Ridge Functions Research Articles

A fuzzy comprehensive evaluation method for assessing the damage degree of cultural relics based on an improved AHP-EWM coupled weight model and a modified ridge function

Due to long-term exposure to outdoor environments, stone artifacts exhibit complex and varying degrees of disease that need to be conserved urgently in order to perpetuate their historical, artistic and scientific values. Owing to the varying degrees of damage inflicted upon cultural relics, discrepancies exist in intervention strategies and conservation materials employed. Hence, it is considerably crucial to conduct a scientific and precise assessment of the preservation status of cultural relics at an early stage for the advancement of cultural relic conservation efforts. In this research, three masonry components of the Hailongtun Fortress site were selected as study objects. A secondary fuzzy comprehensive evaluation system was established through selecting a total of 12 indicators from four aspects, namely, surface change, rock mass structure, physical properties and matrix weathering. The weight of each indicator was determined by using an improved AHP-EWM weight coupling model. The membership degree was calculated by selecting a modified ridge function. The asymmetric nearness method was employed in the final decision-making process so as to correct the error caused by the principle of maximum membership degree. The research shows that the results of the improved evaluation method are in line with the real damage degree of each masonry component, which provides a scientific and quantitative basis for determining the priority of the conservation of masonry components in the Hailongtun Fortress Site. This method is also a reference for carrying out the disease assessment and conservation of other types of cultural relics.

On the representability of a continuous multivariate function by sums of ridge functions

In this paper, new conditions are found for the representability of a continuous multivariate function as a sum of ridge functions. Using these conditions, we give a new proof for the earlier theorem solving the problem, posed by A.Pinkus in his monograph “Ridge Functions”, up to a multivariate polynomial. That is, we show that if a continuous multivariate function has a representation as a sum of arbitrarily behaved ridge functions, then it can be represented as a sum of continuous ridge functions and some multivariate polynomial.

A Sequential Stein's Method for Faster Training of Additive Index Models.

The additive index models (AIMs) can be viewed as a kind of artificial neural networks based on nonparametric activation or so-called ridge functions. Recently, they are shown to achieve enhanced explainability after incorporating various interpretability constraints. However, the training of AIMs by either the backfitting algorithm or the joint stochastic optimization is known to be very slow for especially high dimensional inputs. In this article, we propose a novel sequential approach based on the celebrated Stein's lemma. The proposed SeqStein method can successfully decouple the training of AIMs into two separable steps, namely, the following: 1) Stein's estimation of the projection indices and 2) nonparametric estimation of ridge functions using the smoothing splines. We show through numerical experiments that the SeqStein algorithm is not only more efficient for training AIMs, but also inclined to produce more interpretable models that have smooth ridge functions with sparse and nearly orthogonal projection indices.

Approximation of curve-based sleeve functions in high dimensions

Sleeve functions are generalizations of the well-established ridge functions that play a major role in the theory of partial differential equation, medical imaging, statistics, and neural networks. Where ridge functions are non-linear, univariate functions of the distance to hyperplanes, sleeve functions are based on the squared distance to lower-dimensional manifolds. The present work is a first step to study general sleeve functions by starting with sleeve functions based on finite-length curves. To capture these curve-based sleeve functions, we propose and study a two-step method, where first the outer univariate function—the profile—is recovered, and second, the underlying curve is represented by a polygonal chain. Introducing a concept of well-separation, we ensure that the proposed method always terminates and approximates the true sleeve function with a certain quality. Investigating the local geometry, we study an inexact version of our method and show its success under certain conditions.

Tensor rank reduction via coordinate flows

Recently, there has been a growing interest in efficient numerical algorithms based on tensor networks and low-rank techniques to approximate high-dimensional functions and solutions to high-dimensional PDEs. In this paper, we propose a new tensor rank reduction method based on coordinate transformations that can greatly increase the efficiency of high-dimensional tensor approximation algorithms. The idea is simple: given a multivariate function, determine a coordinate transformation so that the function in the new coordinate system has smaller tensor rank. We restrict our analysis to linear coordinate transformations, which gives rise to a new class of functions that we refer to as tensor ridge functions. Leveraging Riemannian gradient descent on matrix manifolds we develop an algorithm that determines a quasi-optimal linear coordinate transformation for tensor rank reduction. The results we present for rank reduction via linear coordinate transformations open the possibility for generalizations to larger classes of nonlinear transformations.

