Fast Multidimensional Convolution in Low-rank Formats via Cross Approximation
نویسنده
چکیده
We propose new cross-conv algorithm for approximate computation of convolution in different low-rank tensor formats (tensor train, Tucker, Hierarchical Tucker). It has better complexity with respect to the tensor rank than previous approaches. The new algorithm has a high potential impact in different applications. The key idea is based on applying cross approximation in the “frequency domain”, where convolution becomes a simple elementwise product. We illustrate efficiency of our algorithm by computing the three-dimensional Newton potential and by presenting preliminary results for solution of the Hartree-Fock equation on tensor-product grids.
منابع مشابه
Fast Computation of Convolution Operations via Low-Rank Approximation
Methods for the approximation of 2D discrete convolution operations are derived for the case when a low-rank approximation of one of the input matrices is available. Algorithms based on explicit computation and on the Fast Fourier Transform are described. Applications to the computation of cross-correlation and autocorrelation are discussed. Both theory and numerical experiments show that the u...
متن کاملA Direct Solver for the Advection-diffusion Equation Using Green’s Functions and Low-rank Approximation
A new direct solution method for the advection-diffusion equation is presented. By employing a semi-implicit time discretisation, the equation is rewritten as a heat equation with source terms. The solution is obtained by discretely approximating the integral convolution of the associated Green’s function with advective source terms. The heat equation has an exponentially decaying Green’s funct...
متن کاملFast and accurate tensor approximation of a multivariate convolution with linear scaling in dimension
In the present paper we present the tensor-product approximation of multidimensional convolution transform discretized via collocation-projection scheme on the uniform or composite refined grids. Examples of convolving kernels are given by the classical Newton, Slater (exponential) and Yukawa potentials, 1/‖x‖, e−λ‖x‖ and e−λ‖x‖/‖x‖ with x ∈ Rd. For piecewise constant elements on the uniform gr...
متن کاملProjective Low-rank Subspace Clustering via Learning Deep Encoder
Low-rank subspace clustering (LRSC) has been considered as the state-of-the-art method on small datasets. LRSC constructs a desired similarity graph by low-rank representation (LRR), and employs a spectral clustering to segment the data samples. However, effectively applying LRSC into clustering big data becomes a challenge because both LRR and spectral clustering suffer from high computational...
متن کاملA fast algorithm for multilinear operators
This paper introduces a fast algorithm for computing multilinear integrals, which are defined through Fourier multipliers. The algorithm is based on generating a hierarchical decomposition of summation domain into squares, constructing a low-rank approximation for the multiplier function within each square, and applying FFT based fast convolution algorithm for the computation associated with ea...
متن کامل