G. Davis, "Adaptive Nonlinear Approximations," Ph.D. dissertation, Mathematics Department, NYU, September 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....one may be interested in finding the smallest number of vectors in the dictionary whose linear combination approximates the given vector within a given error threshold. This problem, in the general case, is a di#cult combinatorial optimization problem, and has recently been proven to be NP hard [9]. However, an e#cient suboptimal greedy solution to this problem has been discovered by di#erent researchers in di#erent contexts but with basically the same underlying mathematics. In statistics, this greedy algorithm was found and named projection pursuit [10] It was used for computation of ....

G. Davis, Adaptive Nonlinear Approximations.PhD thesis, New York University, Sept. 1994.

....Because of the complexity of the search, however, it is not computationally feasible to derive an optimal sparse expansion that perfectly models a signal. It is likewise not feasible to compute approximate sparse expansions that minimize the error for a given sparsity; this is an NP hard problem [25]. For this reason, it is necessary to narrow the considerations to methods that either derive sparse approximate solutions according to suboptimal criteria or derive exact solutions that are not optimally sparse. Methods of the latter type tend to be computationally costly and to lack an effective ....

....in the mean squared sense because of the term by term greediness of the algorithm. Computing the optimal I term estimate using an overcomplete dictionary requires finding the minimum projection error over all I dimensional dictionary subspaces, which is an NP hard problem as mentioned earlier [25]. To enable representation of a wide range of signal features, a large dictionary of time frequency atoms is used in the matching pursuit algorithm. The computation of the correlations hg; r i i for all g 2 D is thus costly. As noted in [3] this computation can be substantially reduced using an ....

G. Davis. Adaptive Nonlinear Approximations. PhD thesis, New York University, September 1994.

....the representation that the model provides. If a representation is both accurate and compact, i.e. is not data intensive, then it can be concluded that the representation captures the primary or meaningful signal behavior; a compact model in some sense extracts the coherent structure of a signal [38, 39]. This insight suggests that accurate compact representations are applicable to the tasks of compression, denoising, analysis, and signal modification; these are discussed in turn. 1.2.1 Compression It is perhaps obvious that by definition a compact representation is useful for compression. In ....

....devising algorithms that find compact solutions. Such algorithms come in two forms: those that find exact solutions that maximize a compaction metric, either formally or heuristically [42, 67, 68] and those that find sparse approximate solutions that model the signal within some error tolerance [38, 39, 69]. These two paradigms have the same fundamental goal, namely compact modeling, but the frameworks are considerably different; in either case, however, the expansion functions are chosen in a signal adaptive fashion and the algorithms for choosing the functions are decidedly nonlinear. This issue ....

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G. Davis, Adaptive Nonlinear Approximations. PhD thesis, New York University, September 1994.

....the rate associated with f k i g L i=1 may depend on the choice of dictionary elements. Nevertheless, we are discouraged CHAPTER 3. ADAPTIVE EXPANSIONS 20 from attempting to find optimal quantized representations by the following theorem. Theorem 3. 1: Intractability of Optimal Approximation [7] Let k 1 and let D be a dictionary that contains O(N k ) vectors. Let 0 fl 1 fl 2 1 and let L 2 Z such that fl 1 N L fl 2 N . For any given ffl 0 and f 2 H, determining whether an (ffl; L) approximation exists is NP complete. Finding the L optimal approximation is NP hard. ....

....problem. It progressively refines a signal estimate instead of finding L components jointly. Matching pursuit was introduced to the signal processing community in the context of time frequency analysis by Mallat and Zhang [20] Mallat and his students have uncovered many of its properties [7, 8, 9, 40]. 3.2.1 Algorithm Let D = f k g M k=1 ae H be a frame. We impose the additional constraint that k k k = 1 for all k. We will call D our dictionary of vectors. Matching pursuit is an algorithm to represent f 2 H by a linear combination of elements of D. Furthermore, matching pursuit is an ....

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G. Davis, "Adaptive Nonlinear Approximations," Ph.D. dissertation, Mathematics Department, NYU, September 1994.

....optimality, however, is not insured because of the nature of the algorithm. Determining the globally optimal I term expansion based on an overcomplete dictionary requires finding the minimum error over all I dimensional dictionary subspaces, which is not computationally feasible for large I [10]. To enable representation of a wide range of signal features, large dictionaries are used in the matching pursuit algorithm. The computation of the correlations hg; r i i is thus intensive. As noted in [9] however, the computation can be drastically reduced using an update formula derived from ....

G. Davis. Adaptive Nonlinear Approximations. PhD thesis, New York University, 1994.

.... matching pursuit approach does not find the optimal I term expansion; determining the optimal I term expansion based on a non orthogonal dictionary requires finding the minimum projection error over all I dimensional dictionary subspaces, which is not computationally feasible for large I [2]. Though searching for the optimal high dimension subspace is not reasonable, it is reasonable to consider the related problem of finding an optimal low dimension subspace at each iteration of the matching pursuit algorithm. In this variation of the algorithm, the i th iteration consists of ....

