Shannon sampling and function reconstruction from point values

By Steve Smale, Ding-Xuan Zhou. 2004

266 citations recorded by the source corpus.

I first met Rene at the well-known 1956 meeting on topology in Mexico City. He then came to the University of Chicago, where I was starting my job as instructor for the fall of 1956. He, Suzanne, Clara and I became good friends and saw much of each other for many decades, especially at IHES in Paris. Thom’s encouragement and support were important for me, especially in my first years after my Ph.D. I studied his work in cobordism, singularities of maps, and transversality, gaining many insights. I also enjoyed listening to his provocations, for example his disparaging remarks on complex analysis, 19th century mathematics, and Bourbaki. There was also a stormy side in our relationship. Neither of us could hide the pain that our public conflicts over “catastrophe theory” caused. Rene Thom was a great mathematician, leaving his impact on a wide part of mathematics. I will always treasure my memories of him. Steve Smale

Subfields: Applied Mathematics, Computer Vision and Pattern Recognition, Geophysics

View Paper: https://www.semanticscholar.org/paper/96e7b9a4fbad3ed0afd9871ba0ba542c847dc6f6

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Citations in the corpus

Sampling in paley-wiener spaces on combinatorial graphs (2008, background): “It seems that our results can find different applications in signal analysis, imaging, learning theory and discrete tomography [18], [36], [37].”
On Early Stopping in Gradient Descent Learning (2007, methodology): “Such sampling operators are used in a generalization of Shannon Sampling Theorem, [Smale and Zhou 2004].”
Application of integral operator for regularized least-square regression (2009, methodology): “∗Corresponding author Keywords: regularized least square regression, integral operator, reproducing kernel Hilbert space, consistency.”
Spectral Algorithms for Supervised Learning (2008, background): “A mathematical connection between sampling theory and learning theory has been recently proposed by Smale and Zhou (2004, 2005) whereas De Vito, Rosasco, Caponnetto, et al. (2005) and De Vito et al. (2006) gave an inverse problem perspective on learning.”
DISCRETIZATION ERROR ANALYSIS FOR TIKHONOV REGULARIZATION (2006, background): “R d be the sampling operator [28, 29] (Sx,nf)i = f(xi) = 〈f, Kxi〉H i = 1, .”
Geometry on Probability Spaces (2009, methodology): “Note that 1 m S̃ x S̃x is given from data x by a linear algorithm [16].”
Universality of Deep Convolutional Neural Networks (2018, background): “Convolutions are closely related to translation-invariance in speeches and images [3,6,27], and also in some learning algorithms [25,8].”
Learning from Examples as an Inverse Problem (2005, unknown): “For the specific case of regularized least-squares algorithm a functional analytical approach to derive consistency results for regularized least squares was proposed in Cucker and Smale (2002a) and eventually refined in De Vito et al. (2004) and Smale and Zhou (2004b).”
Learning Theory Estimates via Integral Operators and Their Approximations (2007, background): “10) estimates the regularization error [10].”
Frames, Riesz bases, and sampling expansions in Banach spaces via semi-inner products☆ (2011, background): “For studies in RKHS along the latter approach, see, for example, [32,34,39,40].”
Reproducing Kernel Hilbert Spaces Associated with Analytic Translation-Invariant Mercer Kernels (2008, methodology): “Let x be a subset of X. Consider the subspace HK,x = span{Ky : y ∈ x} of HK . Restricting learning processes from HK onto the subspace HK,x may reduce the complexity of learning algorithms [ 15 ].”
Statistical topology via Morse theory, persistence and nonparametric estimation (2009, background): “There is also a growing interest in function estimation over manifolds in the learning theory literature [7, 31, 2]; see also the references cited therein.”
Online Learning as Stochastic Approximation of Regularization Paths: Optimality and Almost-Sure Convergence (2011, background): “(i) generalized smooth spline functions in Sobelev spaces [Wahba 1990], (ii) real analytic functions with bounded bandwidth [Daubechies 1992] and their generalizations [Smale and Zhou 2004], (iii) gaussian processes [Loéve 1948; Parzen 1961].”
Shannon sampling II: Connections to learning theory (2005, methodology): “We continue our study [12] of Shannon sampling and function reconstruction.”