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A family of dissimilarity measures between nodes generalizing both the shortestpath and the commutetime distances
 in Proceedings of the 14th SIGKDD International Conference on Knowledge Discovery and Data Mining
"... This work introduces a new family of linkbased dissimilarity measures between nodes of a weighted directed graph. This measure, called the randomized shortestpath (RSP) dissimilarity, depends on a parameter θ and has the interesting property of reducing, on one end, to the standard shortestpath d ..."
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Cited by 24 (11 self)
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This work introduces a new family of linkbased dissimilarity measures between nodes of a weighted directed graph. This measure, called the randomized shortestpath (RSP) dissimilarity, depends on a parameter θ and has the interesting property of reducing, on one end, to the standard shortestpath distance when θ is large and, on the other end, to the commutetime (or resistance) distance when θ is small (near zero). Intuitively, it corresponds to the expected cost incurred by a random walker in order to reach a destination node from a starting node while maintaining a constant entropy (related to θ) spread in the graph. The parameter θ is therefore biasing gradually the simple random walk on the graph towards the shortestpath policy. By adopting a statistical physics approach and computing a sum over all the possible paths (discrete path integral), it is shown that the RSP dissimilarity from every node to a particular node of interest can be computed efficiently by solving two linear systems of n equations, where n is the number of nodes. On the other hand, the dissimilarity between every couple of nodes is obtained by inverting an n × n matrix. The proposed measure can be used for various graph mining tasks such as computing betweenness centrality, finding dense communities, etc, as shown in the experimental section.
Graph nodes clustering with the sigmoid commutetime kernel: A . . .
 DATA & KNOWLEDGE ENGINEERING
, 2009
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Metric based upscaling
 Communications on Pure and Applied Mathematics
, 2007
"... We consider divergence form elliptic operators in dimension n ≥ 2. Although solutions of these operators are only Hölder continuous, we show that they are differentiable (C 1,α) with respect to harmonic coordinates. It follows that numerical homogenization can be extended to situations where the med ..."
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Cited by 23 (2 self)
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We consider divergence form elliptic operators in dimension n ≥ 2. Although solutions of these operators are only Hölder continuous, we show that they are differentiable (C 1,α) with respect to harmonic coordinates. It follows that numerical homogenization can be extended to situations where the medium has no ergodicity at small scales and is characterized by a continuum of scales by transferring a new metric in addition to traditional averaged (homogenized) quantities from subgrid scales into computational scales and error bounds can be given. This numerical homogenization method can also be used as a compression tool for differential operators. 1 Introduction and main results Let Ω be a bounded and convex domain of class C2. We consider the following benchmark PDE
Riemannian manifold learning for nonlinear dimensionality reduction
, 2006
"... In recent years, nonlinear dimensionality reduction (NLDR) techniques have attracted much attention in visual perception and many other areas of science. We propose an efficient algorithm called Riemannian manifold learning (RML). A Riemannian manifold can be constructed in the form of a simplicial ..."
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Cited by 22 (1 self)
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In recent years, nonlinear dimensionality reduction (NLDR) techniques have attracted much attention in visual perception and many other areas of science. We propose an efficient algorithm called Riemannian manifold learning (RML). A Riemannian manifold can be constructed in the form of a simplicial complex, and thus its intrinsic dimension can be reliably estimated. Then the NLDR problem is solved by constructing Riemannian normal coordinates (RNC). Experimental results demonstrate that our algorithm can learn the data’s intrinsic geometric structure, yielding uniformly distributed and well organized lowdimensional embedding data.
An experimental investigation of kernels on graphs for collaborative . . .
 NEURAL NETWORKS
, 2012
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An Analysis of the Convergence of Graph Laplacians
"... Existing approaches to analyzing the asymptotics of graph Laplacians typically assume a wellbehaved kernel function with smoothness assumptions. We remove the smoothness assumption and generalize the analysis of graph Laplacians to include previously unstudied graphs including kNN graphs. We also i ..."
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Cited by 14 (0 self)
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Existing approaches to analyzing the asymptotics of graph Laplacians typically assume a wellbehaved kernel function with smoothness assumptions. We remove the smoothness assumption and generalize the analysis of graph Laplacians to include previously unstudied graphs including kNN graphs. We also introduce a kernelfree framework to analyze graph constructions with shrinking neighborhoods in general and apply it to analyze locally linear embedding (LLE). We also describe how, for a given limit operator, desirable properties such as a convergent spectrum and sparseness can be achieved by choosing the appropriate graph construction. 1.
