- Email: [email protected]
“Ridge Functions” by Allan Pinkus
Available online at www.sciencedirect.com
ScienceDirect Journal of Approximation Theory (
)
– www.elsevier.com/locate/jat
Book review
“Ridge Functions” by Allan Pinkus Ridge Functions, Allan Pinkus. Cambridge Tracts in Mathematics, Vol. 205. Cambridge University Press (2015). 218 pp., Hardcover, ISBN: 978-1107124394 Reviewer: Pencho Petrushev Ridge functions have been around for quite some time under the name “plane waves” or with no name. The term “ridge functions” was coined by Logan and Shepp in their article of 1975. These are functions on Rn of the form F(x) = f (a · x), where f : R → R is a real-valued function and a · x stands for the inner product of a, x ∈ Rn , a ̸= 0. The generalized ridge functions take the form F(x) = f (Ax), where f : Rd → R and A is a d × n real matrix (1 ≤ d ≤ n − 1). Ridge functions are most intimately related to the Radon transform. However, they naturally appear in various other areas ranging from Harmonic Analysis and PDEs to Computerized Tomography and Neural Networks, and to Approximation Theory, Statistics, and Waring’s problem in Number Theory. Ridge functions and related topics have attracted a significant attention in the last two decades which created the need for a comprehensive treatise on the subject. The book under review fills this void in the literature. It is unique in treating mostly problems related to ridge functions that can only be found in research articles. The book focuses on ridge functions from an Approximation Theory point of view. Thus central are the problems of representation, approximation and interpolation by linear combinations of ridge functions. The author of the book is among the main developers of this theory; a substantial part of the book is based on his results. The book is very well written, accessible and self-contained; it is recommended to graduate students, mathematicians and all who are interested in ridge functions and, in particular, in approximation from linear combinations of ridge functions. Interesting open problems or questions are included in almost all sections. The book also contains a rather complete bibliography. Chapter 1 is an introduction where the author presents his motivation for writing this book and introduces his basic notation. Chapter 2 studies the relationship between the Sobolev smoothness of functions that can be represented as finite sums of ridge functions in Rn and the smoothness of their univariate generating functions (components). As is well known, in general, smooth functions can be represented as a linear combination of nonsmooth functions. However, if the components belong to the linear space of univariate functions that is closed under translation and the directions of the compounding ridge functions are pairwise linearly independent, then the components inherit http://dx.doi.org/10.1016/j.jat.2016.03.003 0021-9045/
2
Book review / Journal of Approximation Theory (
)
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the smoothness of the original function. The same problem is considered for generalized ridge functions. Chapter 3 deals with the problem of uniqueness of the representation of a finite linear combination of ridge functions. It is shown that essentially there is uniqueness of such a representation up to polynomials of a certain degree. This result is also extended to generalized ridge functions. given a function F of the form F(x) = rChapter j4 is devoted to the following problem: j and the functions f from the knowledge of F f (a · x) determine the directions a j j j=1 and r . A simpler problem is to determine the functions f j given F and the directions a j . Methods for solving these problems modulo polynomials of degree r − 2 are presented under natural smoothness assumptions. The same problems in the case of generalized ridge functions are also discussed. Chapter 5 is concerned with ridge functions that are algebraic polynomials. This chapter contains various results on the representation of homogeneous and general algebraic polynomials in n variables as linear combinations of ridge monomials of degree m, namely, (a · x)m , a ∈ Ω ⊂ Rn , as well as on the linear independence and interpolation by ridge monomials. Waring’s problem for polynomials is also considered. Generalized ridge functions that are polynomials are considered as well. The problem of the density of linear combinations of continuous ridge functions on Rn in the topology of uniform convergence on compact subsets of Rn is treated in Chapter 6. First, necessary and sufficient conditions are established in the case of fixed directions. The same problem for generalized ridge functions is also treated. In a natural next step the problem of density in the setting of generalized ridge functions with variable directions is resolved. Chapter 6 also resolves the problem of which continuous functions on Rn can be approximated by linear combinations of continuous