Related papers: Characterizing and Enumerating Walsh-Hadamard Tran…
We propose a novel algorithm for computing the Walsh-Hadamard Transform (WHT) which consists entirely of Haar wavelet transforms. We prove that the algorithm, which we call the Cascading Haar Wavelet (CHW) algorithm, shares precisely the…
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…
Many matrices associated with fast transforms posess a certain low-rank property characterized by the existence of several block partitionings of the matrix, where each block is of low rank. Provided that these partitionings are known,…
This paper presents a multilevel tensor compression algorithm called tensor butterfly algorithm for efficiently representing large-scale and high-dimensional oscillatory integral operators, including Green's functions for wave equations and…
We unify the discrete Fourier transform (DFT), discrete cosine transform (DCT), Walsh-Hadamard, Haar wavelet, Karhunen-Lo\`eve transform, and several others along with their continuous counterparts (Fourier transform, Fourier series,…
We study relationships between permutation statistics and pattern-functions, counting the number of times particular patterns occur in a permutation. This allows us to write several familiar statistics as linear combinations of pattern…
The eigenfunctions of the Laplacian are a natural basis of functions for many tasks in computational mathematics. On the circle and sphere, the eigenfunctions are given by complex periodic exponentials and spherical harmonics, respectively,…
We describe an algorithm for the application of the forward and inverse spherical harmonic transforms. It is based on a new method for rapidly computing the forward and inverse associated Legendre transforms by hierarchically applying the…
A closed formula multiallelic Walsh (or Hadamard) transform is introduced. Basic results are derived, and a statistical interpretation of some of the resulting linear forms is discussed.
Butterfly algorithms are an effective multilevel technique to compress discretizations of integral operators with highly oscillatory kernel functions. The particular version of the butterfly algorithm considered here realizes the transfer…
Fast linear transforms are ubiquitous in machine learning, including the discrete Fourier transform, discrete cosine transform, and other structured transformations such as convolutions. All of these transforms can be represented by dense…
We propose a novel framework for fast integral operations by uncovering hidden geometries in the row and column structures of the underlying operators. This is accomplished through the \texttt{Questionnaire} algorithm, an iterative…
Fast transforms correspond to factorizations of the form $\mathbf{Z} = \mathbf{X}^{(1)} \ldots \mathbf{X}^{(J)}$, where each factor $ \mathbf{X}^{(\ell)}$ is sparse and possibly structured. This paper investigates essential uniqueness of…
A new iterative low complexity algorithm has been presented for computing the Walsh-Hadamard transform (WHT) of an $N$ dimensional signal with a $K$-sparse WHT, where $N$ is a power of two and $K = O(N^\alpha)$, scales sub-linearly in $N$…
In this paper, we propose a novel joint coding-modulation technique based on serial concatenation of orthogonal linear transform, such as discrete Fourier transform (DFT) or Walsh-Hadamard transform (WHT), with memoryless nonlinearity. We…
In this paper, we focus on Fourier analysis and holographic transforms for signal representation. For instance, in the case of image processing, the holographic representation has the property that an arbitrary portion of the transformed…
A new transform over finite fields, the finite field Hartley transform (FFHT), was recently introduced and a number of promising applications on the design of efficient multiple access systems and multilevel spread spectrum sequences were…
Fast Fourier Transform (FFT) is an efficient algorithm to compute the Discrete Fourier Transform (DFT) and its inverse. In this paper, we pay special attention to the description of complex-data FFT. We analyze two common descriptions of…
We accelerate the computation of spherical harmonic transforms, using what is known as the butterfly scheme. This provides a convenient alternative to the approach taken in the second paper from this series on "Fast algorithms for spherical…
We consider the problem of counting motifs in bipartite affiliation networks, such as author-paper, user-product, and actor-movie relations. We focus on counting the number of occurrences of a "butterfly", a complete $2 \times 2$ biclique,…