Ridge Functions Research Articles

Representing a function of two variables as a superposition over angles of ridge functions with a given shape via linear programming

A ridge function with shape function g in the horizontal direction is a function of the form g ( x ) h ( y , 0 ) . Along each horizontal line it has the shape g ( x ) , multiplied by a function h ( y , 0 ) which depends on the y-value of the horizontal line. Similarly a ridge function with shape function g in the vertical direction has the form g ( y ) h ( x , π / 2 ) . For a given shape function g it may or may not be possible to represent an arbitrary function f ( x , y ) as a superposition over all angles of a ridge function with shape g in each direction, where h = h f = h f , g depends on the functions f and g and also on the direction, θ : h = h f , g ( · , θ ) . We show that if g is Gaussian centered at zero then this is always possible and we give the function h f , g for a given f ( x , y ) . For highpass or for odd shapes g, we show it is impossible to represent an arbitrary f ( x , y ) , i.e. in general there is no h f , g . Note that our problem is similar to tomography, where the problem is to invert the Radon transform, except that the use of the word inversion is here somewhat “inverted”: in tomography f ( x , y ) is unknown and we find it by inverting the projections of f; here, f ( x , y ) is known, g ( z ) is known, and h f ( · , θ ) = h f , g ( · , θ ) is the unknown.

Identification of Preisach-Type Hysteresis Operators

Identification of a Preisach-type operator using suitable fixed ridge functions is shown to be a well-posed problem, and its discretization is proved to converge and is implemented by an adaptive manner.

Approximation by polynomials and ridge functions of classes of s-monotone radial functions

We obtain estimates on the order of best approximation by polynomials and ridge functions in the spaces L q of classes of s-monotone radial functions which belong to the space L p , 1 ⩽ q ⩽ p ≤ ∞ .

Coupling volume-of-fluid based interface reconstructions with the extended finite element method

We examine the coupling of the patterned-interface-reconstruction (PIR) algorithm with the extended finite element method (X-FEM) for general multi-material problems over structured and unstructured meshes. The coupled method offers the advantages of allowing for local, element-based reconstructions of the interface, and facilitates the imposition of discrete conservation laws. Of particular note is the use of an interface representation that is volume-of-fluid based, giving rise to a segmented interface representation that is not continuous across element boundaries. In conjunction with such a representation, we employ enrichment with the ridge function for treating material interfaces and an analog to Heaviside enrichment for treating free surfaces. We examine a series of benchmark problems that quantify the convergence aspects of the coupled method and examine the sensitivity to noise in the interface reconstruction. The fidelity of a remapping strategy is also examined for a moving interface problem.

An Infinite-Dimensional Linear Programming Algorithm for Deterministic Semi-Markov Decision Processes on Borel Spaces

We devise an algorithm for solving the infinite-dimensional linear programs that arise from general deterministic semi-Markov decision processes on Borel spaces. The algorithm constructs a sequence of approximate primal-dual solutions that converge to an optimal one. The innovative idea is to approximate the dual solution with continuous piecewise linear ridge functions that naturally represent functions defined on a high-dimensional domain as linear combinations of functions defined on only a single dimension. This approximation gives rise to a primal/dual pair of semi-infinite programs, for which we show strong duality. In addition, we prove various properties of the underlying ridge functions.

Fields of non-linear regression models for atmospheric correction of satellite ocean-color imagery

Remote sensing of ocean color from space, a problem that consists of retrieving spectral marine reflectance from spectral top-of-atmosphere reflectance, is considered as a collection of similar inverse problems continuously indexed by the angular variables influencing the observation process. A general solution is proposed in the form of a field of non-linear regression models over the set T of permitted values for the angular variables, i.e., as a map from T to some function space. Each value of the field is a regression model that performs a direct mapping from the top-of-atmosphere reflectance to the marine reflectance. Since the spectral components of the field take values in the same variable vector space, the retrievals in individual spectral bands are not independent, i.e., the solution is not just a juxtaposition of independent models for each spectral band. A scheme based on ridge functions is developed to approximate this solution to an arbitrary accuracy, and is applied to the retrieval of marine reflectance in Case 1 waters, for which optical properties are only governed by biogenic content. The statistical models are evaluated on synthetic data as well as actual data originating from the SeaWiFS instrument, taking into account noise in the data. Theoretical performance is good in terms of accuracy, robustness, and generalization capabilities, suggesting that the function field methodology might improve atmospheric correction in the presence of absorbing aerosols and provide more accurate estimates of marine reflectance in productive waters. When applied to SeaWiFS imagery acquired off California, the function field methodology gives generally higher estimates of marine reflectance than the standard SeaDAS algorithm, but the values are more realistic.

