## Eignets for function approximation on manifolds (909)

### BibTeX

@MISC{Mhaskar909eignetsfor,

author = {H. N. Mhaskar},

title = {Eignets for function approximation on manifolds},

year = {909}

}

### OpenURL

### Abstract

Let X be a compact, smooth, connected, Riemannian manifold without boundary, G: X × X → R be P a kernel. Analogous to a radial basis function network, an eignet is an expression of the form M j=1 ajG(◦, yj), where aj ∈ R, yj ∈ X, 1 ≤ j ≤ M. We describe a deterministic, universal algorithm for constructing an eignet for approximating functions in L p (µ; X) for a general class of measures µ and kernels G. Our algorithm yields linear operators. Using the minimal separation amongst the centers yj as the cost of approximation, we give modulus of smoothness estimates for the degree of approximation by our eignets, and show by means of a converse theorem that these are the best possible for every individual function. We also give estimates on the coefficients aj in terms of the norm of the eignet. Finally, we demonstrate that if any sequence of eignets satisfies the optimal estimates for the degree of approximation of a smooth function, measured in terms of the minimal separation, then the derivatives of the eignets also approximate the corresponding derivatives of the target function in an optimal manner.

### Citations

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Citation Context ...m norm on [−π, π], and Vm denotes the class of all trigonometric polynomials of order at most m; i.e., expressions of the form ∑ |j|≤m aje ij◦ . The well known equivalence theorem in this case states =-=[8]-=- that if 0 < α < 1, and r ≥ 0 is an integer, then dist (X; f, Vm) = O(m −r−α ) if and only if f has r continuous derivatives and |f (r) (x)−f (r) (y)| = O(|x−y| α ), x, y ∈ R. To cover the case when α... |

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Citation Context ...a low dimensional structure, for example, it lies on a low dimensional manifold in the high dimensional ambient space. Applications of these techniques include document analysis [7], face recognition =-=[18]-=-, semi–supervised learning [2, 1], image processing [12], and cataloguing of galaxies [13]. The special issue [6] of Applied and Computational Harmonic Analysis contains several papers that serve as a... |

156 | Semi-supervised learning on Riemannian manifolds
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Citation Context ...r example, it lies on a low dimensional manifold in the high dimensional ambient space. Applications of these techniques include document analysis [7], face recognition [18], semi–supervised learning =-=[2, 1]-=-, image processing [12], and cataloguing of galaxies [13]. The special issue [6] of Applied and Computational Harmonic Analysis contains several papers that serve as a good introduction to this subjec... |

101 | Towards a theoretical foundation for Laplacian-based manifold methods
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Citation Context ...lace of the heat kernel an approximation by means of a suitable radial basis function, typically a Gaussian. The error in this approximation is investigated in detail by several authors, for example, =-=[23, 31, 3, 4]-=-. In a different idea, Saito [30] has advocated the use of other kernels which commute with the heat kernel, and hence, share the invariant subspaces with it, but for which explict formulas are known.... |

80 |
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Citation Context ... the negative square root of the Laplace–Beltrami operator, then Minakshisundaram and Pleijel have proved an asymptotic expression for the heat kernel in [29], which implies both (3.14) and (2.4). In =-=[19]-=-, Hörmander has obtained uniform asymptotics for the sums ∑ ℓj≤L φ2j (x) for a very general class of elliptic differential operators on a manifold. It will be shown in Lemma 5.2 that these lead to (3.... |

73 |
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Citation Context ...lace of the heat kernel an approximation by means of a suitable radial basis function, typically a Gaussian. The error in this approximation is investigated in detail by several authors, for example, =-=[23, 31, 3, 4]-=-. In a different idea, Saito [30] has advocated the use of other kernels which commute with the heat kernel, and hence, share the invariant subspaces with it, but for which explict formulas are known.... |

44 |
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Citation Context ... eigenfunctions (respectively, eigenvalues) of the negative square root of the Laplace–Beltrami operator, then Minakshisundaram and Pleijel have proved an asymptotic expression for the heat kernel in =-=[29]-=-, which implies both (3.14) and (2.4). In [19], Hörmander has obtained uniform asymptotics for the sums ∑ ℓj≤L φ2j (x) for a very general class of elliptic differential operators on a manifold. It wil... |

41 |
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(Show Context)
Citation Context ...lace of the heat kernel an approximation by means of a suitable radial basis function, typically a Gaussian. The error in this approximation is investigated in detail by several authors, for example, =-=[23, 31, 3, 4]-=-. In a different idea, Saito [30] has advocated the use of other kernels which commute with the heat kernel, and hence, share the invariant subspaces with it, but for which explict formulas are known.... |

36 |
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Citation Context ...ow dimensional manifold in the high dimensional ambient space. Applications of these techniques include document analysis [7], face recognition [18], semi–supervised learning [2, 1], image processing =-=[12]-=-, and cataloguing of galaxies [13]. The special issue [6] of Applied and Computational Harmonic Analysis contains several papers that serve as a good introduction to this subject. An essential ingredi... |