A complex structural variant near SOX3 causes X-linked split-hand/foot malformation.

Split-hand/foot malformation (SHFM) is a congenital limb defect most typically presenting with median clefts in hands and/or feet, that can occur in a syndromic context as well as in isolated form. SHFM is caused by failure to maintain normal apical ectodermal ridge function during limb development. Although several genes and contiguous gene syndromes are implicated in the monogenic etiology of isolated SHFM, the disorder remains genetically unexplained for many families and associated genetic loci. We describe a family with isolated X-linked SHFM, for which the causative variant could be detected after a diagnostic journey of 20 years. We combined well-established approaches including microarray-based copy number variant analysis and fluorescence in situ hybridization coupled with optical genome mapping and whole genome sequencing. This strategy identified a complex structural variant (SV) comprising a 165-kb gain of 15q26.3 material ([GRCh37/hg19] chr15:99795320-99960362dup) inserted in inverted position at the site of a 38-kb deletion on Xq27.1 ([GRCh37/hg19] chrX:139481061-139518989del). In silico analysis suggested that the SV disrupts the regulatory framework on the X chromosome and may lead to SOX3 misexpression. We hypothesize that SOX3 dysregulation in the developing limb disturbed the fine balance between morphogens required for maintaining AER function, resulting in SHFM in this family.

Penalty and Shrinkage Strategies Based on Local Polynomials for Right-Censored Partially Linear Regression.

This study aims to propose modified semiparametric estimators based on six different penalty and shrinkage strategies for the estimation of a right-censored semiparametric regression model. In this context, the methods used to obtain the estimators are ridge, lasso, adaptive lasso, SCAD, MCP, and elasticnet penalty functions. The most important contribution that distinguishes this article from its peers is that it uses the local polynomial method as a smoothing method. The theoretical estimation procedures for the obtained estimators are explained. In addition, a simulation study is performed to see the behavior of the estimators and make a detailed comparison, and hepatocellular carcinoma data are estimated as a real data example. As a result of the study, the estimators based on adaptive lasso and SCAD were more resistant to censorship and outperformed the other four estimators.

Sharp Bounds on the Approximation Rates, Metric Entropy, and n-Widths of Shallow Neural Networks

In this article, we study approximation properties of the variation spaces corresponding to shallow neural networks with a variety of activation functions. We introduce two main tools for estimating the metric entropy, approximation rates, and n-widths of these spaces. First, we introduce the notion of a smoothly parameterized dictionary and give upper bounds on the nonlinear approximation rates, metric entropy, and n-widths of their absolute convex hull. The upper bounds depend upon the order of smoothness of the parameterization. This result is applied to dictionaries of ridge functions corresponding to shallow neural networks, and they improve upon existing results in many cases. Next, we provide a method for lower bounding the metric entropy and n-widths of variation spaces which contain certain classes of ridge functions. This result gives sharp lower bounds on the \(L^2\)-approximation rates, metric entropy, and n-widths for variation spaces corresponding to neural networks with a range of important activation functions, including ReLU\(^k\) activation functions and sigmoidal activation functions with bounded variation.

Prediction of thermal conductivity ofconcrete under variable temperatures in cold regions using projection pursuit regression

A projection pursuit regression (PPR) model was proposed in this paper to predict the thermal conductivity of concrete (TCC) under variable temperatures in cold regions. The PPR model was established based on experimental data obtained from a single–factor experiment. Unlike the conventional PPR model that selected a random form from an empirical distribution function to describe the ridge functions in advance, the PPR model in this paper can directly apply numerical functions to describe the ridge functions obtained by projection. The comparison study indicated that, for the TCC prediction, the PPR model (R = 0.985) outperformed trained model based on Back–propagation (BP) neural network (R = 0.974). The PPR model simulation results demonstrated that the aggregate volume fraction, sand rate, saturation, water–cement ratio, fly ash content, and slag content all had impacts on the TCC under variable temperatures in cold regions. Especially, in temperature-sensitive zone (0 °C to −10 °C), it was found that the TCC of specimens with high water–cement ratio and high saturation degree increase abruptly but other factors did not cause the similar abrupt change of the TCC during the same temperature range. The correlation between the water-cement ratio and temperature–sensitive zone was explained after an in–depth study of the concrete pore structure.

