the surface at every point of the surface. Moreover, the proposed algorithm is simple and easily implementable. Experimental evidence shows that our algorithm exhibits convergence empirically and outperforms cotangentbased methods in providing accurate approximation of the Laplace operator for various meshes.
Persistent Homology  a Survey
 CONTEMPORARY MATHEMATICS
"... Persistent homology is an algebraic tool for measuring topological features of shapes and functions. It casts the multiscale organization we frequently observe in nature into a mathematical formalism. Here we give a record of the short history of persistent homology and present its basic concepts. ..."
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Cited by 36 (1 self)
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Persistent homology is an algebraic tool for measuring topological features of shapes and functions. It casts the multiscale organization we frequently observe in nature into a mathematical formalism. Here we give a record of the short history of persistent homology and present its basic concepts. Besides the mathematics we focus on algorithms and mention the various connections to applications, including to biomolecules, biological networks, data analysis, and geometric modeling.
Manifold reconstruction in arbitrary dimensions using witness complexes
 In Proc. 23rd ACM Sympos. on Comput. Geom
, 2007
"... It is a wellestablished fact that the witness complex is closely related to the restricted Delaunay triangulation in low dimensions. Specifically, it has been proved that the witness complex coincides with the restricted Delaunay triangulation on curves, and is still a subset of it on surfaces, und ..."
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Cited by 31 (7 self)
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It is a wellestablished fact that the witness complex is closely related to the restricted Delaunay triangulation in low dimensions. Specifically, it has been proved that the witness complex coincides with the restricted Delaunay triangulation on curves, and is still a subset of it on surfaces, under mild sampling assumptions. Unfortunately, these results do not extend to higherdimensional manifolds, even under stronger sampling conditions. In this paper, we show how the sets of witnesses and landmarks can be enriched, so that the nice relations that exist between both complexes still hold on higherdimensional manifolds. We also use our structural results to devise an algorithm that reconstructs manifolds of any arbitrary dimension or codimension at different scales. The algorithm combines a farthestpoint refinement scheme with a vertex pumping strategy. It is very simple conceptually, and it does not require the input point sample W to be sparse. Its time complexity is bounded by c(d)W  2, where c(d) is a constant depending solely on the dimension d of the ambient space. 1
Topology and Data
, 2008
"... An important feature of modern science and engineering is that data of various kinds is being produced at an unprecedented rate. This is so in part because of new experimental methods, and in part because of the increase in the availability of high powered computing technology. It is also clear that ..."
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Cited by 30 (0 self)
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An important feature of modern science and engineering is that data of various kinds is being produced at an unprecedented rate. This is so in part because of new experimental methods, and in part because of the increase in the availability of high powered computing technology. It is also clear that the nature of the data
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 ..."
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Cited by 26 (2 self)
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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
Constructing Laplace Operator from Point Clouds in R^d
, 2009
"... We present an algorithm for approximating the LaplaceBeltrami operator from an arbitrary point cloud obtained from a kdimensional manifold embedded in the ddimensional space. We show that this PCD Laplace (PointCloud Data Laplace) operator converges to the LaplaceBeltrami operator on the underl ..."
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Cited by 24 (3 self)
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We present an algorithm for approximating the LaplaceBeltrami operator from an arbitrary point cloud obtained from a kdimensional manifold embedded in the ddimensional space. We show that this PCD Laplace (PointCloud Data Laplace) operator converges to the LaplaceBeltrami operator on the underlying manifold as the point cloud becomes denser. Unlike the previous work, we do not assume that the data samples are independent identically distributed from a probability distribution and do not require a global mesh. The resulting algorithm is easy to implement. We present experimental results indicating that even for point sets sampled from a uniform distribution, PCD Laplace converges faster than the weighted graph Laplacian. We also show that using our PCD Laplacian we can directly estimate certain geometric invariants, such as manifold area.
Reconstruction using witness complexes
 in Proc. 18th ACMSIAM Sympos. on Discrete Algorithms
, 2007
"... We present a novel reconstruction algorithm that, given an input point set sampled from an object S, builds a oneparameter family of complexes that approximate S at different scales. At a high level, our method is very similar in spirit to Chew’s surface meshing algorithm, with one notable differen ..."
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Cited by 20 (9 self)
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We present a novel reconstruction algorithm that, given an input point set sampled from an object S, builds a oneparameter family of complexes that approximate S at different scales. At a high level, our method is very similar in spirit to Chew’s surface meshing algorithm, with one notable difference though: the restricted Delaunay triangulation is replaced by the witness complex, which makes our algorithm applicable in any metric space. To prove its correctness on curves and surfaces, we highlight the relationship between the witness complex and the restricted Delaunay triangulation in 2d and in 3d. Specifically, we prove that both complexes are equal in 2d and closely related in 3d, under some mild sampling assumptions. 1
Compressive Sensing on Manifolds Using a Nonparametric Mixture of Factor Analyzers: Algorithm and Performance Bounds 1
"... Nonparametric Bayesian methods are employed to constitute a mixture of lowrank Gaussians, for data x ∈ RN that are of high dimension N but are constrained to reside in a lowdimensional subregion of RN. The number of mixture components and their rank are inferred automatically from the data. The re ..."
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Cited by 18 (7 self)
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Nonparametric Bayesian methods are employed to constitute a mixture of lowrank Gaussians, for data x ∈ RN that are of high dimension N but are constrained to reside in a lowdimensional subregion of RN. The number of mixture components and their rank are inferred automatically from the data. The resulting algorithm can be used for learning manifolds and for reconstructing signals from manifolds, based on compressive sensing (CS) projection measurements. The statistical CS inversion is performed analytically. We derive the required number of CS random measurements needed for successful reconstruction, based on easily computed quantities, drawing on block–sparsity properties. The proposed methodology is validated on several synthetic and real datasets. I.