## Diffusion Wavelets (2004)

Citations: | 78 - 12 self |

### BibTeX

@MISC{Coifman04diffusionwavelets,

author = {Ronald R. Coifman and Mauro Maggioni},

title = { Diffusion Wavelets},

year = {2004}

}

### Years of Citing Articles

### OpenURL

### Abstract

We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green’s function, in compressed form, and their fast application. Classes of operators satisfying these conditions include diffusion-like operators, in any dimension, on manifolds, graphs, and in non-homogeneous media. In this case our construction can be viewed as a far-reaching generalization of Fast Multipole Methods, achieved through a different point of view, and of the non-standard wavelet representation of Calderón-Zygmund and pseudodifferential operators, achieved through a different multiresolution analysis adapted to the operator. We show how the dyadic powers of an operator can be used to induce a multiresolution analysis, as in classical Littlewood-Paley and wavelet theory, and we show how to construct, with fast and stable algorithms, scaling function and wavelet bases associated to this multiresolution analysis, and the corresponding downsampling operators, and use them to compress the corresponding powers of the operator. This allows to extend multiscale signal processing to general spaces (such as manifolds and graphs) in a very natural way, with corresponding fast algorithms.

### Citations

2756 | Normalized cuts and image segmentation
- Shi, Malik
- 2000
(Show Context)
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2631 | A theory for multiresolution signal decomposition: The wavelet representation
- Mallat
- 1989
(Show Context)
Citation Context ... ⎞ j−1 ∏ = ⎝ ⎠ϕ0,l,k(x) , x ∈ X0 ϕj,l,k(x) = [Φj]Φj−1 ϕj−1,l,k(x) l=0 [Φl]Φl−1 (4.18) This is of course completely analogous to the standard construction of scaling functions in the Euclidean setting =-=[49,49,50,2,51]-=-. This formula also immediately generalizes to arbitrary functions in Vj, extending them from Xj to the whole original space X (see for example Figure 13). A detailed analysis of computational complex... |

1909 |
lectures on wavelets
- Ten
- 1992
(Show Context)
Citation Context ... ⎞ j−1 ∏ = ⎝ ⎠ϕ0,l,k(x) , x ∈ X0 ϕj,l,k(x) = [Φj]Φj−1 ϕj−1,l,k(x) l=0 [Φl]Φl−1 (4.18) This is of course completely analogous to the standard construction of scaling functions in the Euclidean setting =-=[49,49,50,2,51]-=-. This formula also immediately generalizes to arbitrary functions in Vj, extending them from Xj to the whole original space X (see for example Figure 13). A detailed analysis of computational complex... |

1781 | A global geometric framework for nonlinear dimensionality reduction, Science 290 (5500 - Tenenbaum, Silva, et al. - 2000 |

1735 | Orthonormal Bases of Compactly Supported Wavelets
- Daubechies
- 1988
(Show Context)
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1043 |
Spectral Graph Theory
- Chung
- 1997
(Show Context)
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785 | Laplacian eigenmaps for dimensionality reduction and data representation." Neural Computation 15(6
- Belkin, Niyogi
- 2003
(Show Context)
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782 | A Fast Algorithm for Particle Simulations
- Greengard, Rokhlin
- 1987
(Show Context)
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760 | Approximate nearest neighbor - towards removing the curse of dimensionality
- Indyk, Motwani
- 1998
(Show Context)
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554 | A signal processing approach to fair surface design - TAUBIN - 1995 |

531 | Entropy-based algorithms for best-basis selection - Coifman, Wickerhauser - 1992 |

464 | The lifting scheme: A custom-design construction of biorthogonal wavelets - Sweldens - 1996 |

456 | H.: Implicit fairing of irregular meshes using diffusion and curvature flow
- DESBRUN, MEYER, et al.
- 1999
(Show Context)
Citation Context ... in the choice of the diffusion operator, and in the choice of scale at each location) filtering, smoothing and 56denoising of the point set, and in this respect it is most similar to [80]. See also =-=[81]-=- for applications of diffusion to implicit mesh fairing. In image processing and vision, the ideas of scale spaces, through linear and nonlinear diffusion, have had a great influence. In [2] the autho... |

