Results 1  10
of
43
Near Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?
, 2004
"... Suppose we are given a vector f in RN. How many linear measurements do we need to make about f to be able to recover f to within precision ɛ in the Euclidean (ℓ2) metric? Or more exactly, suppose we are interested in a class F of such objects— discrete digital signals, images, etc; how many linear m ..."
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Cited by 832 (16 self)
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Suppose we are given a vector f in RN. How many linear measurements do we need to make about f to be able to recover f to within precision ɛ in the Euclidean (ℓ2) metric? Or more exactly, suppose we are interested in a class F of such objects— discrete digital signals, images, etc; how many linear measurements do we need to recover objects from this class to within accuracy ɛ? This paper shows that if the objects of interest are sparse or compressible in the sense that the reordered entries of a signal f ∈ F decay like a powerlaw (or if the coefficient sequence of f in a fixed basis decays like a powerlaw), then it is possible to reconstruct f to within very high accuracy from a small number of random measurements. typical result is as follows: we rearrange the entries of f (or its coefficients in a fixed basis) in decreasing order of magnitude f  (1) ≥ f  (2) ≥... ≥ f  (N), and define the weakℓp ball as the class F of those elements whose entries obey the power decay law f  (n) ≤ C · n −1/p. We take measurements 〈f, Xk〉, k = 1,..., K, where the Xk are Ndimensional Gaussian
Regularization networks and support vector machines
 Advances in Computational Mathematics
, 2000
"... Regularization Networks and Support Vector Machines are techniques for solving certain problems of learning from examples – in particular the regression problem of approximating a multivariate function from sparse data. Radial Basis Functions, for example, are a special case of both regularization a ..."
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Cited by 266 (33 self)
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Regularization Networks and Support Vector Machines are techniques for solving certain problems of learning from examples – in particular the regression problem of approximating a multivariate function from sparse data. Radial Basis Functions, for example, are a special case of both regularization and Support Vector Machines. We review both formulations in the context of Vapnik’s theory of statistical learning which provides a general foundation for the learning problem, combining functional analysis and statistics. The emphasis is on regression: classification is treated as a special case.
Vector Greedy Algorithms
"... Our objective is to study nonlinear approximation with regard to redundant systems. Redundancy on the one hand offers much promise for greater efficiency in terms of approximation rate, but on the other hand gives rise to highly nontrivial theoretical and practical problems. Greedy type approximati ..."
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Cited by 51 (8 self)
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Our objective is to study nonlinear approximation with regard to redundant systems. Redundancy on the one hand offers much promise for greater efficiency in terms of approximation rate, but on the other hand gives rise to highly nontrivial theoretical and practical problems. Greedy type approximations proved to be convenient and efficient ways of constructing mterm approximants. We introduce and study vector greedy algorithms that are designed with aim of constructing mth greedy approximants simultaneously for a given finite number of elements. We prove convergence theorems and obtain some estimates for the rate of convergence of vector greedy algorithms when elements come from certain classes.
Nonlinear Methods of Approximation
, 2002
"... Our main interest in this paper is nonlinear approximation. The basic idea behind nonlinear approximation is that the elements used in the approximation do not come from a xed linear space but are allowed to depend on the function being approximated. While the scope of this paper is mostly theoretic ..."
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Cited by 51 (4 self)
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Our main interest in this paper is nonlinear approximation. The basic idea behind nonlinear approximation is that the elements used in the approximation do not come from a xed linear space but are allowed to depend on the function being approximated. While the scope of this paper is mostly theoretical, we should note that this form of approximation appears in many numerical applications such as adaptive PDE solvers, compression of images and signals, statistical classication, and so on. The standard problem in this regard is the problem of mterm approximation where one xes a basis and looks to approximate a target function by a linear combination of m terms of the basis. When the basis is a wavelet basis or a basis of other waveforms, then this type of approximation is the starting point for compression algorithms. We are interested in the quantitative aspects of this type of approximation. Namely, we want to understand the properties (usually smoothness) of the function which govern its rate of approximation in some given norm (or metric). We are also interested in stable algorithms for nding good or near best approximations using m terms. Some of our earlier work has introduced and analyzed such algorithms. More recently, there has emerged another more complicated form of nonlinear approximation which we call highly nonlinear approximation. It takes many forms but has the basic ingredient that a basis is replaced by a larger system of functions that is usually redundant. Some types of approximation that fall into this general category are mathematical frames, adaptive pursuit (or greedy algorithms) and adaptive basis selection. Redundancy on the one hand oers much promise for greater eciency in terms of approximation rate, but on the other hand gives rise to h...
