Results 11  20
of
114
Geometry of probability spaces
 Constr. Approx
"... Partial differential equations and the Laplacian operator on domains in Euclidean spaces have played a central role in understanding natural phenomena. However this avenue has been limited in many areas where calculus is obstructed as in singular spaces, and function spaces of functions on a space X ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
(Show Context)
Partial differential equations and the Laplacian operator on domains in Euclidean spaces have played a central role in understanding natural phenomena. However this avenue has been limited in many areas where calculus is obstructed as in singular spaces, and function spaces of functions on a space X where X itself is a function space. Examples of the last
Biorthogonal diffusion wavelets for multiscale representations on manifolds and graphs
 59141M. SPIE, 2005. URL HTTP://LINK.AIP.ORG/LINK/?PSI/5914/59141M/1
, 2005
"... Recent work by some of the authors presented a novel construction of a multiresolution analysis on manifolds and graphs, acted upon by a given symmetric Markov semigroup {T t}t≥0, for which T t has low rank for large t. 1 This includes important classes of diffusionlike operators, in any dimension, ..."
Abstract

Cited by 12 (8 self)
 Add to MetaCart
(Show Context)
Recent work by some of the authors presented a novel construction of a multiresolution analysis on manifolds and graphs, acted upon by a given symmetric Markov semigroup {T t}t≥0, for which T t has low rank for large t. 1 This includes important classes of diffusionlike operators, in any dimension, on manifolds, graphs, and in nonhomogeneous media. The dyadic powers of an operator are used to induce a multiresolution analysis, analogous to classical LittlewoodPaley 14 and wavelet theory, while associated wavelet packets can also be constructed. 2 This extends multiscale function and operator analysis and signal processing to a large class of spaces, such as manifolds and graphs, with efficient algorithms. Powers and functions of T (notably its Green’s function) are efficiently computed, represented and compressed. This construction is related and generalizes certain Fast Multipole Methods, 3 the wavelet representation of CalderónZygmund and pseudodifferential operators, 4 and also relates to algebraic multigrid techniques. 5 The original diffusion wavelet construction yields orthonormal bases for multiresolution spaces {Vj}. The orthogonality requirement has some advantages from the numerical perspective, but several drawbacks in terms of the space and frequency localization of the basis functions. Here we show how to relax this requirement in order to construct biorthogonal bases of diffusion scaling functions and wavelets. This yields more compact representations of the powers of the operator, better localized basis functions. This new construction also applies to non selfadjoint semigroups, arising in many applications.
Geodesic Gaussian kernels for value function approximation
, 2007
"... The leastsquares policy iteration approach works efficiently in value function approximation, given appropriate basis functions. Because of its smoothness, the Gaussian kernel is a popular and useful choice as a basis function. However, it does not allow for discontinuity which typically arises in ..."
Abstract

Cited by 11 (4 self)
 Add to MetaCart
The leastsquares policy iteration approach works efficiently in value function approximation, given appropriate basis functions. Because of its smoothness, the Gaussian kernel is a popular and useful choice as a basis function. However, it does not allow for discontinuity which typically arises in realworld reinforcement learning tasks. In this paper, we propose a new basis function based on geodesic Gaussian kernels, which exploits the nonlinear manifold structure induced by the Markov decision processes. The usefulness of the proposed method is successfully demonstrated in simulated robot arm control and Khepera robot navigation.
A General Framework for Manifold Alignment
"... Manifold alignment has been found to be useful in many fields of machine learning and data mining. In this paper we summarize our work in this area and introduce a general framework for manifold alignment. This framework generates a family of approaches to align manifolds by simultaneously matching ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
Manifold alignment has been found to be useful in many fields of machine learning and data mining. In this paper we summarize our work in this area and introduce a general framework for manifold alignment. This framework generates a family of approaches to align manifolds by simultaneously matching the corresponding instances and preserving the local geometry of each given manifold. Some approaches like semisupervised alignment and manifold projections can be obtained as special cases. Our framework can also solve multiple manifold alignment problems and be adapted to handle the situation when no correspondence information is available. The approaches are described and evaluated both theoretically and experimentally, providing results showing useful knowledge transfer from one domain to another. Novel applications of our methods including identification of topics shared by multiple document collections, and biological structure alignment are discussed in the paper.
Lowrank variance approximation in GMRF models: Single and multiscale approaches
 IEEE TRANSACTIONS ON SIGNAL PROCESSING
, 2008
"... We present a versatile framework for tractable computation of approximate variances in largescale Gaussian Markov random field estimation problems. In addition to its efficiency and simplicity, it also provides accuracy guarantees. Our approach relies on the construction of a certain lowrank alia ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
(Show Context)
We present a versatile framework for tractable computation of approximate variances in largescale Gaussian Markov random field estimation problems. In addition to its efficiency and simplicity, it also provides accuracy guarantees. Our approach relies on the construction of a certain lowrank aliasing matrix with respect to the Markov graph of the model. We first construct this matrix for singlescale models with shortrange correlations and then introduce spliced wavelets and propose a construction for the longrange correlation case, and also for multiscale models. We describe the accuracy guarantees that the approach provides and apply the method to a large interpolation problem from oceanography with sparse, irregular, and noisy measurements, and to a gravity inversion problem.
Stability of Localized Operators
, 2008
"... Let ℓ p,1 ≤ p ≤ ∞, be the space of all psummable sequences and Ca be the convolution operator associated with a summable sequence a. It is known that the ℓ p stability of the convolution operator Ca for different 1 ≤ p ≤ ∞ are equivalent to each other, i.e., if Ca has ℓ pstability for some 1 ≤ p ..."
Abstract

