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17
Diffusion Wavelets
, 2004
"... We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green’s function, in compressed form, and their ..."
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Cited by 152 (18 self)
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We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green’s function, in compressed form, and their fast application. Classes of operators satisfying these conditions include diffusionlike operators, in any dimension, on manifolds, graphs, and in nonhomogeneous media. In this case our construction can be viewed as a farreaching generalization of Fast Multipole Methods, achieved through a different point of view, and of the nonstandard wavelet representation of CalderónZygmund and pseudodifferential operators, achieved through a different multiresolution analysis adapted to the operator. We show how the dyadic powers of an operator can be used to induce a multiresolution analysis, as in classical LittlewoodPaley and wavelet theory, and we show how to construct, with fast and stable algorithms, scaling function and wavelet bases associated to this multiresolution analysis, and the corresponding downsampling operators, and use them to compress the corresponding powers of the operator. This allows to extend multiscale signal processing to general spaces (such as manifolds and graphs) in a very natural way, with corresponding fast algorithms.
Mathematics and the Internet: A Source of Enormous Confusion and Great Potential
"... For many mathematicians and physicists, the Internet has become a popular realworld domain for the application and/or development of new theories related to the organization and behavior of largescale, complex, and dynamic systems. In some cases, the Internet has served both as inspiration and just ..."
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Cited by 46 (7 self)
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For many mathematicians and physicists, the Internet has become a popular realworld domain for the application and/or development of new theories related to the organization and behavior of largescale, complex, and dynamic systems. In some cases, the Internet has served both as inspiration and justification for the popularization of new models and mathematics within the scientific enterprise. For example, scalefree network models of the preferential attachment type [8] have been claimed to describe the Internet’s connectivity structure, resulting in surprisingly general and strong claims about the network’s resilience to random failures of its components and its vulnerability to targeted attacks against its infrastructure [2]. These models have, as their trademark, powerlaw type node degree distributions that drastically distinguish them from the classical ErdősRényi type random graph models [13]. These “scalefree ” network models have attracted significant attention within the scientific community and have been partly responsible for launching and fueling the new field of network science [42, 4]. To date, the main role that mathematics has played in network science has been to put the physicists’ largely empirical findings on solid grounds Walter Willinger is at AT&T LabsResearch in Florham Park, NJ. His email address is walter@research.att. com.
Regularization on graphs with functionadapted diffusion process
, 2006
"... Harmonic analysis and diffusion on discrete data has been shown to lead to stateoftheart algorithms for machine learning tasks, especially in the context of semisupervised and transductive learning. The success of these algorithms rests on the assumption that the function(s) to be studied (learn ..."
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Cited by 38 (8 self)
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Harmonic analysis and diffusion on discrete data has been shown to lead to stateoftheart algorithms for machine learning tasks, especially in the context of semisupervised and transductive learning. The success of these algorithms rests on the assumption that the function(s) to be studied (learned, interpolated, etc.) are smooth with respect to the geometry of the data. In this paper we present a method for modifying the given geometry so the function(s) to be studied are smoother with respect to the modified geometry, and thus more amenable to treatment using harmonic analysis methods. Among the many possible applications, we consider the problems of image denoising and transductive classification. In both settings, our approach improves on standard diffusion based methods.
Diffusion polynomial frames on metric measure spaces. submitted
, 2006
"... We construct a multiscale tight frame based on an arbitrary orthonormal basis for the L 2 space of an arbitrary sigma finite measure space. The approximation properties of the resulting multiscale are studied in the context of Besov approximation spaces, which are characterized both in terms of suit ..."
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Cited by 23 (6 self)
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We construct a multiscale tight frame based on an arbitrary orthonormal basis for the L 2 space of an arbitrary sigma finite measure space. The approximation properties of the resulting multiscale are studied in the context of Besov approximation spaces, which are characterized both in terms of suitable K–functionals and the frame transforms. The only major condition required is the uniform boundedness of a summabilility operator. We give sufficient conditions for this to hold in the context of a very general class of metric measure spaces. The theory is illustrated using the approximation of characteristic functions of caps on a dumbell manifold, and applied to the problem of recognition of hand–written digits. Our methods outperforms comparable methods for semi–supervised learning.
Ghash: towards fast kernelbased similarity search in large graph databases
 In EDBT
, 2009
"... Structured data including sets, sequences, trees and graphs, pose significant challenges to fundamental aspects of data management such as efficient storage, indexing, and similarity search. With the fast accumulation of graph databases, similarity search in graph databases has emerged as an impor ..."
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Cited by 10 (0 self)
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Structured data including sets, sequences, trees and graphs, pose significant challenges to fundamental aspects of data management such as efficient storage, indexing, and similarity search. With the fast accumulation of graph databases, similarity search in graph databases has emerged as an important research topic. Graph similarity search has applications in a wide range of domains including cheminformatics, bioinformatics, sensor network management, social network management, and XML documents, among others. Most of the current graph indexing methods focus on subgraph query processing, i.e. determining the set of database graphs that contains the query graph and hence do not directly support similarity search. In data mining and machine learning, various graph kernel functions have been designed
Research on Online Social Networks: Time to Face the Real Challenges
"... Online Social Networks (OSNs) provide a unique opportunity for researchers to study how a combination of technological, economical, and social forces have been conspiring to provide a service that has attracted the largest user population in the history of the Internet. With more than half a billion ..."
