Multivariate Regression and Machine Learning with Sums of Separable Functions

By Gregory Beylkin, Jochen Garcke, Martin J. Mohlenkamp. 2009

127 citations recorded by the source corpus.

Subfields: Computational Mathematics, Computational Mechanics, Control and Systems Engineering

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mathematics

In Computational Mechanics

In Control and Systems Engineering

Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “…• computational tasks in micromagnetics [80,81]; • rational approximation problems [242]; • computational homogenization [42,96,176]; • computational ﬁnance [134,137]; • the approximation of stationary states of stochastic automata networks [38]; • multivariate regression and machine learning [25].”
Inductive Bias of Deep Convolutional Networks through Pooling Geometry (2016, background): “Since then, it has been employed for various applications, including quantum chemistry ([13]), particle engineering ([11]), machine learning ([3]), and more.”
Non-intrusive low-rank separated approximation of high-dimensional stochastic models (2012, methodology): “Consequently, they have been extensively applied to several areas including image compression and classification [66, 30], telecommunication [68, 67, 48], neuroscience [54, 29, 6], chemometrics [14], and data mining [8, 9, 44, 42, 71, 35, 1, 7, 2, 43].”
Limits to Depth Efficiencies of Self-Attention (2020, methodology): “The separation rank, introduced in Beylkin and Mohlenkamp [2002] for high-dimensional numerical analysis, was employed for various applications, e.g. , chemistry [Harrison et al., 2003], particle engineering [Hackbusch, 2006], and machine learning [Beylkin et al., 2009].”
Tensors-structured Numerical Methods in Scientific Computing: Survey on Recent Advances (2012, background): “The employment of separation of variables in machine learning is addressed in [6].”
Mathematik in den Naturwissenschaften Leipzig Tensor-Structured Galerkin Approximation of parametric and stochastic Elliptic PDEs (2010, background): “Tensor-structured techniques in higher dimensions [7, 13, 15, 23, 26] have already demonstrated their solid potential in electronic structure calculations [3, 12, 23, 24, 25], in PDEs with random input [34, 37, 35, 4], and in machine learning [1].”
Which transformer architecture fits my data? A vocabulary bottleneck in self-attention (2021, methodology): “The separation rank, introduced in Beylkin & Mohlenkamp (2002) for high-dimensional numerical analysis, was employed for various applications, e.g., chemistry (Harrison et al., 2003), particle engineering (Hackbusch, 2006), and machine learning (Beylkin et al., 2009).”
Numerical construction of Green's functions in high dimensional elliptic problems with variable coefficients and analysis of renewable energy data via sparse and separable approximations (2012, background): “Although we do not discuss it here, a more general approach to multi-way regression may be found in [18].”
Black Box Low Tensor-Rank Approximation Using Fiber-Crosses (2009, background): “The only reference—to our knowledge—for an alternative approach in the literature is the three-dimensional cross approximation [13] (see also [2] for low rank approximation on given point sets and [8] for an application).”
O(dlog N)-Quantics Approximation of N-d Tensors in High-Dimensional Numerical Modeling (2011, background): “Using separation of variables in machine learning is addressed in [3].”
A least-squares approximation of partial differential equations with high-dimensional random inputs (2009, methodology): “When large samples of stochastic processes representing the uncertainty in the system are available, a more consistent approach is to use procedures of [5].”