ridge functions with given directions (no density is required). The density of the so called ridgelets is established in the last section of Chapter 6. Chapter 7 contains results on the closure in L p (K ), K ⊂ Rn , of spaces generated by ridge functions or generalized ridge functions. More explicitly, assuming that K ⊂ Rn is bounded, denote by L p (A; K ) the set of all functions in L p (K ) of the form f (Ax). The problem is, under what conditions on K and the d × n matrices A j , j = 1, . . . , r , the space r p j p j=1 L (A ; K ) is closed in L (K ). Various positive results are presented without proof and several counterexamples where closedness does not hold are presented. The interesting case of C(K ) is also considered. Chapter 8 treats the fundamental questions of existence and characterization of best approximations from finite linear combinations of ridge functions. Approximation from the subspace rj=1 L p (A j ; K ) in L p (K ), p ∈ (1, ∞), with K ⊂ Rn of finite measure is considered. If the above subspace is closed then an element of best approximation always exists and its uniqueness follows by the general result presented in this chapter. A characterization of the element of best approximation is established. This characterization in the interesting case p = 2 appears in a particularly nice form and is presented separately. Several particular examples of this theory are also presented. The last section of Chapter 8 deals with the case of C(K ), where results are known only when approximating from linear combinations of two ridge functions. Chapter 9 contains algorithms for finding best approximations from spaces of linear combinations of ridge functions. The main problem considered here is for approximation over some domains in Rn from the linear space of all functions of the form rj=1 f j (A j x), where the functions f j : Rd → R are free and A j , j = 1, . . . , r , are fixed d × n matrices. The major
Book review / Journal of Approximation Theory (
)
–
3
assumption is that the operator P j of best approximation from the space M(A j ) of functions of the form f (A j x) with f : Rd → R is computable. First, approximation algorithms are developed in a Hilbert space setting. Second, greedy-type algorithms are considered, which allows to deal with infinitely many directions. Further, approximation algorithms are considered in uniform convex and uniform smooth Banach spaces. All these algorithms are developed in general settings and applicable to approximation from ridge functions. The last section of Chapter 9 contains a discussion of the Diliberto–Straus algorithms for approximation from ridge functions with two directions in C(K ) with K a compact convex set in Rn . Chapter 10 deals with integral representations of functions on Rn with kernels that are ridge functions. The Fourier transform and its inverse provide a simple example of such a representation. Another integral representation considered in the book involves the kernel |(y − x) · a|k and the Laplace operator. An integral representation of functions on Rn that n/2 involves the ridge Gegenbauer polynomials Cm (a · x) is considered in more detail and an associated Parseval identity is derived. More attention is also paid to the so called continuous ridgelet transform and the associated Parseval identity. Chapter 11 is concerned with point interpolation by ridge functions. It deals with the problem of characterization of all points x1 , . . . , xk in Rn such that for given d × n matrices A1 , . . . , Ak d and r arbitraryj bj 1 , . . . , bk ∈ R there exist functions f j : R → R, j = 1, . . . , k such that j=1 f j (A x ) = b j , j = 1, . . . , k. Necessary and sufficient conditions for solution of this problem are given in the case of two directions. For more than two directions but only in R2 an exact geometric characterization is derived for a large set of points where interpolation is not always possible. Chapter 12 focuses on the problem of interpolation on straight lines by ridge functions. The question is: Given vectors ai ∈ Rn , i = 1, . . . , r and b j , c j ∈ Rn , b j ̸= 0, j = 1, . . . , m, and arbitrary functions g j (t) do there exist univariate functions f i such that ri=1 f i (ai ·(tb j +c j )) = g j (t), j = 1, . . . , m. It is first shown that interpolation by ridge functions on an arbitrary set X ⊂ Rn is possible if and only if it is possible on every finite point set {x1 , . . . , xk } ⊂ X . The ridge function interpolation on a single line is considered in the second section. The third section deals with ridge function interpolation on two lines. Exact conditions for this kind of interpolation are derived. Further, these conditions are reduced to more tangible geometric conditions in dimension n = 2. Finally, it is shown that it is never possible to interpolate on the union of three lines by ridge functions with two directions.
Pencho Petrushev University of South Carolina, United States E-mail address: [email protected].