Certified dimension reduction in nonlinear Bayesian inverse problems

We propose a dimension reduction technique for Bayesian inverse problems with nonlinear forward operators, non-Gaussian priors, and non-Gaussian observation noise. The likelihood function is approximated by a ridge function, i.e., a map which depends nontrivially only on a few linear combinations of the parameters. We build this ridge approximation by minimizing an upper bound on the Kullback–Leibler divergence between the posterior distribution and its approximation. This bound, obtained via logarithmic Sobolev inequalities, allows one to certify the error of the posterior approximation. Computing the bound requires computing the second moment matrix of the gradient of the log-likelihood function. In practice, a sample-based approximation of the upper bound is then required. We provide an analysis that enables control of the posterior approximation error due to this sampling. Numerical and theoretical comparisons with existing methods illustrate the benefits of the proposed methodology.

Linear/Ridge expansions: enhancing linear approximations by ridge functions

We consider approximations formed by the sum of a linear combination of given functions enhanced by ridge functions—a Linear/Ridge expansion. For an explicitly or implicitly given objective function, we reformulate finding a best Linear/Ridge expansion in terms of an optimization problem. We introduce a particle grid algorithm for its solution. Several numerical results underline the flexibility, robustness and efficiency of the algorithm. One particular source of motivation is model reduction of parameterized transport or wave equations. We show that the particle grid algorithm is able to find a Linear/Ridge expansion as an efficient nonlinear model reduction.

Inpainting of regular textures using ridge functions

Abstract In this paper, we propose a method for inpainting regular textures composed of oriented patterns and completion of images distorted by long stripes. We have developed a model fit to describe a regular texture as a sum of a few ridge functions. The ridge functions are known in tomography as a result of a back-projection operator applied to projections, or samples of the Radon transform. A computational scheme for the singular value decomposition of a regular texture into the sum of informative ridge functions is presented. The numerical experiments on various images demonstrate the efficiency of the techniques proposed.

Artificial neural network based response surface for data-driven dimensional analysis

The classical dimensional analysis method has limitations in determining the uniqueness and relative importance of the dimensionless quantities. A machine-learning based dimensional analysis method is proposed to address the limitations. The proposed method identifies unique and relevant dimensionless quantities by combining an artificial neural network with the data-driven dimensional analysis. We employ a fully connected neural network to construct the ridge function for the response surface in a physical system. The gradient of the response surface for active subspace analysis is computed based on a finite difference approximation. An effective approach is proposed to determine the independent variables of experimental measurements or numerical simulations for computing the gradient of the response surface. The proposed method is validated by analyzing benchmark pipe flows and a fluid-structure interaction system. The dominant dimensionless quantities obtained by the proposed method are consistent with those reported in the literature. The proposed method has the advantage of identifying the relatively important dimensionless quantities without referring to the complex theoretical equations.

Recovery of Regular Ridge Functions on the Ball

We consider the problem of the uniform (in \(L_\infty \)) recovery of ridge functions \(f(x)=\varphi (\langle a,x\rangle )\), \(x\in B_2^n\), using noisy evaluations \(y_1\approx f(x^1),\ldots ,y_N\approx f(x^N)\). It is known that for classes of functions \(\varphi \) of finite smoothness the problem suffers from the curse of dimensionality: in order to provide good accuracy for the recovery it is necessary to make exponential number of evaluations. We prove that if \(\varphi \) is analytic in a neighborhood of \([-1,1]\) and the noise is very small, \(\varepsilon \leqslant \exp (-c\log ^2n)\), then there is an efficient algorithm that recovers f with good accuracy using \(O(n\log ^2n)\) function evaluations.

