Interpolation of sparse high-dimensional data
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Interpolation of sparse high-dimensional data Thomas C. H. Lux1 Kirk Cameron1
· Layne T. Watson1 · Tyler H. Chang1 · Yili Hong1 ·
Received: 30 March 2019 / Accepted: 28 October 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Increases in the quantity of available data have allowed all fields of science to generate more accurate models of multivariate phenomena. Regression and interpolation become challenging when the dimension of data is large, especially while maintaining tractable computational complexity. Regression is a popular approach to solving approximation problems with high dimension; however, there are often advantages to interpolation. This paper presents a novel and insightful error bound for (piecewise) linear interpolation in arbitrary dimension and contrasts the performance of some interpolation techniques with popular regression techniques. Empirical results demonstrate the viability of interpolation for moderately high-dimensional approximation problems, and encourage broader application of interpolants to multivariate approximation in science. Keywords Approximation · Regression · Interpolation · High dimension · Error bound
1 Introduction Regression and interpolation are problems of considerable importance that find applications across many fields of science. Pollution and air quality analysis [19], energy consumption management [30], and student performance prediction [16, 33] are a few of many interdisciplinary applications of multivariate regression for predictive analysis. As discussed later, these techniques can also be applied to prediction problems related to forest fire risk assessment [15], Parkinson’s patient clinical evaluations [49], local rainfall and weather [51], credit card transactions [43], and high performance computing (HPC) file input/output (I/O) [34].
Thomas C. H. Lux
[email protected] 1
Virginia Polytechnic Institute and State University (VPI, SU), Blacksburg, VA 24061, USA
Numerical Algorithms
Regression and interpolation have a considerable theoretical base in one dimension [11]. Common techniques include Lagrangian interpolation, piecewise linear interpolation, and more generally spline interpolation via B-splines [5]. Tensor products of B-splines [50] or other basis functions have an unfortunate exponential scaling with increasing dimension. Exponential scaling prohibits tensor products from being reasonably applied beyond three-dimensional data. In order to address this dimensional scaling challenge, C. de Boor and others proposed box splines [6], of which one of the approximation techniques in this work is composed [35]. The theoretical foundation of low-dimensional interpolation allows the construction of strong error bounds that are lacking in high-dimensional problems. A large body of literature exists studying the construction and application of sparse grids [8, 12], which attempt to address the problem of dimensional scaling. The sparse grid point sets combined with specific approximation functions allow for the cons
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