Orthogonal polynomials on the disc

We consider the space P n of orthogonal polynomials of degree n on the unit disc for a general radially symmetric weight function. We show that there exists a single orthogonal polynomial whose rotations through the angles j π n + 1 , j = 0 , 1 , … , n forms an orthonormal basis for P n , and compute all such polynomials explicitly. This generalises the orthonormal basis of Logan and Shepp for the Legendre polynomials on the disc. Furthermore, such a polynomial reflects the rotational symmetry of the weight in a deeper way: its rotations under other subgroups of the group of rotations forms a tight frame for P n , with a continuous version also holding. Along the way, we show that other frame decompositions with natural symmetries exist, and consider a number of structural properties of P n including the form of the monomial orthogonal polynomials, and whether or not P n contains ridge functions.

Polynomial bases for subspaces of vector fields in the unit ball. Method of ridge functions

This paper deals with the construction of orthogonal polynomial bases for particular subspaces of vector fields defined in the unit ball of ℝ3. Our approach uses vector spherical harmonics to construct orthogonal sets of specific solenoidal and potential vector fields by means of ridge functions. It is shown that the approach leads to bases according to the subspaces induced by the Helmholtz–Hodge decomposition of square integrable vector fields.

Representation of multivariate functions by sums of ridge functions

In this paper, we find geometric means of deciding if any continuous multivariate function can be represented by sums of two continuous ridge functions.

Characterization of an extremal sum of ridge functions

The approximation problem considered in the paper is to approximate a continuous multivariate function f ( x ) = f ( x 1 , … , x d ) by sums of two ridge functions in the uniform norm. We give a necessary and sufficient condition for a sum of two ridge functions to be a best approximation to f ( x ) . This main result is next used in a special case to obtain an explicit formula for the approximation error and to construct one best approximation. The problem of well approximation by such sums is also considered.

Nonasymptotic Bounds on the L 2 Error of Neural Network Regression Estimates

The estimation of multivariate regression functions from bounded i.i.d. data is considered. The L 2 error with integration with respect to the design measure is used as an error criterion. The distribution of the design is assumed to be concentrated on a finite set. Neural network estimates are defined by minimizing the empirical L 2 risk over various sets of feedforward neural networks. Nonasymptotic bounds on the L 2 error of these estimates are presented. The results imply that neural networks are able to adapt to additive regression functions and to regression functions which are a sum of ridge functions, and hence are able to circumvent the curse of dimensionality in these cases.

Remote sensing of phytoplankton chlorophyll-a concentration by use of ridge function fields

A methodology is presented for retrieving phytoplankton chlorophyll-a concentration from space. The data to be inverted, namely, vectors of top-of-atmosphere reflectance in the solar spectrum, are treated as explanatory variables conditioned by angular geometry. This approach leads to a continuum of inverse problems, i.e., a collection of similar inverse problems continuously indexed by the angular variables. The resolution of the continuum of inverse problems is studied from the least-squares viewpoint and yields a solution expressed as a function field over the set of permitted values for the angular variables, i.e., a map defined on that set and valued in a subspace of a function space. The function fields of interest, for reasons of approximation theory, are those valued in nested sequences of subspaces, such as ridge function approximation spaces, the union of which is dense. Ridge function fields constructed on synthetic yet realistic data for case I waters handle well situations of both weakly and strongly absorbing aerosols, and they are robust to noise, showing improvement in accuracy compared with classic inversion techniques. The methodology is applied to actual imagery from the Sea-Viewing Wide Field-of-View Sensor (SeaWiFS); noise in the data are taken into account. The chlorophyll-a concentration obtained with the function field methodology differs from that obtained by use of the standard SeaWiFS algorithm by 15.7% on average. The results empirically validate the underlying hypothesis that the inversion is solved in a least-squares sense. They also show that large levels of noise can be managed if the noise distribution is known or estimated.

Where there?s a Will there?s a way ? the research of Will Light

This paper describes the work of Will Light, who died on December 8th, 2002. It highlights his contributions in the study of minimal norm projections, tensor product approximation (including the convergence of the Diliberto–Straus algorithm), proximinality, radial and ridge function approximation, both via quasi-interpolation and interpolation. My aim is not only to describe the impact of Will’s work, but also to convey some of his impact as a person.

Approximation by ridge function fields over compact sets

We study the approximation of a continuous function field over a compact set T by a continuous field of ridge approximants over T, named ridge function fields. We first give general density results about function fields and show how they apply to ridge function fields. We next discuss the parameterization of sets of ridge function fields and give additional density results for a class of continuous ridge function fields that admits a weak parameterization. Finally, we discuss the construction of the elements in that class.

Fields of nonlinear regression models for inversion of satellite data

A solution is provided to a common inverse problem in satellite remote sensing, the retrieval of a variable y from a vector x of explanatory variables influenced by a vector t of conditioning variables. The solution is in the general form of a field of nonlinear regression models, i.e., the relation between y and x is modeled as a map from some space to a subset of a function space. Elementary yet important mathematical results are presented for fields of shifted ridge functions, selected for their approximation properties. These fields are shown to span a dense set and to inherit the approximation properties of shifted ridge functions. A serious mathematical difficulty regarding the practical construction of continuous fields of shifted ridge functions is pointed out; it is circumvented while providing grounding to a large class of construction methodologies. Within this class, a construction scheme that builds upon multilinear interpolation is described. When applied to the retrieval of upper‐ocean chlorophyll‐a concentration from space, the solution shows potential for improved accuracy compared with existing algorithms.