434 |
Fast wavelet transforms and numerical algorithms
- Beylkin, Coifman, et al.
- 1992
(Show Context)
Citation Context ...ent levels of specificity. Our construction can be related to Fast Multipole Methods [1], and to the wavelet representation for Calderón-Zygmund integral operators and pseudodifferential operators of =-=[2]-=-, but from a “dual” perspective. We start from a semigroup {T t }, associated to a diffusion process (e.g. T = e −ǫ∆ ), rather than from the Green’s operator, since the latter is not available in the ... |

423 | The lifting scheme: A construction of second generation wavelets - Sweldens - 1997 |

349 | Multiresolution analysis for surfaces of arbitrary topological type
- Lounsbery, DeRose, et al.
- 1997
(Show Context)
Citation Context ...nt of the meshes. An improved construction of a subdivision scheme with good regularity properties for arbitrary topologies is proposed in [87] and its applications to surface compression studied. In =-=[88]-=- the authors propose a construction of semi-orthogonal and biorthogonal wavelets based on regular subdivision schemes on samples from surfaces of arbitrary topology. The geometry of the subdivision sc... |

321 |
Heat Kernels and Spectral Theory
- Davies
- 1989
(Show Context)
Citation Context ...a manifold with smooth boundary, and V a nonnegative potential function. This type of operators are called Schrödinger operators, and are well studied objects in mathematical physics. See for example =-=[14,15]-=-. A particular example is when X represents a cloud of points generated by a (stochastic) process driven by a Langevin equation (e.g. a protein configuration in a solvent), and the diffusion operator ... |

307 |
Prolate Spheroidal Wave Functions, Fourier Analysis, and Uncertainty- V: The Discrete
- Slepian
- 1978
(Show Context)
Citation Context ... ) : supp. ˆ f ⊆ Bc(0) } = 〈{e i<ξ,·> : |ξ| ≤ c}〉, the space of band-limited functions with band c, and then let En be BL2n. The { ˜ θn,i}i in this case are generalized prolate spherical functions of =-=[72,73]-=-, called geometric harmonics in [4,3]. (ii) Let En be one the approximation spaces Vn of some classical wavelet multiresolution analysis in Rn , like the one associated to Meyer wavelets. (iii) Let { ... |

305 | Using the Nyström method to speed up kernel machines - Williams, Seeger |

280 |
Ondelettes et opérateurs
- Meyer
- 1990
(Show Context)
Citation Context ... ⎞ j−1 ∏ = ⎝ ⎠ϕ0,l,k(x) , x ∈ X0 ϕj,l,k(x) = [Φj]Φj−1 ϕj−1,l,k(x) l=0 [Φl]Φl−1 (4.18) This is of course completely analogous to the standard construction of scaling functions in the Euclidean setting =-=[49,49,50,2,51]-=-. This formula also immediately generalizes to arbitrary functions in Vj, extending them from Xj to the whole original space X (see for example Figure 13). A detailed analysis of computational complex... |

239 | Spherical wavelets: Efficiently representing functions on the sphere - Schröder, Sweldens - 1995 |

233 | Locality preserving projections
- He, Niyogi
- 2003
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Citation Context ...or any subset σ(T) ′ ⊆ σ(T) we can consider the map of metric spaces Ξ (t) σ(T) ′ : (X, d (t) ) → (R |σ(T)′ | , dEuc.) x ↦→ ( λ t 2 ξλ(x) ) λ∈σ(T) ′ (4.5) which is in particular cases called eigenmap =-=[33,34]-=-, and is a form of local multidimensional scaling (see also [35,3,4]). By the definition of d (t) , this map is an isometry when σ(T) ′ = σ(T), and an approximation to an isometry when σ(T) ′ � σ(T). ... |

228 | Multiresolution signal processing for meshes
- Guskov, Sweldens, et al.
- 1999
(Show Context)
Citation Context ... adaptive (both in the choice of the diffusion operator, and in the choice of scale at each location) filtering, smoothing and 56denoising of the point set, and in this respect it is most similar to =-=[80]-=-. See also [81] for applications of diffusion to implicit mesh fairing. In image processing and vision, the ideas of scale spaces, through linear and nonlinear diffusion, have had a great influence. I... |