Cited by 9 (6 self)
 Add to MetaCart
Let ℓ p,1 ≤ p ≤ ∞, be the space of all psummable sequences and Ca be the convolution operator associated with a summable sequence a. It is known that the ℓ p stability of the convolution operator Ca for different 1 ≤ p ≤ ∞ are equivalent to each other, i.e., if Ca has ℓ pstability for some 1 ≤ p ≤ ∞ then Ca has ℓ qstability for all 1 ≤ q ≤ ∞. In the study of spline approximation, wavelet analysis, timefrequency analysis, and sampling, there are many localized operators of nonconvolution type whose stability is one of the basic assumptions. In this paper, we consider the stability of those localized operators including infinite matrices in the Sjöstrand class, synthesis operators with generating functions enveloped by shifts of a function in the Wiener amalgam space, and integral operators with kernels having certain regularity and decay at infinity. We show that the ℓ p stability (or L pstability) of those three classes of localized operators are equivalent to each other, and we also prove that the left inverse of those localized operators are well localized.
Capturing ridge functions in high dimensions from point queries,” preprint
, 2010
"... Constructing a good approximation to a function of many variables suffers from the “curse of dimensionality”. Namely, functions on R N with smoothness of order s can in general be captured with accuracy at most O(n −s/N) using linear spaces or nonlinear manifolds of dimension n. If N is large and s ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
(Show Context)
Constructing a good approximation to a function of many variables suffers from the “curse of dimensionality”. Namely, functions on R N with smoothness of order s can in general be captured with accuracy at most O(n −s/N) using linear spaces or nonlinear manifolds of dimension n. If N is large and s is not, then n has to be chosen inordinately large for good accuracy. The large value of N often precludes reasonable numerical procedures. On the other hand, there is the common belief that real world problems in high dimensions have as their solution, functions which are more amenable to numerical recovery. This has led to the introduction of models for these functions that do not depend on smoothness alone but also involve some form of variable reduction. In these models it is assumed that, although the function depends on N variables, only a small number of them are significant. Another variant of this principle is that the function lives on a low dimensional manifold. Since the dominant variables (respectively the manifold) are unknown, this leads to new problems of how to organize point queries to capture such functions. The present paper studies where to query the values of a ridge function f(x) = g(a · x) when both a ∈ R N and g ∈ C[0, 1] are unknown. We establish estimates on how well f can be approximated using these point queries under the assumptions that g ∈ C s [0, 1]. We also study the role of sparsity or compressibility of a in such query problems. 1
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 ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
(Show Context)
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.
Relation extraction with relation topics
 In EMNLP
, 2011
"... This paper describes a novel approach to the semantic relation detection problem. Instead of relying only on the training instances for a new relation, we leverage the knowledge learned from previously trained relation detectors. Specifically, we detect a new semantic relation by projecting the new ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
(Show Context)
This paper describes a novel approach to the semantic relation detection problem. Instead of relying only on the training instances for a new relation, we leverage the knowledge learned from previously trained relation detectors. Specifically, we detect a new semantic relation by projecting the new relation’s training instances onto a lower dimension topic space constructed from existing relation detectors through a three step process. First, we construct a large relation repository of more than 7,000 relations from Wikipedia. Second, we construct a set of nonredundant relation topics defined at multiple scales from the relation repository to characterize the existing relations. Similar to the topics defined over words, each relation topic is an interpretable multinomial distribution over the existing relations. Third, we integrate the relation topics in a kernel function, and use it together with SVM to construct detectors for new relations. The experimental results on Wikipedia and ACE data have confirmed that backgroundknowledgebased topics generated from the Wikipedia relation repository can significantly improve the performance over the stateoftheart relation detection approaches. 1