Fast Computation of Convolution Operations via Low-Rank Approximation
نویسندگان
چکیده
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 use of low-rank approximations makes it possible to determine accurate approximations of convolutive operations at competitive speed.
منابع مشابه
Fast Multidimensional Convolution in Low-rank Formats via Cross Approximation
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”,...
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 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 کاملMultilevel Toeplitz Matrices Generated by Tensor-Structured Vectors and Convolution with Logarithmic Complexity
We consider two operations in the QTT format: composition of a multilevel Toeplitz matrix generated by a given multidimensional vector and convolution of two given multidimensional vectors. We show that low-rank QTT structure of the input is preserved in the output and propose efficient algorithms for these operations in the QTT format. For a d-dimensional 2n× . . .×2n-vector x given in a QTT r...
متن کاملFast Computation of Voigt Functions via Fourier Transforms
This work presents a method of computing Voigt functions and their derivatives, to high accuracy, on a uniform grid. It is based on an adaptation of Fourier-transform based convolution. The relative error of the result decreases as the fourth power of the computational effort. Because of its use of highly vectorizable operations for its core, it can be implemented very efficiently in scripting ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2013