223 | A course on wavelets - Hernandez, Weiss - 1996 |

215 | Interpolating Subdivision for Meshes with Arbitrary Topology
- Zorin, Schröder, et al.
- 1996
(Show Context)
Citation Context ...gularity assumption are made in the construction and refinement of the meshes. An improved construction of a subdivision scheme with good regularity properties for arbitrary topologies is proposed in =-=[87]-=- and its applications to surface compression studied. In [88] the authors propose a construction of semi-orthogonal and biorthogonal wavelets based on regular subdivision schemes on samples from surfa... |

166 |
Analyse harmonique non-commutative sur certains espaces homogènes
- Coifman, Weiss
- 1971
(Show Context)
Citation Context ...from the discretization of continuous and infinite-dimensional problems, we introduce the following definitions. Definition 3 A quasi-metric measure space (X, d, µ) is said to be of homogeneous type =-=[21,6]-=- if µ is a non-negative Borel measure and there exists a constant CX > 0 such that for every x ∈ X, δ > 0, µ(B2δ(x)) ≤ CXµ(Bδ(x)) (3.1) We assume µ(Bδ(x)) < ∞ for all x ∈ X, δ > 0, and we will work on... |

158 | Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps - Coifman, Lafon, et al. |

157 |
Lectures on Analysis on Metric Spaces
- Heinonen
- 2001
(Show Context)
Citation Context ...∫ e X −νρ′ (x,y) dµ(y) < E . (4.14) PROOF. Let ρ ′ be a metric, topologically equivalent to ρ, such that µ({y ∈ X : ρ ′ (x, y) < r}) ≤ CXr d , for some d > 0. The existence of such a ρ ′ is proved in =-=[48]-=-. We have: ∫ ∫ e −νρ′ (x,y) dµ(y) = e −νρ′ (x,y) dµ(y) X {y:ρ′ (x,y)≤1} + ∑ ∫ e −νρ′ (x,y) dµ(y) j≥0 {y:ρ′(x,y)∈(2j ,2j+1 ]} ≤ µ(B1(x)) + ∑ ≤ E(ν) . j≥0 e −ν2j 2 (j+1)d One can apply Proposition 25 fo... |

152 | A sparse matrix arithmetic based on H-matrices. Part I: Introduction to Hmatrices
- Hackbusch
- 1999
(Show Context)
Citation Context ... a function f ∈ L 2 (X, µ) be the set of all coefficients {〈f, Ψj〉}j=0,...,J . All these coefficients can be computed in time O(n k ). Material relevant to the following two sections is in the papers =-=[44,45,64]-=- (and references therein) which present matrix compression techniques with applications to numerical functional calculus and eigenfunction computations. 436.1 Compressed eigenfunction computation The... |

148 |
Matrix Computations, The Johns Hopkins
- Golub, Loan
- 1989
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Citation Context ...imes write ˜ϕ˜xk for ˜ϕk. Let V = 〈{˜ϕk} k∈ ˜K 〉 . We want to build an orthonormal basis Φ = {ϕk}k∈K whose span is ǫ-close to V . Out of the many possible solutions to this standard problem (see e.g. =-=[57,58]-=- and references therein as a starting point), we seek one for which the ϕk’s have small support (ideally of the same order as the support of ˜ϕk). Standard orthonormalization in general may completely... |

125 | Multivariate splines - Chui - 1988 |

124 |
and metrics defined by vector fields I: Basic properties
- Nagel, Stein, et al.
- 1985
(Show Context)
Citation Context ...r forms and their powers (see e.g. [24]). (ii) Compact Riemannian manifolds of bounded curvature, with the geodesic metric, or also with respect to metrics induced by certain classes of vector fields =-=[25]-=-. (iii) Finite graphs of bounded degree with shortest path distance, in particular k-regular graphs, where the degree of a vertex is defined as dx = ∑ wyx , y∼x where y ∼ x means there is an edge betw... |

