Numerical operator calculus in higher dimensions

By Gregory Beylkin, Martin J. Mohlenkamp. 2002

335 citations recorded by the source corpus.

Subfields: Computational Mathematics, Computational Theory and Mathematics, Signal Processing

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “Other examples of combining iterative methods with low-rank truncation include: • the (shift-and-invert) power method combined with CP [26,27]; • the (shift-and-invert) power and Lanczos methods combined with CP and Tucker [11,114]; • a restarted Lanczos method combined with TT [126]; • conjugate…”
Inductive Bias of Deep Convolutional Networks through Pooling Geometry (2016, background): “The concept of separation rank was introduced in [2] for numerical treatment of high-dimensional functions.”
A Link between the Canonical Decomposition in Multilinear Algebra and Simultaneous Matrix Diagonalization (2006, background): “Moreover, decomposition is finding its way into scientific computing, where it leads to a way around the curse of dimensionality [ 4 , 5].”
Adaptive low rank wavelet methods and applications to two-electron Schrödinger equations (2012, background): “Separable expansions for potential terms generated by exponential sum approximations have been used for a variety of related purposes, for instance, in [109, 25, 48] as part of an integration scheme for higher-dimensional integrals of potentials with wavelets, in [12, 79, 60] for the approximation o”
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).”
Algorithms for Numerical Analysis in High Dimensions (2005, methodology): “Basic linear algebra operations, such as matrix-vector multiplication, are straightforward and were described in [3], but other operations are not as simple.”
Tensors-structured Numerical Methods in Scientific Computing: Survey on Recent Advances (2012, background): “+ xd) over N ⊗d tensor grid in Rd, is exactly 2 (see also [79]), while its canonical rank is estimated by R = d [5].”
Tensor-Train Decomposition (2011, background): “the one considered in [3, 4], where the canonical format was used.”
Non-intrusive low-rank separated approximation of high-dimensional stochastic models (2012, methodology): “Additionally, as described in [8, 9], such a normalization of ck may be used to control the loss of significant digits in the evaluation of the separated representation.”
Decompositions of a Higher-Order Tensor in Block Terms - Part II: Definitions and Uniqueness (2008, background): “Moreover, the decomposition is finding its way to scientific computing, where it leads to a way around the curse of dimensionality [2, 3, 24, 25, 33].”
Tensor Decompositions and Applications (2009, background): “Beylkin and Mohlenkamp [22, 23] apply CP to operators and conjecture that the optimal separation rank (which is the rank of an ϵ approximation) is low for many common operators.”
On approximation of functions by exponential sums (2005, methodology): “This approximation provides an example of an analytic construction of a separate resentation as introduced in [9] and discussed in [10,23].”
Tucker Dimensionality Reduction of Three-Dimensional Arrays in Linear Time (2008, background): “One can mention signal processing, statistics [3, 1, 4], chemometrics [5], face recognition [7], solving multidimensional integral and differential equations [6](a very comprehensive list of references on the subject can be found on a Three-mode company web site, [2]).”
FFT-based Kronecker product approximation to micromagnetic long-range interactions (2012, methodology): “Also approximation of operators like the multiparticle Schrödinger operator [7] or Newton potential [18] can be done by using the canonical tensor format in order to overcome the curse of dimensionality .”
A regularized Newton method for the efficient approximation of tensors represented in the canonical tensor format (2012, methodology): “the ALS method [1] and the regularized Newton method [7].”
Tensor properties of multilevel Toeplitz and related matrices (2006, methodology): “Separation of variables is a topic of permanent interest in approximation theory (cf [1]).”
QTT approximation of elliptic solution operators in higher dimensions (2011, background): “[4, 5, 16] are based on the canonical decomposition [3, 6, 18].”
Numerical solution of the Hartree-Fock equation by multilevel tensor-structured methods (2011, neutral): “6.1 Brief summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.2 Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140”