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Cited by 10 (1 self)
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Online Social Networks (OSNs) provide a unique opportunity for researchers to study how a combination of technological, economical, and social forces have been conspiring to provide a service that has attracted the largest user population in the history of the Internet. With more than half a billion of users and counting, OSNs have the potential to impact almost every aspect of networking, including measurements and performance modeling/analysis, network architecture and system design, and privacy and user behavior, to name just a few. However, much of the existing OSN research literature seems to have lost sight of this unique opportunity and has avoided dealing with the new challenges posed by OSNs. We argue in this position paper that it is high time for OSN researcher to exploit and face these opportunities and challenges to provide a basic understanding of the OSN ecosystem as a whole. Such an understanding has to reflect the key role users play in this system and must focus on the system’s dynamics, purpose and functionality when trying to illuminate the main technological, economic, and social forces at work in the current OSN revolution. 1.
GRAPH WAVELET ALIGNMENT KERNELS FOR DRUG VIRTUAL SCREENING
, 2008
"... In this paper we introduce a novel graph classification algorithm and demonstrate its efficacy in drug design. In our method, we use graphs to model chemical structures and apply a wavelet analysis of graphs to create features capturing graph local topology. We design a novel graph kernel function t ..."
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Cited by 8 (2 self)
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In this paper we introduce a novel graph classification algorithm and demonstrate its efficacy in drug design. In our method, we use graphs to model chemical structures and apply a wavelet analysis of graphs to create features capturing graph local topology. We design a novel graph kernel function to utilize the created feature to build predictive models for chemicals. We call the new graph kernel a graph waveletalignment kernel. We have evaluated the efficacy of the waveletalignment kernel using a set of chemical structureactivity prediction benchmarks. Our results indicate that the use of the kernel function yields performance profiles comparable to, and sometimes exceeding that of the existing stateoftheart chemical classification approaches. In addition, our results also show that the use of wavelet functions significantly decreases the computational costs for graph kernel computation with more than 10 fold speed up.
Dimensionality Reduction for Hyperspectral Data
, 2008
"... This thesis is about dimensionality reduction for hyperspectral data. Special emphasis is given to dimensionality reduction techniques known as kernel eigenmap methods and manifold learning algorithms. Kernel eigenmap methods require a nearest neighbor or a radius parameter be set. A new algorithm t ..."
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Cited by 2 (0 self)
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This thesis is about dimensionality reduction for hyperspectral data. Special emphasis is given to dimensionality reduction techniques known as kernel eigenmap methods and manifold learning algorithms. Kernel eigenmap methods require a nearest neighbor or a radius parameter be set. A new algorithm that does not require these neighborhood parameters is given. Most kernel eigenmap methods use the eigenvectors of the kernel as coordinates for the data. An algorithm that uses the frame potential along with subspace frames to create nonorthogonal coordinates is given. The algorithms are demonstrated on hyperspectral data. The last two chapters include analysis of representation systems for LIDAR data and motion blur estimation, respectively.
Hierarchical InNetwork Attribute Compression via Importance Sampling
"... Abstract—Many realworld complex systems can be modeled as dynamic networks with realvalued vertex/edge attributes. Examples include users ’ opinions in social networks and average speeds in a road system. When managing these large dynamic networks, compressing attribute values becomes a key requi ..."
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Abstract—Many realworld complex systems can be modeled as dynamic networks with realvalued vertex/edge attributes. Examples include users ’ opinions in social networks and average speeds in a road system. When managing these large dynamic networks, compressing attribute values becomes a key requirement, since it enables the answering of attributebased queries regarding a node/edge or network region based on a compact representation of the data. To address this problem, we introduce a lossy network compression scheme called Slice Tree (ST), which partitions a network into smooth regions with respect to node/edge values and compresses each value as the average of its region. ST applies a compact representation for network partitions, called slices, that are defined as a center node and radius distance. We propose an importance sampling algorithm to efficiently prune the search space of candidate slices in the ST construction by biasing the sampling process towards the node values that most affect the compression error. The effectiveness of ST in terms of compression error, compression rate, and running time is demonstrated using synthetic and real datasets. ST scales to millionnode instances and removes up to 87 % of the error in attribute values with a 103 compression ratio. We also illustrate how ST captures relevant phenomena in real networks, such as research collaboration patterns and traffic congestions. I.
unknown title
"... We describe signal processing tools to extract structure and information from arbitrary digital data sets. In particular heterogeneous multisensor measurements which involve corrupt data, either noisy or with missing entries present formidable challenges. We sketch methodologies for using the netwo ..."
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We describe signal processing tools to extract structure and information from arbitrary digital data sets. In particular heterogeneous multisensor measurements which involve corrupt data, either noisy or with missing entries present formidable challenges. We sketch methodologies for using the network of inferences and similarities between the data points to create robust nonlinear estimators for missing or noisy entries. These methods enable coherent fusion of data from a multiplicity of sources, generalizing signal processing to a non linear setting. Since they provide empirical data models they could also potentially extend analog to digital conversion schemes like “sigma delta”.