15,168 research outputs found

    Composite Cyclotomic Fourier Transforms with Reduced Complexities

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    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(211)(2^{11}). 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

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    We prove lower bounds of order nlognn\log n for both the problem to multiply polynomials of degree nn, 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

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    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

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    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

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    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

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    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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