"A Course in Approximation Theory"

This textbook is designed for graduate students in mathematics, physics, engineering, and computer science. Its purpose is to guide the reader in exploring contemporary approximation theory. The emphasis is on multi-variable approximation theory, i.e., the approximation of functions in several variables, as opposed to the classical theory of functions in one variable. Most of the topics in the book, heretofore accessible only through research papers, are treated here from the basics to the currently active research, often motivated by practical problems arising in diverse applications such as science, engineering, geophysics, and business and economics. Among these topics are projections, interpolation paradigms, positive definite functions, interpolation theorems of Schoenberg and Micchelli, tomography, artificial neural networks, wavelets, thin-plate splines, box splines, ridge functions, and convolutions. An important and valuable feature of the book is the bibliography of almost 600 items directing the reader to important books and research papers. There are 438 problems and exercises scattered through the book allowing the student reader to get a better understanding of the subject.

Estimation of heterogeneous aquifer parameters from piezometric data using ridge functions and neural networks

Inverse modeling aims to estimate transmissivity and other parameters needed by distributed aquifer models, using piezometric measurements. While these parameters are highly variable in space, the two-dimensional aquifer area is essentially empty of measurements (“curse of dimensionality”). To address this problem, a representation of the two-dimensional transmissivity map based on ridge functions and neural networks is introduced and applied to inverse aquifer modeling. The proposed representation has good expressive power, i.e. it is concise and convergences quickly as the number of parameters are increased, and it is expected to express complex transmissivity variations with relatively few parameters which can be estimated from the piezometric measurements. A simple regularization that can dampen erratic high frequency terms in the estimated parameters is suggested. Several examples indicate that the proposed parameterization can handle diverse types of transmissivity variations while it is particularly suited when the true transmissivity map exhibits specific sorts of heterogeneity with large anisotropies or abrupt changes along lines.

Ridgelets: estimating with ridge functions

Feedforward neural networks, projection pursuit regression, and more generally, estimation via ridge functions have been proposed as an approach to bypass the curse of dimensionality and are now becoming widely applied to approximation or prediction in applied sciences. To address problems inherent to these methods--ranging from the construction of neural networks to their efficiency and capability--Candès [Appl. Comput. armon. Anal. 6 (1999) 197-218] developed a new system that allows the representation of arbitrary functions as superpositions of specific ridge functions, the ridgelets. In a nonparametric regression setting, this article suggests expanding noisy data into a ridgelet series and applying a scalar nonlinearity to the coefficients (damping); this is unlike existing approaches based on stepwise additions of elements. The procedure is simple, constructive, stable and spatially adaptive--and fast algorithms have been developed to implement it. The ridgelet estimator is nearly optimal for estimating functions with certain kinds of spatial inhomogeneities. In addition, ridgelets help to identify new classes of estimands--corresponding to a new notion of smoothness--that are well suited for ridge functions estimation. While the results are stated in a decision theoretic framework, numerical experiments are also presented to illustrate the practical performance of the methodology.

Multiscale analysis of waves reflected by complex interfaces: Basic principles and experiments

This paper considers the reflection of waves by multiscale interfaces in the framework of the wavelet transform. First, we show how the wavelet transform is efficient to detect and characterize abrupt changes present in a signal. Locally homogeneous abrupt changes have conspicuous cone‐like signatures in the wavelet transform from which their regularity may be obtained. Multiscale clusters of nearby singularities produce a hierarchical arrangement of conical patterns where the multiscale structure of the cluster may be identified. Second, the wavelet response is introduced as a natural extension of the wavelet transform when the signal to be analyzed (i.e., the velocity structure of the medium) can only be remotely probed by propagating wavelets into the medium instead of being directly convolved as in the wavelet transform. The reflected waves produced by the incident wavelets onto the reflectors present in the medium constitute the wavelet response. We show that both transforms are equivalent when multiple scattering is neglected and that cone‐like features and ridge functions can be recognized in the wavelet response as well. Experimental applications of the acoustical wavelet response show how useful information can be obtained about remote multiscale reflectors. A first experiment implements the synthetic cases discussed before and concerns the characterization of planar reflectors with finite thicknesses. Another experiment concerns the multiscale characterization of a complex interface constituted by the surface of a layer of monodisperse glass beads immersed in water.

Tomographic reconstruction by maximum entropy in the mean: unconstrained reconstructions

In this note we present a solution to the tomographic reconstruction problem using the method of maximum entropy in the mean. When there are no convex constraints on the solution, the Hilbert setup combined with a Gaussian a priori measure on the Hilbert space yields solutions which resemble back projections or reconstructions by ridge functions.