Communicated by Andrei Mart´ınez-Finkelshtein
ScienceDirect Journal of Approximation Theory (
)
– www.elsevier.com/locate/jat
Book review
“Ridge Functions” by Allan Pinkus Ridge Functions, Allan Pinkus. Cambridge Tracts in Mathematics, Vol. 205. Cambridge University Press (2015). 218 pp., Hardcover, ISBN: 978-1107124394 Reviewer: Pencho Petrushev Ridge functions have been around for quite some time under the name “plane waves” or with no name. The term “ridge functions” was coined by Logan and Shepp in their article of 1975. These are functions on Rn of the form F(x) = f (a · x), where f : R → R is a real-valued function and a · x stands for the inner product of a, x ∈ Rn , a ̸= 0. The generalized ridge functions take the form F(x) = f (Ax), where f : Rd → R and A is a d × n real matrix (1 ≤ d ≤ n − 1). Ridge functions are most intimately related to the Radon transform. However, they naturally appear in various other areas ranging from Harmonic Analysis and PDEs to Computerized Tomography and Neural Networks, and to Approximation Theory, Statistics, and Waring’s problem in Number Theory. Ridge functions and related topics have attracted a significant attention in the last two decades which created the need for a comprehensive treatise on the subject. The book under review fills this void in the literature. It is unique in treating mostly problems related to ridge functions that can only be found in research articles. The book focuses on ridge functions from an Approximation Theory point of view. Thus central are the problems of representation, approximation and interpolation by linear combinations of ridge functions. The author of the book is among the main developers of this theory; a substantial part of the book is based on his results. The book is very well written, accessible and self-contained; it is recommended to graduate students, mathematicians and all who are interested in ridge functions and, in particular, in approximation from linear combinations of ridge functions. Interesting open problems or questions are included in almost all sections. The book also contains a rather complete bibliography. Chapter 1 is an introduction where the author presents his motivation for writing this book and introduces his basic notation. Chapter 2 studies the relationship between the Sobolev smoothness of functions that can be represented as finite sums of ridge functions in Rn and the smoothness of their univariate generating functions (components). As is well known, in general, smooth functions can be represented as a linear combination of nonsmooth functions. However, if the components belong to the linear space of univariate functions that is closed under translation and the directions of the compounding ridge functions are pairwise linearly independent, then the components inherit http://dx.doi.org/10.1016/j.jat.2016.03.003 0021-9045/
2
Book review / Journal of Approximation Theory (
)
–
the smoothness of the original function. The same problem is considered for generalized ridge functions. Chapter 3 deals with the problem of uniqueness of the representation of a finite linear combination of ridge functions. It is shown that essentially there is uniqueness of such a representation up to polynomials of a certain degree. This result is also extended to generalized ridge functions. given a function F of the form F(x) = rChapter j4 is devoted to the following problem: j and the functions f from the knowledge of F f (a · x) determine the directions a j j j=1 and r . A simpler problem is to determine the functions f j given F and the directions a j . Methods for solving these problems modulo polynomials of degree r − 2 are presented under natural smoothness assumptions. The same problems in the case of generalized ridge functions are also discussed. Chapter 5 is concerned with ridge functions that are algebraic polynomials. This chapter contains various results on the representation of homogeneous and general algebraic polynomials in n variables as linear combinations of ridge monomials of degree m, namely, (a · x)m , a ∈ Ω ⊂ Rn , as well as on the linear independence and interpolation by ridge monomials. Waring’s problem for polynomials is also considered. Generalized ridge functions that are polynomials are considered as well. The problem of the density of linear combinations of continuous ridge functions on Rn in the topology of uniform convergence on compact subsets of Rn is treated in Chapter 6. First, necessary and sufficient conditions are established in the case of fixed directions. The same problem for generalized ridge functions is also treated. In a natural next step the problem of density in the setting of generalized ridge functions with variable directions is resolved. Chapter 6 also resolves the problem of which continuous functions on Rn can be approximated by linear combinations of continuous ridge functions with given directions (no density is required). The density