Related papers: Learning Fast Algorithms for Linear Transforms Usi…
The Fourier Transform is one of the most important linear transformations used in science and engineering. Cooley and Tukey's Fast Fourier Transform (FFT) from 1964 is a method for computing this transformation in time $O(n\log n)$.…
A polynomial transform is the multiplication of an input vector $x\in\C^n$ by a matrix $\PT_{b,\alpha}\in\C^{n\times n},$ whose $(k,\ell)$-th element is defined as $p_\ell(\alpha_k)$ for polynomials $p_\ell(x)\in\C[x]$ from a list…
Fast Fourier transform algorithms are an arsenal of effective tools for solving various problems of analysis and high-speed processing of signals of various natures. Almost all of these algorithms are designed to process sequences of…
It is demonstrated is this letter that linear multistep methods for integrating ordinary differential equations can be used to develop a family of fast forward scattering algorithms with higher orders of convergence. Excluding the cost of…
This paper is concerned with the fast computation of Fourier integral operators of the general form $\int_{\R^d} e^{2\pi\i \Phi(x,k)} f(k) d k$, where $k$ is a frequency variable, $\Phi(x,k)$ is a phase function obeying a standard…
The special unitary group SU(2) plays a fundamental role in the description of symmetries in quantum mechanics, theoretical physics, and spherical signal processing. In this paper, we address the computational challenges of performing…
One of the main computational bottlenecks when working with kernel based learning is dealing with the large and typically dense kernel matrix. Techniques dealing with fast approximations of the matrix vector product for these kernel…
The paper introduces the butterfly factorization as a data-sparse approximation for the matrices that satisfy a complementary low-rank property. The factorization can be constructed efficiently if either fast algorithms for applying the…
We introduce a new kind of linear transform named Deformable Butterfly (DeBut) that generalizes the conventional butterfly matrices and can be adapted to various input-output dimensions. It inherits the fine-to-coarse-grained learnable…
Discrete transforms such as the discrete Fourier transform (DFT) or the discrete Hartley transform (DHT) furnish an indispensable tool in signal processing. The successful application of transform techniques relies on the existence of the…
Fast Fourier Transforms (FFT) are widely used to reduce memory and computational costs in deep learning. However, existing implementations, including standard FFT and real FFT (rFFT), cannot achieve true in-place computation. In particular,…
We present a fast and approximate multifrontal solver for large-scale sparse linear systems arising from finite-difference, finite-volume or finite-element discretization of high-frequency wave equations. The proposed solver leverages the…
This paper proposes to use Fast Fourier Transformation-based U-Net (a refined fully convolutional networks) and perform image convolution in neural networks. Leveraging the Fast Fourier Transformation, it reduces the image convolution costs…
The Fast Fourier Transform (FFT) over a finite field $\mathbb{F}_q$ computes evaluations of a given polynomial of degree less than $n$ at a specifically chosen set of $n$ distinct evaluation points in $\mathbb{F}_q$. If $q$ or $q-1$ is a…
This paper introduces the multidimensional butterfly factorization as a data-sparse representation of multidimensional kernel matrices that satisfy the complementary low-rank property. This factorization approximates such a kernel matrix of…
Fourier representations play a central role in operator learning methods for partial differential equations and are increasingly being explored in quantum machine learning architectures. The classical fast Fourier transform (FFT),…
Convolutional networks are one of the most widely employed architectures in computer vision and machine learning. In order to leverage their ability to learn complex functions, large amounts of data are required for training. Training a…
This paper concerns the fast evaluation of the matvec $g=Kf$ for $K\in \mathbb{C}^{N\times N}$, which is the discretization of the oscillatory integral transform $g(x) = \int K(x,\xi) f(\xi)d\xi$ with a kernel function…
The fast Fourier transform (FFT) is one of the most successful numerical algorithms of the 20th century and has found numerous applications in many branches of computational science and engineering. The FFT algorithm can be derived from a…
Normalizing flows model complex probability distributions using maps obtained by composing invertible layers. Special linear layers such as masked and 1x1 convolutions play a key role in existing architectures because they increase…