## Approximation of functions over redundant dictionaries using coherence (2003)

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Venue: | Proc. of SODA |

Citations: | 70 - 4 self |

### BibTeX

@INPROCEEDINGS{Gilbert03approximationof,

author = {Anna C. Gilbert and S. Muthukrishnan and Martin J. Strauss},

title = {Approximation of functions over redundant dictionaries using coherence},

booktitle = {Proc. of SODA},

year = {2003},

pages = {243--252}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

1722 |
Ten Lectures on Wavelets
- Daubechies
- 1992
(Show Context)
Citation Context ...search in nonlinear approximation theory has mostly focused on specific dictionaries. Mathematicians have studied specific dictionaries-- spikes and sinusoids [4], wavelet packets [5, 25, 24], frames =-=[6]-=-, time/frequency tilings with algebraic hierarchical structure [19]--and presented individual algorithms (some provable, some heuristic) for constructing the best B-term approximation for any given fu... |

1654 | Atomic decomposition by basis pursuit
- Chen, Donoho, et al.
- 1998
(Show Context)
Citation Context ...ctor, however large. As a consequence, research in nonlinear approximation theory has mostly focused on specific dictionaries. Mathematicians have studied specific dictionaries-- spikes and sinusoids =-=[4]-=-, wavelet packets [5, 25, 24], frames [6], time/frequency tilings with algebraic hierarchical structure [19]--and presented individual algorithms (some provable, some heuristic) for constructing the b... |

1047 | Matching Pursuit With Time-Frequency Dictionaries
- Mallat, Zhang
- 1993
(Show Context)
Citation Context ...¢ : ¢ E A}, and repeat this process a total of B times. This technique is called orthogonal matching pursuit, OMP. Both these approaches have been studied extensively in the literature (MP appears in =-=[18]-=- and OMP appears in [21]). However, there are two drawbacks in the state of art, as follows: The first issue is of proving bounds on the error of the solution. 1 If the dictionary is an orthonormal ba... |

713 | Approximate nearest neighbors: towards removing the curse of dimensionality, Proceedings of the thirtieth annual ACM symposium on Theory of computing
- Indyk, Motwani
- 1998
(Show Context)
Citation Context ...ctionary T~ that has the largest inner product with the signal is equivalent to the Farthest Neighbors Problem, which faces the same "dimensionality bottleneck" as the Nearest Neighbors Problem [14]- =-=[15]-=-, [17]. Designing efficient algorithms for this problem is still an open challenge, in particular, since the local search heuristics rely on finding the optimal such ¢'s. This is a serious drawback be... |

701 | The space complexity of approximating the frequency moments
- ALON, MATIAS, et al.
- 1999
(Show Context)
Citation Context ... ~, A find a representation for the normalized signal, and scale back up. For the normalized signal, we can get the approximate dot product operations from approximate ~2 distance operations. • Using =-=[1]-=-, :randomly project the signal A and the dictionary vectors, using a randomly-chosen linear function h from a particular family, into a log ISI/~ °(1) dimensional space so the ~2 norm between any pair... |

628 | Constructive Approximation
- DeVore, Lorentz
- 1993
(Show Context)
Citation Context ...compression and statistical estimation with its applications to classification. It is a rich area with significant mathematical achievements and successful applications, both classical and novel; see =-=[8]-=-, [9] for good surveys, and The Journal of Approximation Theory [16] and Constructive Approximation [2] for current mathematical developments. There are two approaches to mathematical approximation th... |

492 | Entropy-based algorithms for best basis selection
- Coifman, Wickerhauser
- 1992
(Show Context)
Citation Context ...As a consequence, research in nonlinear approximation theory has mostly focused on specific dictionaries. Mathematicians have studied specific dictionaries-- spikes and sinusoids [4], wavelet packets =-=[5, 25, 24]-=-, frames [6], time/frequency tilings with algebraic hierarchical structure [19]--and presented individual algorithms (some provable, some heuristic) for constructing the best B-term approximation for ... |

350 | Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition
- Pati, Rezaiifar, et al.
- 1993
(Show Context)
Citation Context ...his process a total of B times. This technique is called orthogonal matching pursuit, OMP. Both these approaches have been studied extensively in the literature (MP appears in [18] and OMP appears in =-=[21]-=-). However, there are two drawbacks in the state of art, as follows: The first issue is of proving bounds on the error of the solution. 1 If the dictionary is an orthonormal basis, then the local sear... |

288 | Hardness vs. randomness
- Nisan, Wigderson
- 1994
(Show Context)
Citation Context ...l it and hence our main algorithmic result applies to them too. Without focusing on any application, we focus on constructing large redundant dictionaries of small coherence. Using the NisanWigderson =-=[20]-=- combinatorial design, we obtain such a dictionary of size exponential in 2We assume that entries in A are bounded by N °(D. The general result requires an additional factor in time and space polynomi... |

190 | Efficient search for approximate nearest neighbor in highdimensional spaces
- Kushilevitz, Ostrovsky, et al.
- 1998
(Show Context)
Citation Context ...ry T~ that has the largest inner product with the signal is equivalent to the Farthest Neighbors Problem, which faces the same "dimensionality bottleneck" as the Nearest Neighbors Problem [14]- [15], =-=[17]-=-. Designing efficient algorithms for this problem is still an open challenge, in particular, since the local search heuristics rely on finding the optimal such ¢'s. This is a serious drawback because ... |

105 |
small-space algorithms for approximate histogram maintenance
- Fast
- 2002
(Show Context)
Citation Context ...cently. That gives the overall result. There are some similarities between this work and our earlier work on finding near-optimal representations of functions over Fourier [13] and Haar wavelet bases =-=[12]-=-, but also some crucial differences. Both Fourier and wavelet bases are non-redundant dictionaries, and so efficient algorithms already existed for finding even optimal representations over them. The ... |