30 | Introduction to the theory of weighted polynomial approximation - Mhaskar - 1996 |

18 |
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(Show Context)
Citation Context ...ions (2.4) and (2.5) can be deduced from the bounds on the spectral functions ∑ ℓj≤L φ2 ∑ j (x), ℓj≤L (∂φj) 2 (x) proved by Bin Xu [32] (cf. [14]), and the finite speed of wave propagation. Kordyukov =-=[22]-=- has proved similar estimates in the case when X has bounded geometry, and φk’s are eigenfunctions of a general, second order, strictly elliptic partial differential operator. Other examples, where µ ... |

15 | Multiscale geometric analysis for 3d catalogs
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Citation Context ...h dimensional ambient space. Applications of these techniques include document analysis [7], face recognition [18], semi–supervised learning [2, 1], image processing [12], and cataloguing of galaxies =-=[13]-=-. The special issue [6] of Applied and Computational Harmonic Analysis contains several papers that serve as a good introduction to this subject. An essential ingredient in these techniques is the not... |

15 |
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Citation Context ... one may obtain wx’s so as to minimize ( ∑ ∑ ∫ ) 2 wxφk(x) − φkdµ . X ℓk≤L x∈C Efficient numerical algorithms for computing the weights in the context of the unit sphere can be found, for example, in =-=[24, 21, 15]-=-. Some of these ideas can be adopted in this context, but our main focus in this paper is of a theoretical nature, and we will not comment further on this issue in this paper. In view of (2.7), (2.1),... |

13 | Diffusion polynomial frames on metric measure spaces
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Citation Context ...al one for smooth functions, since the Ktφj ̸= φj except when ℓj = 0. In this paper, for L > 0, an element of ΠL := span {φj : ℓj ≤ L} will be called a diffusion polynomial of degree at most L, as in =-=[25]-=-. In [28, 25], we have developed a different multiscale analysis based on Π 2 j as the scaling spaces. We have obtained a Littlewood–Paley expansion, valid for functions in all L p spaces including p ... |

11 | Efficient reconstruction of functions on the sphere from scattered data
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(Show Context)
Citation Context ... one may obtain wx’s so as to minimize ( ∑ ∑ ∫ ) 2 wxφk(x) − φkdµ . X ℓk≤L x∈C Efficient numerical algorithms for computing the weights in the context of the unit sphere can be found, for example, in =-=[24, 21, 15]-=-. Some of these ideas can be adopted in this context, but our main focus in this paper is of a theoretical nature, and we will not comment further on this issue in this paper. In view of (2.7), (2.1),... |

10 |
Regularization and regression on large graphs
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- 2004
(Show Context)
Citation Context ...r example, it lies on a low dimensional manifold in the high dimensional ambient space. Applications of these techniques include document analysis [7], face recognition [18], semi–supervised learning =-=[2, 1]-=-, image processing [12], and cataloguing of galaxies [13]. The special issue [6] of Applied and Computational Harmonic Analysis contains several papers that serve as a good introduction to this subjec... |

9 |
Heat kernels and function theory on metric measure spaces, in: “Heat kernels and analysis on manifolds, graphs, and metric spaces
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(Show Context)
Citation Context ...eometry, and φk’s are eigenfunctions of a general, second order, strictly elliptic partial differential operator. Other examples, where µ is not the Riemannian measure on X are given by Grigor´yan in =-=[17]-=-. The bounds on the heat kernel are closely connected with the measures of the balls B(x, r). For example, it is proved in [17] that the conditions (2.3), (2.1), and (2.4) imply that µ(B(x, r)) ≥ cr α... |

8 |
Polynomial frames: a fast tour, in “Approximation Theory
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- 2004
(Show Context)
Citation Context ...r smooth functions, since the Ktφj ̸= φj except when ℓj = 0. In this paper, for L > 0, an element of ΠL := span {φj : ℓj ≤ L} will be called a diffusion polynomial of degree at most L, as in [25]. In =-=[28, 25]-=-, we have developed a different multiscale analysis based on Π 2 j as the scaling spaces. We have obtained a Littlewood–Paley expansion, valid for functions in all L p spaces including p = 1, ∞. This ... |

7 | Universal local parametrizations via heat kernels and eigenfunctions of the laplacian. Submitted
- Jones, Maggioni, et al.
- 2008
(Show Context)
Citation Context ...ng fast multipole techniques. The diffusion wavelets and wavelet packets can be obtained by applying Gram Schmidt procedure to the kernels K2−j. On a more theoretical side, Jones, Maggioni, and Schul =-=[20]-=- have recently proved that the heat kernel can be used to construct a ∗ Department of Mathematics, California State University, Los Angeles, California, 90032, USA, email: hmhaska@calstatela.edu. The ... |

7 |
When is approximation by Gaussian networks necessarily a linear process
- Mhaskar
(Show Context)
Citation Context ...omplexity, and what smoothness classes are characterized by a given rate of convergence of dist (X; f, Vm) to 0 as m → ∞. In the context of approximation by Gaussian networks, we have demonstrated in =-=[27, 26]-=- that a satisfactory theory can be developed by using the minimal separation amongst the centers as the measurement of model complexity, with the smoothness classes defined in terms of certain weighte... |