Polynomial ridge flowfield estimation

Computational fluid dynamics plays a key role in the design process across many industries. Recently, there has been increasing interest in data-driven methods in order to exploit the large volume of data generated by such computations. This paper introduces the idea of using spatially correlated polynomial ridge functions for rapid flowfield estimation. Dimension reducing ridge functions are obtained for numerous points within training flowfields. The functions can then be used to predict flow variables for new, previously unseen, flowfields. Their dimension reducing nature alleviates the problems associated with visualizing high-dimensional datasets, enabling improved understanding of design spaces and potentially providing valuable physical insights. The proposed framework is computationally efficient; consisting of either readily parallelizable tasks or linear algebra operations. To further reduce the computational cost, ridge functions need only be computed at a small number of subsampled locations. The flow physics encoded within covariance matrices obtained from the training flowfields can then be used to predict flow quantities, conditional upon those predicted by the ridge functions at the sampled points. To demonstrate the efficacy of the framework, the incompressible flow around an ensemble of airfoils is used as a test case. The ridge functions’ predictive accuracy is found to be competitive with a state-of-the-art convolutional neural network. The local ridge functions can also be reused to obtain surrogate models for integral quantities, avoiding the need for long-term storage of the training data. Finally, use of the ridge framework with varying boundary conditions is demonstrated on a transonic wing.

On the Frontier Between Kalman Conjecture and Markus-Yamabe Conjecture

The Kalman conjecture and the Markus-Yamabe conjecture are revisited. These conjectures make statements about global stability of certain classes of nonlinear systems. Both conjectures are known to be false in the general case, but in the context of traction control, an interesting special case of the Markus-Yamabe conjecture becomes apparent. In this letter, two methods are proposed to analyze general Lur'e systems with respect to their input and output dimension, resulting in a new open problem in the context of the Markus-Yamabe conjecture. The connection to ridge functions and traction control systems is discussed. It is shown that although both the Kalman conjecture, for single-input, single-output Lur'e systems and the Markus-Yamabe conjecture are true up to a specific system order, the problem is still open if different input and output dimensions of a Lur'e formulation are considered.

On the Representation by Bivariate Ridge Functions

On the Representation by Bivariate Ridge Functions

Modeling Hydration Kinetics of the Portland-Cement-Based Cementitious Systems with Mortar Blends by Non-Assumptive Projection Pursuit Regression

The non-assumptive projection pursuit regression (NA-PPR) method, which is an exploratory data analysis method for analyzing high-dimensional data, was applied to predict the hydration heat of Portland-cement-based cementitious systems with mortar blends. The hydration mechanisms of cementitious systems were investigated by the Krstulovic–Dabic model, which was based on the proposed NA-PPR model of hydration heat. The new ridge functions and three-layer-grouping iterative optimization method were used to develop the NA-PPR model to improve the performance and address the non-uniqueness problem of parameter selection and subjective assumptions. It was shown that the proposed NA-PPR model is a promising non-assumptive method, with favorable prediction accuracy and a visible functional expression. Moreover, the Krstulovic–Dabic model combined with the NA-PPR model was able to well simulate the hydration process of the Portland-cement-based cementitious materials systems with mortar blends.

On the representation by bivariate ridge functions

UDC 517.5 We consider the problem of representation of a bivariate function by sums of ridge functions. It is shown that if a function of a certain smoothness class is represented by a sum of finitely many arbitrarily behaved ridge functions, then it can also be represented by a sum of ridge functions of the same smoothness class. As an example, this result is applied to a homogeneous constant coefficient partial differential equation.

On the population least‐squares criterion in the monotone single index model

Monotone single index models have gained increasing popularity over the past decades due to their flexibility and versatile use in diverse areas. Semi‐parametric estimators such as the least squares and maximum likelihood estimators of the unknown index and monotone ridge function were considered to make inference in such models without having to choose some tuning parameter. Description of the asymptotic behavior of those estimators crucially depends on acquiring a good understanding of the optimization problems associated with the corresponding population criteria. In this paper, we give several insights into these criteria by proving existence of minimizers thereof over general classes of parameters. In order to describe these minimizers, we prove different results which give the direction of variation of the population criteria in general and in the special case where the common distribution of the covariates is Gaussian. A complementary simulation study was performed and whose results give support to our main theorems.