100 | Wedgelets: Nearly-minimax estimation of edges
- Donoho
- 1999
(Show Context)
Citation Context ...ous best [4] of O(N 3'5) time via Linear Programming. • We explore the concept of it-coherent redundant dictionaries further, adding to the growing knowledge of redundant dictionary construction [3], =-=[10]-=-- [11]. Known redundant dictionaries are of size only O(N) or O(Nlog(N)); it is desirable to construct larger dictionaries if "natural" properties of a specific application can still be captured. We c... |

79 | Near-optimal sparse fourier representations via sampling
- Gilbert, Guha, et al.
- 2002
(Show Context)
Citation Context ...s that have been designed recently. That gives the overall result. There are some similarities between this work and our earlier work on finding near-optimal representations of functions over Fourier =-=[13]-=- and Haar wavelet bases [12], but also some crucial differences. Both Fourier and wavelet bases are non-redundant dictionaries, and so efficient algorithms already existed for finding even optimal rep... |

77 |
Ridgelets: Theory and Applications
- Candes
- 1998
(Show Context)
Citation Context ...previous best [4] of O(N 3'5) time via Linear Programming. • We explore the concept of it-coherent redundant dictionaries further, adding to the growing knowledge of redundant dictionary construction =-=[3]-=-, [10]- [11]. Known redundant dictionaries are of size only O(N) or O(Nlog(N)); it is desirable to construct larger dictionaries if "natural" properties of a specific application can still be captured... |

54 |
Greedy adaptive approximation
- Davis, Mallat, et al.
- 1997
(Show Context)
Citation Context ...ion A, parameter B and a prespecified dictionary :D, finding the optimal representation of B terms. 1.2 State of the Art The general highly-non-linear approximation problem of our interest is NP-hard =-=[7]-=-. In fact, the proof there implicitly shows that the problem is NP-hard even to determine whether the optimal error is zero for the given B; hence, unless P=NP, no polynomial time algorithm exists tha... |

54 |
High-Dimensional Computational Geometry
- Indyk
(Show Context)
Citation Context ...the dictionary T~ that has the largest inner product with the signal is equivalent to the Farthest Neighbors Problem, which faces the same "dimensionality bottleneck" as the Nearest Neighbors Problem =-=[14]-=-- [15], [17]. Designing efficient algorithms for this problem is still an open challenge, in particular, since the local search heuristics rely on finding the optimal such ¢'s. This is a serious drawb... |

36 | The best m-term approximation and greedy algorithms - Temlyakov - 1998 |

18 | Beamlet pyramids: a new form of multiresolution analysis, suited for extracting lines, curves and objects from very noisy image data
- Donoho, Huo
- 2000
(Show Context)
Citation Context ...tice is to take several orthonormal bases and combine them, and one gets the most out of this combination if the bases were as orthogonal to each other as possible. (Using sinusoids and spikes, as in =-=[11]-=-, is an example.) Dictionaries generated in this manner have small coherence, thus motivating our work here. Coherence has been discussed as an important notion in the context of local search heuristi... |

18 |
Greedy algorithms and M-term approximation with regard to redundant dictionaries
- Temlyakov
- 1999
(Show Context)
Citation Context ...ancy on the one hand offers much promise ]or greater e~iciency in terms o] approximation rate, but on the other hand, gives rise to highly nontrivial theoretical and practical problems," (quoted from =-=[23]-=-). In fact, the mathematics of the theory of highly nonlinear approximation is only now emerging, and algorithmic complexity of these problems is wide open. The subject of this paper is the central al... |

13 | Best approximation with walsh atoms
- Villemoes
- 1997
(Show Context)
Citation Context ...As a consequence, research in nonlinear approximation theory has mostly focused on specific dictionaries. Mathematicians have studied specific dictionaries-- spikes and sinusoids [4], wavelet packets =-=[5, 25, 24]-=-, frames [6], time/frequency tilings with algebraic hierarchical structure [19]--and presented individual algorithms (some provable, some heuristic) for constructing the best B-term approximation for ... |

7 |
Villemoes, “A fast algorithm for adapted time-frequency tilings
- Thiele, F
- 1996
(Show Context)
Citation Context ...cific dictionaries. Mathematicians have studied specific dictionaries-- spikes and sinusoids [4], wavelet packets [5, 25, 24], frames [6], time/frequency tilings with algebraic hierarchical structure =-=[19]-=---and presented individual algorithms (some provable, some heuristic) for constructing the best B-term approximation for any given function. Also, certain "mathematical recipes" such as algorithms for... |

4 | approximation with walsh atoms - Best - 1997 |

1 |
The best m-terra approximation and greedy algorithms
- Temlyakov
- 1998
(Show Context)
Citation Context ...c) for constructing the best B-term approximation for any given function. Also, certain "mathematical recipes" such as algorithms for infinite dimensional Hilbert and Banach spaces have been proposed =-=[22]-=-, [23]. Save these exceptions, no provable, algorithmic results--running time for finding provably approximate B-term representations for inputs over general redundant dictionaries-- are currently kno... |

1 |
approximation with Walsh atoms. Constructive Approximation
- Best
- 1997
(Show Context)
Citation Context ...As a consequence, research in nonlinear approximation theory has mostly focused on specific dictionaries. Mathematicians have studied specific dictionaries-- spikes and sinusoids [4], wavelet packets =-=[5, 25, 24]-=-, frames [6], time/frequency tilings with algebraic hierarchical structure [19]--and presented individual algorithms (some provable, some heuristic) for constructing the best B-term approximation for ... |