6 | Localized linear polynomial operators and quadrature formulas on the sphere
- Gia, Mhaskar
(Show Context)
Citation Context ... P(y)dµ(y) = ∑ wxP(x), P ∈ ΠL. (2.14) X x∈C A simple way to find the weights wx is to solve the least square problem of minimizing ∑ w2 x with the constraints ∑ x∈C wxφk(x) = ∫ X φkdµ, k = 0, · · ·,L =-=[24]-=-. Alternately, one may obtain wx’s so as to minimize ( ∑ ∑ ∫ ) 2 wxφk(x) − φkdµ . X ℓk≤L x∈C Efficient numerical algorithms for computing the weights in the context of the unit sphere can be found, fo... |

5 | Data analysis and representation on a general domain using eigenfunctions of laplacian
- Saito
- 2008
(Show Context)
Citation Context ... means of a suitable radial basis function, typically a Gaussian. The error in this approximation is investigated in detail by several authors, for example, [23, 31, 3, 4]. In a different idea, Saito =-=[30]-=- has advocated the use of other kernels which commute with the heat kernel, and hence, share the invariant subspaces with it, but for which explict formulas are known. Several applications, especially... |

5 |
Bin: Derivatives of spectral function and Sobolev norms of eigenfunctions on a closed Riemannian manifold
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(Show Context)
Citation Context ...alues) of the square root of the negative Laplacian on X, the assumptions (2.4) and (2.5) can be deduced from the bounds on the spectral functions ∑ ℓj≤L φ2 ∑ j (x), ℓj≤L (∂φj) 2 (x) proved by Bin Xu =-=[32]-=- (cf. [14]), and the finite speed of wave propagation. Kordyukov [22] has proved similar estimates in the case when X has bounded geometry, and φk’s are eigenfunctions of a general, second order, stri... |

3 |
Special Issue: Diffusion maps and wavelets
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(Show Context)
Citation Context ...ace. Applications of these techniques include document analysis [7], face recognition [18], semi–supervised learning [2, 1], image processing [12], and cataloguing of galaxies [13]. The special issue =-=[6]-=- of Applied and Computational Harmonic Analysis contains several papers that serve as a good introduction to this subject. An essential ingredient in these techniques is the notion of a heat kernel Kt... |

3 |
Local Paley Wiener theorems for analytic functions on the unit sphere, Inverse Problems
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(Show Context)
Citation Context ...re has several potential applications: signal processing, Paley Wiener theorems in inverse problems, computer vision, imaging, geo-remote sensing, among others, and that further hints can be found in =-=[11, 9, 10, 33, 34]-=-. The paper is organized as follows. In Section 2, we will describe the general set up, including the conditions on the manifold, the system {φj}, the kernel G, etc., including some basic facts. The m... |

3 |
Clustering of cancer tissues using diffusion maps and fuzzy ART with gene expression data,” in preparation
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(Show Context)
Citation Context ...re has several potential applications: signal processing, Paley Wiener theorems in inverse problems, computer vision, imaging, geo-remote sensing, among others, and that further hints can be found in =-=[11, 9, 10, 33, 34]-=-. The paper is organized as follows. In Section 2, we will describe the general set up, including the conditions on the manifold, the system {φj}, the kernel G, etc., including some basic facts. The m... |

2 | On bounds for diffusion, discrepancy and fill distance metrics. Principal manifolds for data visualization and dimension reduction, 261–270, Lect
- Damelin
- 2008
(Show Context)
Citation Context ...re has several potential applications: signal processing, Paley Wiener theorems in inverse problems, computer vision, imaging, geo-remote sensing, among others, and that further hints can be found in =-=[11, 9, 10, 33, 34]-=-. The paper is organized as follows. In Section 2, we will describe the general set up, including the conditions on the manifold, the system {φj}, the kernel G, etc., including some basic facts. The m... |

2 |
A Walk through Energy
- Damelin
(Show Context)
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2 |
A Bernstein inequality for diffusion polynomials corresponding to a generalized heat kernel, Manuscript
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(Show Context)
Citation Context ...te system, |∂yKt(x, y)| ≤ c1t −α/2−1 exp(−cρ(x, y) 2 /t), t ∈ (0, 1], x, y ∈ X. (2.5) We note that our assumptions imply that Kt(x, y) is well defined for all x, y ∈ X and t ∈ (0, 1]. It is proved in =-=[14]-=- that (2.4) implies that ∑ φ 2 j (x) ≤ cLα , L > 0. (2.6) ℓj≤L In the case when φk’s (respectively, ℓk’s) are the eigenfunctions (respectively, eigenvalues) of the square root of the negative Laplacia... |

2 |
Wunsch II, Applications of diffusion maps in gene expression data-based cancer diagnosis analysis
- Xu, Damelin, et al.
- 2007
(Show Context)
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