of the so called ridgelets is established in the last section of Chapter 6. Chapter 7 contains results on the closure in L p (K ), K ⊂ Rn , of spaces generated by ridge functions or generalized ridge functions. More explicitly, assuming that K ⊂ Rn is bounded, denote by L p (A; K ) the set of all functions in L p (K ) of the form f (Ax). The problem is, under what conditions on K and the d × n matrices A j , j = 1, . . . , r , the space r p j p j=1 L (A ; K ) is closed in L (K ). Various positive results are presented without proof and several counterexamples where closedness does not hold are presented. The interesting case of C(K ) is also considered. Chapter 8 treats the fundamental questions of existence and characterization of best approximations from finite linear combinations of ridge functions. Approximation from the subspace rj=1 L p (A j ; K ) in L p (K ), p ∈ (1, ∞), with K ⊂ Rn of finite measure is considered. If the above subspace is closed then an element of best approximation always exists and its uniqueness follows by the general result presented in this chapter. A characterization of the element of best approximation is established. This characterization in the interesting case p = 2 appears in a particularly nice form and is presented separately. Several particular examples of this theory are also presented. The last section of Chapter 8 deals with the case of C(K ), where results are known only when approximating from linear combinations of two ridge functions. Chapter 9 contains algorithms for finding best approximations from spaces of linear combinations of ridge functions. The main problem considered here is for approximation over some domains in Rn from the linear space of all functions of the form rj=1 f j (A j x), where the functions f j : Rd → R are free and A j , j = 1, . . . , r , are fixed d × n matrices. The major
Book review / Journal of Approximation Theory (
)
–
3
assumption is that the operator P j of best approximation from the space M(A j ) of functions of the form f (A j x) with f : Rd → R is computable. First, approximation algorithms are developed in a Hilbert space setting. Second, greedy-type algorithms are considered, which allows to deal with infinitely many directions. Further, approximation algorithms are considered in uniform convex and uniform smooth Banach spaces. All these algorithms are developed in general settings and applicable to approximation from ridge functions. The last section of Chapter 9 contains a discussion of the Diliberto–Straus algorithms for approximation from ridge functions with two directions in C(K ) with K a compact convex set in Rn . Chapter 10 deals with integral representations of functions on Rn with kernels that are ridge functions. The Fourier transform and its inverse provide a simple example of such a representation. Another integral representation considered in the book involves the kernel |(y − x) · a|k and the Laplace operator. An integral representation of functions on Rn that n/2 involves the ridge Gegenbauer polynomials Cm (a · x) is considered in more detail and an associated Parseval identity is derived. More attention is also paid to the so called continuous ridgelet transform and the associated Parseval identity. Chapter 11 is concerned with point interpolation by ridge functions. It deals with the problem of characterization of all points x1 , . . . , xk in Rn such that for given d × n matrices A1 , . . . , Ak d and r arbitraryj bj 1 , . . . , bk ∈ R there exist functions f j : R → R, j = 1, . . . , k such that j=1 f j (A x ) = b j , j = 1, . . . , k. Necessary and sufficient conditions for solution of this problem are given in the case of two directions. For more than two directions but only in R2 an exact geometric characterization is derived for a large set of points where interpolation is not always possible. Chapter 12 focuses on the problem of interpolation on straight lines by ridge functions. The question is: Given vectors ai ∈ Rn , i = 1, . . . , r and b j , c j ∈ Rn , b j ̸= 0, j = 1, . . . , m, and arbitrary functions g j (t) do there exist univariate functions f i such that ri=1 f i (ai ·(tb j +c j )) = g j (t), j = 1, . . . , m. It is first shown that interpolation by ridge functions on an arbitrary set X ⊂ Rn is possible if and only if it is possible on every finite point set {x1 , . . . , xk } ⊂ X . The ridge function interpolation on a single line is considered in the second section. The third section deals with ridge function interpolation on two lines. Exact conditions for this kind of interpolation are derived. Further, these conditions are reduced to more tangible geometric conditions in dimension n = 2. Finally, it is shown that it is never possible to interpolate on the union of three lines by ridge functions with two directions.
Pencho Petrushev University of South Carolina, United States E-mail address: [email protected].
Communicated by Andrei Mart´ınez-Finkelshtein