121 | R.: Fast multiscale image segmentation - Sharon, Brandt, et al. |

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- Ham, Lee, et al.
- 2004
(Show Context)
Citation Context ... rephrasing, in our situation, of the well-known fact that the top k singular vectors span the best approximating k dimensional subspace to the domain of a linear operator, in the L 2 sense. See e.g. =-=[36]-=- and references therein for a comparison between different dimensionality reduction techniques that can be cast in this framework. These ideas are related to various techniques used for nonlinear dime... |

115 |
Topics in harmonic analysis related to the Littlewood–Paley theory
- Stein
- 1970
(Show Context)
Citation Context ...is, in particular in harmonic analysis, scientific computation, image processing, among many others. Much of the development of multiscale analysis and Littlewood Paley theory can already be found in =-=[21,5]-=-. Here we are merging the two points of views. Harmonic analysis of clouds of points has been considered in the past by [77–79] where the eigenfunctions of the Laplace operator, or multiresolution con... |

103 | Out-of-sample extensions for LLE, Isomap, MDS, eigenmaps, and spectral clustering
- Bengio, Paiement, et al.
- 2003
(Show Context)
Citation Context ...it is also possible to use the Nyström extension [74–76], which however has the disadvantages described above. For a survey of extension techniques in the context of nonlinear dimension reduction see =-=[38]-=- and references therein, as well as [4,3]. Example 43 We illustrate the construction above by extending one of the scaling functions on the dumbell-shaped manifold already considered in subsection 2.6... |

103 | Reorthogonalization and stable algorithms for updating the Gram–Schmidt QR factorization - Daniel, Gragg, et al. - 1976 |

102 | Hessian eigenmaps: New locally linear embedding techniques for high-dimensional data
- DL, Grimes
(Show Context)
Citation Context ...s Ξ (t) σ(T) ′ : (X, d (t) ) → (R |σ(T)′ | , dEuc.) x ↦→ ( λ t 2 ξλ(x) ) λ∈σ(T) ′ (4.5) which is in particular cases called eigenmap [33,34], and is a form of local multidimensional scaling (see also =-=[35,3,4]-=-). By the definition of d (t) , this map is an isometry when σ(T) ′ = σ(T), and an approximation to an isometry when σ(T) ′ � σ(T). If σ(T) ′ is the set of the first n top eigenvalues, and if ∫ X K(x,... |

100 |
Efficient algorithms for computing a strong rank-revealing QR factorization
- Gu, Eisenstat
- 1996
(Show Context)
Citation Context ...lting basis functions. On the other hand we can construct quite artifical examples in which any choice of λ < 1 leads to loss of precision. 5.4 Rank Revealing QR factorizations We refer the reader to =-=[63]-=-, and references therein, for details regarding the numerical and algorithmic aspects related to the rank-revealing factorizations discussed in this section. Definition 38 A partial QR factorization o... |

100 | Ideal de-noising in an orthonormal basis chosen from a library of bases
- Donoho, Johnstone
(Show Context)
Citation Context ...al and complex. Furthermore, the original image I itself can be viewed as a function on G, and hence it can be analyzed as a function on G, and denoised by using standard wavelet denoising algorithms =-=[69]-=- adapted to the context of diffusion wavelets, such as best basis algorithms [12]. This denoising naturally preserves edges: the “filtering” happens mainly on and along the edges and not across them, ... |

85 |
A T (b) theorem with remarks on analytic capacity and the Cauchy Integral
- Christ
- 1990
(Show Context)
Citation Context ...tex is defined as dx = ∑ wyx , y∼x where y ∼ x means there is an edge between y and x. In quite general settings, one can construct the following analogue of the dyadic cubes in the Euclidean setting =-=[26,27]-=-: 15Theorem 6 Let (X, ρ, µ) be a space of homogeneous type as above. There exists a collection of open subsets Q = { {Qj,k}k∈Kj and constants δX > 1, η > 0 and c1, c2 ∈ (0, ∞), depending only on AX, ... |

