Ridge FunctionsRidge functions are a rich class of simple multivariate functions which have found applications in a variety of areas. These include partial differential equations (where they are sometimes termed 'plane waves'), computerised tomography, projection pursuit in the analysis of large multivariate data sets, the MLP model in neural networks, Waring's problem over linear forms, and approximation theory. Ridge Functions is the first book devoted to studying them as entities in and of themselves. The author describes their central properties and provides a solid theoretical foundation for researchers working in areas such as approximation or data science. He also includes an extensive bibliography and discusses some of the unresolved questions that may set the course for future research in the field. |
Contents
Smoothness | 12 |
Uniqueness | 19 |
Identifying Functions and Directions | 28 |
Polynomial Ridge Functions | 36 |
Density and Representation | 60 |
Closure | 77 |
Existence and Characterization of Best Approximations | 90 |
Approximation Algorithms | 105 |
Integral Representations | 141 |
Interpolation at Points | 152 |
Interpolation on Lines | 168 |
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Common terms and phrases
a e Q algebraic polynomials approximation to G arbitrary Assume assumption Banach space best approximation operator C(Rn Cand`es Cauchy Functional Equation Chapter closed path closure combinations of ridge compact subsets consider convergence on compact convex Corollary defined denote dense in C(R distinct equations example exists f(Ax finite number follows function G Gegenbauer polynomials given Hilbert space holds homogeneous polynomial hyperplanes implies integral interpolation Ismailov Lebesgue measure Lemma linear combinations linear functional linear subspace linearly independent directions LP(A Math matrix multivariate NI-property with respect non-trivial non-zero obtain orthogonal p e H pairwise linearly independent Pinkus polynomial of degree problem proof of Theorem Proposition prove q e H rate of convergence result ridge functions ridge monomials ridgelets Section set of points smooth span straight lines Theorem 9.5 topology of uniform uniform convergence uniform norm unique variables zero