Tree Approximation and Optimal Encoding
 J. Appl. Comp. Harm. Anal
, 2000
"... Tree approximation is a new form of nonlinear approximation which appears naturally in some applications such as image processing and adaptive numerical methods. It is somewhat more restrictive than the usual nterm approximation. We show that the restrictions of tree approximation cost little in ..."
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Cited by 44 (7 self)
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Tree approximation is a new form of nonlinear approximation which appears naturally in some applications such as image processing and adaptive numerical methods. It is somewhat more restrictive than the usual nterm approximation. We show that the restrictions of tree approximation cost little in terms of rates of approximation. We then use that result to design encoders for compression. These encoders are universal (they apply to general functions) and progressive (increasing accuracy is obtained by sending bit stream increments). We show optimality of the encoders in the sense that they provide upper estimates for the Kolmogorov entropy of Besov balls. AMS subject classication: 41A25, 41A46, 65F99, 65N12, 65N55. Key Words: compression, nterm approximation, encoding, Kolmogorov entropy . 1 Introduction Wavelets are utilized in many applications including image/signal processing and numerical methods for PDEs. Their usefulness stems in part from the fact that they provide ...
Matched sourcechannel communication for field estimation in wireless sensor networks
 Proc. the Fouth Int. Symposium on Information Processing in Sensor Networks
, 2005
"... Abstract — Sensing, processing and communication must be jointly optimized for efficient operation of resourcelimited wireless sensor networks. We propose a novel sourcechannel matching approach for distributed field estimation that naturally integrates these basic operations and facilitates a uni ..."
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Cited by 34 (10 self)
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Abstract — Sensing, processing and communication must be jointly optimized for efficient operation of resourcelimited wireless sensor networks. We propose a novel sourcechannel matching approach for distributed field estimation that naturally integrates these basic operations and facilitates a unified analysis of the impact of key parameters (number of nodes, power, field complexity) on estimation accuracy. At the heart of our approach is a distributed sourcechannel communication architecture that matches the spatial scale of field coherence with the spatial scale of node synchronization for phasecoherent communication: the sensor field is uniformly partitioned into multiple cells and the nodes in each cell coherently communicate simple statistics of their measurements to the destination via a dedicated noisy multiple access channel (MAC). Essentially, the optimal field estimate in each cell is implicitly computed at the destination via the coherent spatial averaging inherent in the MAC, resulting in optimal powerdistortion scaling with the number
Fully adaptive multiresolution finite volume schemes for conservation laws
 Math. Comp
, 2003
"... Abstract. The use of multiresolution decompositions in the context of finite volume schemes for conservation laws was first proposed by A. Harten for the purpose of accelerating the evaluation of numerical fluxes through an adaptive computation. In this approach the solution is still represented at ..."
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Cited by 26 (13 self)
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Abstract. The use of multiresolution decompositions in the context of finite volume schemes for conservation laws was first proposed by A. Harten for the purpose of accelerating the evaluation of numerical fluxes through an adaptive computation. In this approach the solution is still represented at each time step on the finest grid, resulting in an inherent limitation of the potential gain in memory space and computational time. The present paper is concerned with the development and the numerical analysis of fully adaptive multiresolution schemes, in which the solution is represented and computed in a dynamically evolved adaptive grid. A crucial problem is then the accurate computation of the flux without the full knowledge of fine grid cell averages. Several solutions to this problem are proposed, analyzed, and compared in terms of accuracy and complexity. 1.