84 |
Algebraic multigrid (AMG) for automatic multigrid solution with application to geodetic computations
- Brandt, McCormick, et al.
- 1984
(Show Context)
Citation Context ...lized “bump functions”, or atoms, at that scale. Such ideas have many applications in numerical analysis, especially to problems motivated by physics (matrix compression [42–45], multigrid techniques =-=[46,47]-=-, etc...). We avoid the computation of the eigenfunctions, nevertheless the approximation spaces ˜ Vj that we build will be ǫ-approximations of the true Vj’s. 4.4 Orthogonalization and downsampling A ... |

83 | Measured descent: A new embedding method for finite metrics. Geometric And Functional Analysis
- Krauthgamer, Lee, et al.
- 2005
(Show Context)
Citation Context ...tions Ψ = {ψx}x∈Γ with a supporting set Γ. We pick a maximal net of points x0,0, . . ., x0,k in Γ which is 2αδ-separated (this is similar to the construction of lattices on spaces of homogeneous type =-=[11,61,26]-=-). In order to do this, we pick x0,0 ∈ Γ, then pick x0,1 to be a closest point in Γ which is at distance at least 2δ from x0,0, and so on: after x0,0, . . ., x0,l have been picked, let x0,l+1 ∈ Γ be a... |

81 | Calderón–Zygmund and multilinear operators, Cambridge - Coifman, Meyer, et al. - 1997 |

79 |
Using manifold structure for partially labeled classification
- Belkin, Niyogi
- 2003
(Show Context)
Citation Context ...s induced by a diffusion semigroup In all that follows, we will assume, mainly for simplicity, that {T t } is a compact semigroup with γ-strong decay, that acts η-locally on X. We refer the reader to =-=[3,33,18]-=- and references therein for some motivations and applications of the ideas presented in this section. Being positive definite, T t induces the diffusion metric d (t) √ ∑ (x, y) = = λ∈σ(T) λ t (ξλ(x) −... |

76 |
Diffusion maps and geometric harmonics
- Lafon
- 2004
(Show Context)
Citation Context ...t Riemannian manifolds can be approximated on a finite number of points which are realizations of a random variable taking values on the manifold according to a certain probability distribution as in =-=[65,66,4,3]-=-. The construction starts with assuming that we have a data set Γ which is obtained by drawing according to some unknown distribution p, and whose range is a smooth manifold Γ. Given a kernel that sat... |

66 | Projective Methods for Stiff Differential Equations: problems with gaps in their eigenvalue spectrum - Gear, Kevrekidis - 2004 |

64 |
Lipschitz functions on spaces of homogeneous type
- Macias, Segovia
- 1979
(Show Context)
Citation Context ... a topology equivalent to the one induced by d, so that the δ-balls in the ρ metric have measure approximately δ: it is enough to define ρ(x, y) as the measure of the smallest ball containing x and y =-=[22]-=-. One can also assume some Höldersmoothness for ρ, in the sense that |ρ(x, y) − ρ(x ′ , y)| ≤ LXρ(x, x ′ ) β [ρ(x, y) + ρ(x ′ , y)] 1−β for some LX > 0, β ∈ (0, 1), c > 0, see [7,8,22,23]. Definition ... |

63 |
A boundedness criterion for generalized Calderón–Zygmund operators
- David, Journé
- 1984
(Show Context)
Citation Context ... containing x and y [22]. One can also assume some Höldersmoothness for ρ, in the sense that |ρ(x, y) − ρ(x ′ , y)| ≤ LXρ(x, x ′ ) β [ρ(x, y) + ρ(x ′ , y)] 1−β for some LX > 0, β ∈ (0, 1), c > 0, see =-=[7,8,22,23]-=-. Definition 4 Let (X, ρ, µ) be a space of homogeneous type, and γ > 0. A subset {xi}i of X is a γ-lattice if ρ(xi, xj) > 1 2clatγ, for all i ̸= j, and if for every y ∈ X, there exists i(y) such that ... |

47 | Spectral Partitioning with Indefinite Kernels Using the Nyström Extension - Belongie, Fowlkes, et al. - 2002 |