Greedy Approximation

This first book on greedy approximation gives a systematic presentation of the fundamental results. It also contains an introduction to two hot topics in numerical mathematics: learning theory and compressed sensing. Nonlinear approximation is becoming increasingly important, especially since two types are frequently employed in applications: adaptive methods are used in PDE solvers, while m-term approximation is used in image/signal/data processing, as well as in the design of neural networks. The fundamental question of nonlinear approximation is how to devise good constructive methods (algorithms) and recent results have established that greedy type algorithms may be the solution. The author has drawn on his own teaching experience to write a book ideally suited to graduate courses. The reader does not require a broad background to understand the material. Important open problems are included to give students and professionals alike ideas for further research.

Vladimir Temlyakov is Carolina Distinguished Professor in the Department of Mathematics at the University of South Carolina.

Preface	p. ix
Greedy approximation with regard to bases	p. 1
Introduction	p. 1
Schauder bases in Banach spaces	p. 6
Greedy bases	p. 15
Quasi-greedy and almost greedy bases	p. 33
Weak Greedy Algorithms with respect to bases	p. 39
Thresholding and minimal systems	p. 43
Greedy approximation with respect to the trigonometric system	p. 47
Greedy-type bases; direct and inverse theorems	p. 58
Some further results	p. 63
Systems L_p-equivalent to the Haar basis	p. 68
Open problems	p. 76
Greedy approximation with respect to dictionaries: Hilbert spaces	p. 77
Introduction	p. 77
Convergence	p. 84
Rate of convergence	p. 89
Greedy algorithms for systems that are not dictionaries	p. 97
Greedy approximation with respect to ¿-quasi-orthogonal dictionaries	p. 101
Lebesgue-type inequalities for greedy approximation	p. 111
Saturation property of greedy-type algorithms	p. 122
Some further remarks	p. 135
Open problems	p. 141
Entropy	p. 143
Introduction: definitions and some simple properties	p. 143
Finite dimensional spaces	p. 144
Trigonometric polynomials and volume estimates	p. 151
The function classes	p. 165
General inequalities	p. 168
Some further remarks	p. 175
Open problems	p. 182
Approximation in learning theory	p. 183
Introduction	p. 183
Some basic concepts of probability theory	p. 189
Improper function learning; upper estimates	p. 206
Proper function learning; upper estimates	p. 235
The lower estimates	p. 253
Application of greedy algorithms in learning theory	p. 270
Approximation in compressed sensing	p. 277
Introduction	p. 277
Equivalence of three approximation properties of the compressed sensing matrix	p. 283
Construction of a good matrix	p. 287
Dealing with noisy data	p. 294
First results on exact recovery of sparse signals; the Orthogonal Greedy Algorithm	p. 298
Exact recovery of sparse signals; the Subspace Pursuit Algorithm	p. 305
On the size of incoherent systems	p. 314
Restricted Isometry Property for random matrices	p. 327
Some further remarks	p. 330
Open problems	p. 332
Greedy approximation with respect to dictionaries: Banach spaces	p. 334
Introduction	p. 334
The Weak Chebyshev Greedy Algorithm	p. 340
Relaxation; co-convex approximation	p. 347
Free relaxation	p. 350
Fixed relaxation	p. 354
Thresholding algorithms	p. 359
Greedy expansions	p. 363
Relaxation; X-greedy algorithms	p. 378
Incoherent dictionaries and exact recovery	p. 381
Greedy algorithms with approximate evaluations and restricted search	p. 385
An application of greedy algorithms for the discrepancy estimates	p. 390
Open problems	p. 404
References	p. 405
Index	p. 415
Table of Contents provided by Ingram. All Rights Reserved.

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