Fast generalized DFTs for all finite groups

For any finite group G, we give an arithmetic algorithm to compute generalized Discrete Fourier Transforms (DFTs) with respect to G, using O(|G|^(ω/2+ ϵ)) operations, for any ϵ > 0. Here, ω is the exponent of matrix multiplication

Fast generalized DFTs for all finite groups

For any finite group $G$, we give an arithmetic algorithm to compute generalized Discrete Fourier Transforms (DFTs) with respect to $G$, using $O(|G|^{\omega/2 + \epsilon})$ operations, for any $\epsilon > 0$. Here, $\omega$ is the exponent of matrix multiplication

Fast generalized DFTs for all finite groups

For any finite group G, we give an arithmetic algorithm to compute generalized Discrete Fourier Transforms (DFTs) with respect to G, using O(|G|^(ω/2+ ϵ)) operations, for any ϵ > 0. Here, ω is the exponent of matrix multiplication

A new algorithm for fast generalized DFTs

We give an new arithmetic algorithm to compute the generalized Discrete Fourier Transform (DFT) over finite groups $G$. The new algorithm uses $O(|G|^{\omega/2 + o(1)})$ operations to compute the generalized DFT over finite groups of Lie type, including the linear, orthogonal, and symplectic families and their variants, as well as all finite simple groups of Lie type. Here $\omega$ is the exponent of matrix multiplication, so the exponent $\omega/2$ is optimal if $\omega = 2$. Previously, "exponent one" algorithms were known for supersolvable groups and the symmetric and alternating groups. No exponent one algorithms were known (even under the assumption $\omega = 2$) for families of linear groups of fixed dimension, and indeed the previous best-known algorithm for $SL_2(F_q)$ had exponent $4/3$ despite being the focus of significant effort. We unconditionally achieve exponent at most $1.19$ for this group, and exponent one if $\omega = 2$. Our algorithm also yields an improved exponent for computing the generalized DFT over general finite groups $G$, which beats the longstanding previous best upper bound, for any $\omega$. In particular, assuming $\omega = 2$, we achieve exponent $\sqrt{2}$, while the previous best was $3/2$

Lemma for Linear Feedback Shift Registers and DFTs Applied to Affine Variety Codes

In this paper, we establish a lemma in algebraic coding theory that frequently appears in the encoding and decoding of, e.g., Reed-Solomon codes, algebraic geometry codes, and affine variety codes. Our lemma corresponds to the non-systematic encoding of affine variety codes, and can be stated by giving a canonical linear map as the composition of an extension through linear feedback shift registers from a Grobner basis and a generalized inverse discrete Fourier transform. We clarify that our lemma yields the error-value estimation in the fast erasure-and-error decoding of a class of dual affine variety codes. Moreover, we show that systematic encoding corresponds to a special case of erasure-only decoding. The lemma enables us to reduce the computational complexity of error-evaluation from O(n^3) using Gaussian elimination to O(qn^2) with some mild conditions on n and q, where n is the code length and q is the finite-field size.Comment: 37 pages, 1 column, 10 figures, 2 tables, resubmitted to IEEE Transactions on Information Theory on Jan. 8, 201

Algebraic Signal Processing Theory: Cooley-Tukey Type Algorithms for DCTs and DSTs

This paper presents a systematic methodology based on the algebraic theory of signal processing to classify and derive fast algorithms for linear transforms. Instead of manipulating the entries of transform matrices, our approach derives the algorithms by stepwise decomposition of the associated signal models, or polynomial algebras. This decomposition is based on two generic methods or algebraic principles that generalize the well-known Cooley-Tukey FFT and make the algorithms' derivations concise and transparent. Application to the 16 discrete cosine and sine transforms yields a large class of fast algorithms, many of which have not been found before.Comment: 31 pages, more information at http://www.ece.cmu.edu/~smar

Algebraic Signal Processing Theory: Cooley-Tukey Type Algorithms for Polynomial Transforms Based on Induction

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 $b=\{p_0(x),\dots,p_{n-1}(x)\}$ and sample points \alpha_k\in\C from a list $\alpha=\{\alpha_0,\dots,\alpha_{n-1}\}$. Such transforms find applications in the areas of signal processing, data compression, and function interpolation. Important examples include the discrete Fourier and cosine transforms. In this paper we introduce a novel technique to derive fast algorithms for polynomial transforms. The technique uses the relationship between polynomial transforms and the representation theory of polynomial algebras. Specifically, we derive algorithms by decomposing the regular modules of these algebras as a stepwise induction. As an application, we derive novel $O(n\log{n})$ general-radix algorithms for the discrete Fourier transform and the discrete cosine transform of type 4.Comment: 19 pages. Submitted to SIAM Journal on Matrix Analysis and Application

Determining Angular Frequency from a video with a Generalized Fast Fourier Transform

Suppose we are given a video of a rotating object and suppose we want to determine the rate of rotation solely from the video itself and its known frame rate. In this thesis, we present a new mathematical operator called the Geometric Sum Transform (GST) that can help one determine the angular frequency of the object in question. The GST is a generalization of the discrete Fourier transform (DFT) and as such, the two transforms have much in common. However, whereas the DFT is applied to a sequence of scalars, the GST can be applied to a sequence of vectors. Most importantly, we show that the GST, like the DFT, can (1) be used to estimate frequency and (2) can be computed surprisingly quickly. Indeed, we provide a Fast Geometric Sum Transform (FGST) algorithm that computes the GST in O(N logN) matrix-vector multiplications, where N is the number of images in the video sequence. This is a vast improvement over the O(N2) such multiplications required for a direct computation of the GST. The remainder of this thesis is devoted to proving other properties of the GST and giving proof-of-concept numerical examples