Joint SourceChannel Communication for Distributed Estimation in Sensor Networks
"... Power and bandwidth are scarce resources in dense wireless sensor networks and it is widely recognized that joint optimization of the operations of sensing, processing and communication can result in significant savings in the use of network resources. In this paper, a distributed joint sourcechan ..."
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Cited by 26 (3 self)
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Power and bandwidth are scarce resources in dense wireless sensor networks and it is widely recognized that joint optimization of the operations of sensing, processing and communication can result in significant savings in the use of network resources. In this paper, a distributed joint sourcechannel communication architecture is proposed for energyefficient estimation of sensor field data at a distant destination and the corresponding relationships between power, distortion, and latency are analyzed as a function of number of sensor nodes. The approach is applicable to a broad class of sensed signal fields and is based on distributed computation of appropriately chosen projections of sensor data at the destination – phasecoherent transmissions from the sensor nodes enable exploitation of the distributed beamforming gain for energy efficiency. Random projections are used when little or no prior knowledge is available about the signal field. Distinct features of the proposed scheme include: 1) processing and communication are combined into one distributed projection operation; 2) it virtually eliminates the need for innetwork processing and communication; 3) given sufficient prior knowledge about the sensed data, consistent estimation is possible with increasing sensor density even with vanishing total network power; and 4) consistent signal estimation is possible with power and latency requirements growing at most sublinearly with the number of sensor nodes even when little or no prior knowledge about the sensed data is assumed at the sensor nodes.
Nonlinear piecewise polynomial approximation beyond Besov spaces
 Appl. Comput. Harmonic Anal
"... We study nonlinear nterm approximation in Lp(R2) (0 < p < ∞) from Courant elements or (discontinuous) piecewise polynomials generated by multilevel nested triangulations of R2 which allow arbitrarily sharp angles. To characterize the rate of approximation we introduce and develop three families of ..."
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Cited by 19 (4 self)
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We study nonlinear nterm approximation in Lp(R2) (0 < p < ∞) from Courant elements or (discontinuous) piecewise polynomials generated by multilevel nested triangulations of R2 which allow arbitrarily sharp angles. To characterize the rate of approximation we introduce and develop three families of smoothness spaces generated by multilevel nested triangulations. We call them Bspaces because they can be viewed as generalizations of Besov spaces. We use the Bspaces to prove Jackson and Bernstein estimates for nterm piecewise polynomial approximation and consequently characterize the corresponding approximation spaces by interpolation. We also develop methods for nterm piecewise polynomial approximation which capture the rates of the best approximation.
Approximation algorithms for wavelet transform coding of data streams
 In SODA ’06: Proceedings of the seventeenth annual ACMSIAM symposium on Discrete algorithm
, 2006
"... Abstract — This paper addresses the problem of finding a Bterm wavelet representation of a given discrete function f ∈ R n whose distance from f is minimized. The problem is well understood when we seek to minimize the Euclidean distance between f and its representation. The first known algorithms f ..."
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Cited by 19 (7 self)
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Abstract — This paper addresses the problem of finding a Bterm wavelet representation of a given discrete function f ∈ R n whose distance from f is minimized. The problem is well understood when we seek to minimize the Euclidean distance between f and its representation. The first known algorithms for finding provably approximate representations minimizing general ℓp distances (including ℓ∞) under a wide variety of compactly supported wavelet bases are presented in this paper. For the Haar basis, a polynomial time approximation scheme is demonstrated. These algorithms are applicable in the onepass sublinearspace data stream model of computation. They generalize naturally to multiple dimensions and weighted norms. A universal representation that provides a provable approximation guarantee under all pnorms simultaneously; and the first approximation algorithms for bitbudget versions of the problem, known as adaptive quantization, are also presented. Further, it is shown that the algorithms presented here can be used to select a basis from a treestructured dictionary of bases and find a Bterm representation of the given function that provably approximates its best dictionarybasis representation. Index Terms — Adaptive quantization, best basis selection, compactly supported wavelets, nonlinear approximation, sparse representation, streaming algorithms, transform coding, universal representation. I.