15,168 research outputs found
Composite Cyclotomic Fourier Transforms with Reduced Complexities
Discrete Fourier transforms~(DFTs) over finite fields have widespread
applications in digital communication and storage systems. Hence, reducing the
computational complexities of DFTs is of great significance. Recently proposed
cyclotomic fast Fourier transforms (CFFTs) are promising due to their low
multiplicative complexities. Unfortunately, there are two issues with CFFTs:
(1) they rely on efficient short cyclic convolution algorithms, which has not
been investigated thoroughly yet, and (2) they have very high additive
complexities when directly implemented. In this paper, we address both issues.
One of the main contributions of this paper is efficient bilinear 11-point
cyclic convolution algorithms, which allow us to construct CFFTs over
GF. The other main contribution of this paper is that we propose
composite cyclotomic Fourier transforms (CCFTs). In comparison to previously
proposed fast Fourier transforms, our CCFTs achieve lower overall complexities
for moderate to long lengths, and the improvement significantly increases as
the length grows. Our 2047-point and 4095-point CCFTs are also first efficient
DFTs of such lengths to the best of our knowledge. Finally, our CCFTs are also
advantageous for hardware implementations due to their regular and modular
structure.Comment: submitted to IEEE trans on Signal Processin
Lower Bounds on the Bounded Coefficient Complexity of Bilinear Maps
We prove lower bounds of order for both the problem to multiply
polynomials of degree , and to divide polynomials with remainder, in the
model of bounded coefficient arithmetic circuits over the complex numbers.
These lower bounds are optimal up to order of magnitude. The proof uses a
recent idea of R. Raz [Proc. 34th STOC 2002] proposed for matrix
multiplication. It reduces the linear problem to multiply a random circulant
matrix with a vector to the bilinear problem of cyclic convolution. We treat
the arising linear problem by extending J. Morgenstern's bound [J. ACM 20, pp.
305-306, 1973] in a unitarily invariant way. This establishes a new lower bound
on the bounded coefficient complexity of linear forms in terms of the singular
values of the corresponding matrix. In addition, we extend these lower bounds
for linear and bilinear maps to a model of circuits that allows a restricted
number of unbounded scalar multiplications.Comment: 19 page
Convolutional Dictionary Learning through Tensor Factorization
Tensor methods have emerged as a powerful paradigm for consistent learning of
many latent variable models such as topic models, independent component
analysis and dictionary learning. Model parameters are estimated via CP
decomposition of the observed higher order input moments. However, in many
domains, additional invariances such as shift invariances exist, enforced via
models such as convolutional dictionary learning. In this paper, we develop
novel tensor decomposition algorithms for parameter estimation of convolutional
models. Our algorithm is based on the popular alternating least squares method,
but with efficient projections onto the space of stacked circulant matrices.
Our method is embarrassingly parallel and consists of simple operations such as
fast Fourier transforms and matrix multiplications. Our algorithm converges to
the dictionary much faster and more accurately compared to the alternating
minimization over filters and activation maps
Fast Digital Convolutions using Bit-Shifts
An exact, one-to-one transform is presented that not only allows digital
circular convolutions, but is free from multiplications and quantisation errors
for transform lengths of arbitrary powers of two. The transform is analogous to
the Discrete Fourier Transform, with the canonical harmonics replaced by a set
of cyclic integers computed using only bit-shifts and additions modulo a prime
number. The prime number may be selected to occupy contemporary word sizes or
to be very large for cryptographic or data hiding applications. The transform
is an extension of the Rader Transforms via Carmichael's Theorem. These
properties allow for exact convolutions that are impervious to numerical
overflow and to utilise Fast Fourier Transform algorithms.Comment: 4 pages, 2 figures, submitted to IEEE Signal Processing Letter
A search for concentric rings with unusual variance in the 7-year WMAP temperature maps using a fast convolution approach
We present a method for the computation of the variance of cosmic microwave
background (CMB) temperature maps on azimuthally symmetric patches using a fast
convolution approach. As an example of the application of the method, we show
results for the search for concentric rings with unusual variance in the 7-year
WMAP data. We re-analyse claims concerning the unusual variance profile of
rings centred at two locations on the sky that have recently drawn special
attention in the context of the conformal cyclic cosmology scenario proposed by
Penrose (2009). We extend this analysis to rings with larger radii and centred
on other points of the sky. Using the fast convolution technique enables us to
perform this search with higher resolution and a wider range of radii than in
previous studies. We show that for one of the two special points rings with
radii larger than 10 degrees have systematically lower variance in comparison
to the concordance LambdaCDM model predictions. However, we show that this
deviation is caused by the multipoles up to order l=7. Therefore, the deficit
of power for concentric rings with larger radii is yet another manifestation of
the well-known anomalous CMB distribution on large angular scales. Furthermore,
low variance rings can be easily found centred on other points in the sky. In
addition, we show also the results of a search for extremely high variance
rings. As for the low variance rings, some anomalies seem to be related to the
anomalous distribution of the low-order multipoles of the WMAP CMB maps. As
such our results are not consistent with the conformal cyclic cosmology
scenario.Comment: 12 pages, 11 figures, 1 table. Published in MNRAS. This research was
supported by the Agence Nationale de la Recherche (ANR-08-CEXC-0002-01
FFT Interpolation from Nonuniform Samples Lying in a Regular Grid
This paper presents a method to interpolate a periodic band-limited signal
from its samples lying at nonuniform positions in a regular grid, which is
based on the FFT and has the same complexity order as this last algorithm. This
kind of interpolation is usually termed "the missing samples problem" in the
literature, and there exists a wide variety of iterative and direct methods for
its solution. The one presented in this paper is a direct method that exploits
the properties of the so-called erasure polynomial, and it provides a
significant improvement on the most efficient method in the literature, which
seems to be the burst error recovery (BER) technique of Marvasti's et al. The
numerical stability and complexity of the method are evaluated numerically and
compared with the pseudo-inverse and BER solutions.Comment: Submitted to the IEEE Transactions on Signal Processin
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