## Diffusion Wavelets (2004)

Citations: | 76 - 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

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Citation Context ...omputer science (e.g. network design, distributed processing). We believe our construction will have applications in all these settings. As a starting point, see for example [13] for an overview, and =-=[28,29]-=- for particular applications. (g) One parameter groups (continuous or discrete) of dilations in Rn , or other motion groups (e.g. the Galilei group or Heisenberg-type groups), act on square-integrable... |

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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... |

1795 |
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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... |

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Citation Context ...function bases. This could follow the ideas of “dual” operators in the reproducing formula on space of homogeneous type [52,6], and also, exactly as in the classical biorthogonal wavelet construction =-=[53]-=-, we would have two ladders of approximation subspaces, with wavelet subspaces giving the oblique projection onto their corresponding duals. Work on this construction is presented in [54] and more is ... |

1025 |
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Citation Context ..., and I − T could be a Laplacian on X. The Laplacian is associated to a natural diffusion process and random walk P, and T is a self-adjoint operator conjugate to the Markov matrix P. See for example =-=[13]-=- and references therein. Also, T could be the heat kernel on a weighted graph. (ii) X could represent the discretization of a domain or manifold and T = e −ǫ(∆+V ) , where ∆ is an elliptic partial dif... |

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Citation Context ...coordinates 3,4,5, and on the right coordinates 4,5,6. We can compute the eigenvalues 0 = λ0 ≤ λ1 ≤ · · · ≤ λn ≤ . . . of L and the corresponding eigenvectors ξ0, ξ1, . . .,ξn, . . .. As described in =-=[18,19,3,20]-=- and in Section 4.2, these eigenfunctions can be used to define, for each n ≥ 0 and t ≥ 0, the nonlinear embedding (Figure 5) Ξ (t) n : X → Rn defined by ( x ↦→ λ t 2 i ξi(x) ) i=1,...,n . 11Multisca... |

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Citation Context ...est neighbor information needed. This is a highly non-trivial encoding, especially in high dimensions: the literature on fast approximate nearest neighbors and range searches is vast, see for example =-=[60]-=- and references therein as a starting point. In the following subsection we will present an algorithm that does not use the geometry of the supports and does not need any knowledge of nearest neighbor... |

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Citation Context ...the data set. As an example, for a body of text documents our construction leads to a directory structure at different 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 di... |

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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... |

408 |
The fast wavelet transform and numerical algorithms
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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 ... |

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

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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... |

297 | Using the Nystrom method to speed up kernel machines - Williams, Seeger - 2001 |

283 |
Prolate spheroidal wave functions, Fourier analysis, and uncertainty - V: The discrete case
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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 { ... |

277 |
Heat Kernels and Spectral Theory
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(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 ... |

253 |
Ondelettes et opérateurs
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(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... |

232 | Spherical wavelets: Efficiently representing functions on the sphere - SCHRÖDER, SWELDENS - 1995 |

226 | Locality preserving projections
- He, Niyogi
(Show Context)
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). ... |

217 | Multiresolution signal processing for meshes
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(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... |

206 | Interpolating subdivision for meshes with arbitrary topology
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(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... |

179 | A first course on wavelets - Hernández, Weiss - 1996 |

158 | Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps - Coifman, Lafon, et al. - 2005 |

147 |
Analyse harmonique non commutative sur certains expaces homogenes
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(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... |

143 |
Matrix Computations. The Johns Hopkins
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(Show Context)
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... |

138 |
A sparse matrix arithmetic based on H-matrices. part i: Introduction to Hmatrices
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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... |

126 |
Lectures on Analysis on Metric Spaces
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(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... |

122 | Fast Multiscale Image Segmentation - Sharon, Brandt, et al. - 2000 |

117 | Cardinal interpolation by multivariate splines - Chui, Jetter, et al. - 1987 |

116 | A kernel view of the dimensionality reduction of manifolds
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(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... |

110 |
Balls and metrics defined by vector fields I: Basic properties
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(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... |

102 | Out-of-sample extensions for LLE, isomap, MDS, eigenmaps, and spectral clustering
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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... |

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

101 |
Topics in harmonic analysis related to the Littlewood-Paley theory
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(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... |

100 | Hessian eigenmaps: new locally linear embedding techniques for highdimensional data
- Donoho, Grimes
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(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 | Ideal de-noising in an orthonormal basis chosen from a libary of bases
- Donoho, Johnstone
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(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, ... |

96 |
Efficient algorithms for computing a strong rank-revealing QR factorization
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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... |

83 | Measured descent: A new embedding method for finite metrics
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(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... |

82 |
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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 ... |

78 |
Using Manifold Structure for Partially Labeled Classication
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(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) −... |

77 |
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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... |

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68 |
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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, ... |

59 |
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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 ... |

58 | Projective methods for stiff differential equations: problems with gaps in their eigenvalue spectrum - Gear, Kevrekidis |

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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 ... |

47 | Spectral partitioning with indefinite kernels using the Nyström extension - Belongie, Fowlkes, et al. - 2002 |