G. Davis. Adaptive Nonlinear Approximations. PhD thesis, New York University, 1994.

....for transients occurring more or less randomly in the signal [1] A general solution for the above limitations can be achieved by using a redundant set of waveforms instead of orthonormal basis. However, the problem of choosing the waveforms that would best explain the signal s variance is NP hard [2]. Matching Pursuit, introduced by Mallat and Zhang in 1993 [3] provides a sub optimal solution. Detection of sleep spindles in EEG was traditionally performed by visual analysis, although several automatic methods were tuned to reproduce visual detection. However, most of them inherited the main ....

Davis G. "Adaptive Nonlinear Approximations" - a dissertation: Courant Institute of Mathematical Sciences, New York University, September 1994 ftp://cs.nyu.edu/pub/wave/report/DissertationGDavis.ps.Z

....recursively with the standard WFA coding algorithm of [1] i.e. the MCPE is subdivided into several range blocks which are approximated with a linear combination of domain images. The mathematical background of this approximation problem (so called orthogonal matching pursuit) was studied in [8] [9] the application to fractal coding was discussed in [4] Mallat et al. suggested a greedy algorithm to compute a linear combination for a given range with an over complete set of dictionary vectors (i.e. the domain pool) Starting with the empty set the linear combination is generated step by ....

G. Davis. Adaptive Nonlinear Approximations. PhD thesis, New York University, 1994.

....to solve an NP hard problem grows faster than any polynomial in the input size [13] Because of the difficulty of computing optimal expansions, we turn to suboptimal algorithms. In section 3 we review the performance of greedy algorithms, called matching pursuits, that were introduced in [24] [7]. We describe a fast implementation of these algorithms, and we give numerical examples for a dictionary composed of waveforms that are well localized in time and frequency. Such dictionaries are particularly important for audio signal processing. In our numerical experiments we find that the rate ....

....length constraints. In H 2 , the orthogonal complement of H 1 in H, we construct a (0; ff 2 N Gamma n) approximation problem which has a unique solution. The combined approximation problem in H will be equivalent to the exact cover problem and will have the requisite M and dictionary size [7]. The optimal approximation criterion of definition 2.1 has number of undesirable properties which are responsible for its NP completeness. The elements contained in the expansions are unstable in that functions which are only slightly different can have optimal expansions containing completely ....

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G. Davis "Adaptive Nonlinear Approximations," Ph.D. dissertation, Department of Mathematics, New York University, 1994.

....5 we prove that for general dictionaries, the problem of finding M vector optimal approximations is NP hard. Because of the numerical intractability of computing optimal expansions, in section 3 we review the performance of greedy algorithms, called matching pursuits, that were introduced in [32] [13]. We also describe a fast numerical implementation of these algorithms, and we give numerical examples for a dictionary composed of waveforms that are well localized in time and frequency. Such dictionaries are particularly important for audio signal processing. In our numerical experiments we ....

....length constraints. In H 2 , the orthogonal complement of H 1 in H, we construct a (0; ff 2 N Gamma n) approximation problem which has a unique solution. The combined approximation problem in H will be equivalent to the exact cover problem and will have the requisite M and dictionary size [13]. The optimal approximation criterion of definition 2.1 has number of undesirable properties which are partly responsible for its NP completeness. The elements contained in the expansions are unstable in that functions which are only slightly different can have optimal expansions containing ....

[Article contains additional citation context not shown here]

G. Davis "Adaptive Nonlinear Approximations," Ph.D. dissertation, Department of Mathematics, New York University, 1994.

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G. Davis, "Adaptive Nonlinear Approximations," Ph.D. dissertation, Mathematics Department, NYU, September 1994.

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G. Davis. Adaptive nonlinear approximations. PhD, dissertation, New York University, 1994.

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Davis G., Adaptive Nonlinear Approximations. PhD thesis, New York University, September 1994.

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G. Davis, "Adaptive nonlinear approximations," Ph.D. dissertation, New York Univ., New York, NY, 1994.

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G. Davis, "Adaptive nonlinear approximations," Ph.D. dissertation, New York Univ., Sept. 1994.

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G. Davis, "Adaptive nonlinear approximations," Ph.D. dissertation, New York Univ., New York, NY, Sept. 1994.

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G. Davis, "Adaptive nonlinear approximations," Ph.D. dissertation, New York Univ., New York, NY, Sept. 1994.

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G. Davis. Adaptive nonlinear approximations. PhD, dissertation, New York University, 1994.

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G. Davis. Adaptive nonlinear approximations. PhD, dissertation, New York University, 1994.

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Davis G., Adaptive Nonlinear Approximations, Ph.D. thesis, New York University, September 1994.

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Davis G., Adaptive Nonlinear Approximations, Ph.D. thesis, New York University, September 1994.

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G. Davis, Adaptive Nonlinear Approximations. PhD thesis, New York University, Sept. 1994.

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