Nonlinear Laplacian spectral analysis for time series with intermittency and lowfrequency variability
 Proc. Natl. Acad. Sci
"... We present a technique for spatiotemporal data analysis called nonlinear Laplacian spectral analysis (NLSA), which generalizes singular spectrum analysis (SSA) to take into account the nonlinear manifold structure of complex datasets. The key principle underlying NLSA is that the functions used to ..."
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Cited by 11 (5 self)
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We present a technique for spatiotemporal data analysis called nonlinear Laplacian spectral analysis (NLSA), which generalizes singular spectrum analysis (SSA) to take into account the nonlinear manifold structure of complex datasets. The key principle underlying NLSA is that the functions used to represent temporal patterns should exhibit a degree of smoothness on the nonlinear data manifold M; a constraint absent from classical SSA. NLSA enforces such a notion of smoothness by requiring that temporal patterns belong in lowdimensional Hilbert spaces Vl spanned by the leading l LaplaceBeltrami eigenfunctions on M. These eigenfunctions can be evaluated efficiently in high ambientspace dimensions using sparse graphtheoretic algorithms. Moreover, they provide orthonormal bases to expand a family of linear maps, whose singular value decomposition leads to sets of spatiotemporal patterns at progressively finer resolution on the data manifold. The Riemannian measure of M and an adaptive graph kernel width enhances the capability of NLSA to detect important nonlinear
Learning binary hash codes for largescale image search
 MACHINE LEARNING FOR COMPUTER VISION
, 2013
"... Algorithms to rapidly search massive image or video collections are critical for many vision applications, including visual search, contentbased retrieval, and nonparametric models for object recognition. Recent work shows that learned binary projections are a powerful way to index large collect ..."
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Algorithms to rapidly search massive image or video collections are critical for many vision applications, including visual search, contentbased retrieval, and nonparametric models for object recognition. Recent work shows that learned binary projections are a powerful way to index large collections according to their content. The basic idea is to formulate the projections so as to approximately preserve a given similarity function of interest. Having done so, one can then search the data efficiently using hash tables, or by exploring the Hamming ball volume around a novel query. Both enable sublinear time retrieval with respect to the database size. Further, depending on the design of the projections, in some cases it is possible to bound the number of database examples that must be searched in order to achieve a given level of accuracy. This chapter overviews data structures for fast search with binary codes, and then describes several supervised and unsupervised strategies for generating the codes. In particular, we review supervised methods that integrate metric learning, boosting, and neural networks into the hash key construction, and unsupervised methods based on spectral analysis or kernelized random projections that compute affinitypreserving binary codes. Whether learning from explicit semantic supervision or exploiting the structure among unlabeled data, these methods make scalable retrieval possible for a variety of robust visual similarity measures. We focus on defining the algorithms, and illustrate the main points with results using millions of images.
2D TOMOGRAPHY FROM NOISY PROJECTIONS TAKEN AT UNKNOWN RANDOM DIRECTIONS
"... Abstract. Computerized Tomography (CT) is a standard method for obtaining internal structure of objects from their projection images. While CT reconstruction requires the knowledge of the imaging directions, there are some situations in which the imaging directions are unknown, for example, when ima ..."
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Abstract. Computerized Tomography (CT) is a standard method for obtaining internal structure of objects from their projection images. While CT reconstruction requires the knowledge of the imaging directions, there are some situations in which the imaging directions are unknown, for example, when imaging a moving object. It is therefore desirable to design a reconstruction method from projection images taken at unknown directions. Recently, it was shown that the imaging directions can be obtained by the diffusion map framework. Another difficulty arises from the fact that projections are often contaminated by noise, practically limiting all current methods, including the diffusion map approach. In this paper, we introduce two denoising steps that allow reconstructions at much lower signaltonoise ratios (SNR) when combined with the diffusion map framework. The first denoising step consists of using the singular value decomposition (SVD) in order to find an adaptive basis for the projection data set, leading to improved similarities between different projections. In the second step, we denoise the graph of similarities using the Jaccard index, which is a widely used measure in network analysis. Using this combination of SVD, Jaccard index and diffusion map, we are able to reconstruct the 2D SheppLogan phantom from simulative noisy projections at SNRs well below their currently reported threshold values. Although the focus of this paper is the 2D CT reconstruction problem, we believe that the combination of SVD, Jaccard index graph denoising and diffusion maps is potentially useful in other signal processing and image analysis applications. Key words. Computerized Tomography (CT), Diffusion maps, manifold graph denoising, singular value decomposition (SVD), Jaccard index, small world graphs, SheppLogan phantom. 1. Introduction. Transmission Computerized Tomography (CT