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31
Data Streams: Algorithms and Applications
, 2005
"... In the data stream scenario, input arrives very rapidly and there is limited memory to store the input. Algorithms have to work with one or few passes over the data, space less than linear in the input size or time significantly less than the input size. In the past few years, a new theory has emerg ..."
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Cited by 384 (21 self)
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In the data stream scenario, input arrives very rapidly and there is limited memory to store the input. Algorithms have to work with one or few passes over the data, space less than linear in the input size or time significantly less than the input size. In the past few years, a new theory has emerged for reasoning about algorithms that work within these constraints on space, time, and number of passes. Some of the methods rely on metric embeddings, pseudorandom computations, sparse approximation theory and communication complexity. The applications for this scenario include IP network traffic analysis, mining text message streams and processing massive data sets in general. Researchers in Theoretical Computer Science, Databases, IP Networking and Computer Systems are working on the data stream challenges. This article is an overview and survey of data stream algorithmics and is an updated version of [175].1
Approximation of functions over redundant dictionaries using coherence
 Proc. of SODA
, 2003
"... ..."
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.
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 ...
Maximal Spaces with given rate of convergence for thresholding algorithms
, 1999
"... this paper is to discuss the existence and the nature of maximal spaces in the context of nonlinear methods based on thresholding (or shrinkage) procedures. Before going further, some remarks should be made: ..."
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Cited by 36 (7 self)
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this paper is to discuss the existence and the nature of maximal spaces in the context of nonlinear methods based on thresholding (or shrinkage) procedures. Before going further, some remarks should be made:
Simultaneous approximation by greedy algorithms
, 2003
"... Abstract. We study nonlinear mterm approximation with regard to a redundant dictionary D in a Hilbert space H. It is known that the Pure Greedy Algorithm (or, more generally, the Weak Greedy Algorithm) provides for each f ∈ H and any dictionary D an expansion into a series f = cj(f)ϕj(f), ϕj(f) ∈ ..."
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Cited by 18 (1 self)
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Abstract. We study nonlinear mterm approximation with regard to a redundant dictionary D in a Hilbert space H. It is known that the Pure Greedy Algorithm (or, more generally, the Weak Greedy Algorithm) provides for each f ∈ H and any dictionary D an expansion into a series f = cj(f)ϕj(f), ϕj(f) ∈ D, j = 1, 2,... j=1 with the Parseval property: ‖f ‖ 2 = ∑ j cj(f)  2. Following the paper of A. Lutoborski and the second author [21] we study analogs of the above expansions for a given finite number of functions f 1,..., f N with a requirement that the dictionary elements ϕj of these expansions are the same for all f i, i = 1,..., N. We study convergence and rate of convergence of such expansions which we call simultanious expansions. 1.
Numerical techniques based on radial basis functions
 Curve and Surface Fitting: SaintMalo 1999
, 2000
"... Radial basis functions are tools for reconstruction of multivariate functions from scattered data. This includes, for instance, reconstruction of surfaces from large sets of measurements, and solving partial differential equations by collocation. The resulting very large linear N x N systems require ..."
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Cited by 14 (4 self)
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Radial basis functions are tools for reconstruction of multivariate functions from scattered data. This includes, for instance, reconstruction of surfaces from large sets of measurements, and solving partial differential equations by collocation. The resulting very large linear N x N systems require efficient techniques for their solution, preferably of O(N) or O(N log N) computational complexity. This contribution describes some special lines of research towards this future goal. Theoretical results are accompanied by numerical examples, and various open problems are pointed out.
Mathematical methods for supervised learning
 Found. Comput. Math
, 2004
"... In honor of Steve Smale’s 75th birthday with the warmest regards of the authors Let ρ be an unknown Borel measure defined on the space Z: = X × Y with X ⊂ IR d and Y = [−M,M]. Given a set z of m samples zi = (xi,yi) drawn according to ρ, the problem of estimating a regression function fρ using thes ..."
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Cited by 12 (2 self)
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In honor of Steve Smale’s 75th birthday with the warmest regards of the authors Let ρ be an unknown Borel measure defined on the space Z: = X × Y with X ⊂ IR d and Y = [−M,M]. Given a set z of m samples zi = (xi,yi) drawn according to ρ, the problem of estimating a regression function fρ using these samples is considered. The main focus is to understand what is the rate of approximation, measured either in expectation or probability, that can be obtained under a given prior fρ ∈ Θ, i.e. under the assumption that fρ is in the set Θ, and what are possible algorithms for obtaining optimal or semioptimal (up to logarithms) results. The optimal rate of decay in terms of m is established for many priors given either in terms of smoothness of fρ or its rate of approximation measured in one of several ways. This optimal rate is determined by two types of results. Upper bounds are established using various tools in approximation such as entropy, widths, and linear and nonlinear approximation. Lower bounds are proved using KullbackLeibler information together with Fano inequalities and a certain type of entropy. A distinction is drawn between algorithms which employ knowledge of the prior in the construction of the estimator and those that do not. Algorithms of the second type which are universally optimal for a certain range of priors are given. 1
Adaptive Greedy Algorithm for Solving Large RBF Collocation Problems
 Numer. Algorithms
, 2003
"... The solution of operator equations with radial basis functions by collocation in scattered points leads to large linear systems which often are nonsparse and illconditioned. But one can try to use only a subset of the data for the actual collocation, leaving the rest of the data points for error c ..."
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Cited by 10 (5 self)
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The solution of operator equations with radial basis functions by collocation in scattered points leads to large linear systems which often are nonsparse and illconditioned. But one can try to use only a subset of the data for the actual collocation, leaving the rest of the data points for error checking. This amounts to finding "sparse" approximate solutions of general linear systems arising from collocation. This contribution proposes an adaptive greedy method with proven (but slow) linear convergence to the full solution of the collocation equations. The collocation matrix need not be stored, and the progress of the method can be controlled by a variety of parameters. Some numerical examples are given.