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Fourier meets möbius: fast subset convolution

2007
*
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing - STOC '07
*

We present a

doi:10.1145/1250790.1250801
dblp:conf/stoc/BjorklundHKK07
fatcat:hvtaeoesqfdo5hnekpxequsb5e
*fast*algorithm for the*subset**convolution*problem: given functions f and g defined on the lattice of*subsets*of an n-element set N , compute their*subset**convolution*f * g, defined for all ... In a first attempt to improve upon the direct evaluation, the*convolution*analogy suggests the natural approach to evaluate (1) as a product of some type of*Fourier*transforms of f and g via a*fast**Fourier*... Using*fast**subset**convolution*we obtain yet anotherÕ(2 n ) algorithm. ...##
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Fourier meets Möbius: fast subset convolution
[article]

2006
*
arXiv
*
pre-print

We present a

arXiv:cs/0611101v1
fatcat:zvrm7bgjwbew7mw5pocjriqsku
*fast*algorithm for the*subset**convolution*problem: given functions f and g defined on the lattice of*subsets*of an n-element set N, compute their*subset**convolution*f*g, defined for all S⊆ ... Via*Möbius*transform and inversion, our algorithm evaluates the*subset**convolution*in O(n^2 2^n) additions and multiplications, substantially improving upon the straightforward O(3^n) algorithm. ... In a first attempt to improve upon the direct evaluation, the*convolution*analogy suggests the natural approach to evaluate (1) as a product of some type of*Fourier*transforms of f and g via a*fast**Fourier*...##
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Fast Algorithms for Join Operations on Tree Decompositions
[article]

2020
*
arXiv
*
pre-print

In this paper, we review two different approaches that have appeared in the literature about computations for the join nodes: one using

arXiv:2006.01588v1
fatcat:36zw37waejhrhjaib4tpquhbyu
*fast*zeta and*Möbius*transforms and one using*fast**Fourier*transforms ... We combine these approaches to obtain new, faster algorithms for a broad class of vertex*subset*problems known as the [σ,ρ]-domination problems. ... One such method is using*fast*zeta and*Möbius*transforms in a way that is similar to the well-known*fast**subset**convolution*algorithm by Björklund et al [2] . ...##
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Fourier inversion for finite inverse semigroups
[article]

2013
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arXiv
*
pre-print

Finally, we give

arXiv:1212.6462v2
fatcat:akf63gi2gbenbexzqt6ema2a4i
*fast*inverse*Fourier*transforms for the symmetric inverse monoid and its wreath product by arbitrary finite groups, as well as*fast**Fourier*and inverse*Fourier*transforms for the planar ... Next, we describe a general approach to the construction of*fast*inverse*Fourier*transforms for finite inverse semigroups complementary to an approach to FFTs given in previous work. ... Among other applications, algorithms for computing*fast**Fourier*transforms (FFTs) and*fast*inverse*Fourier*transforms (FIFTs) give rise to efficient algorithms for computing the*convolution*of functions ...##
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Clifford Algebras Meet Tree Decompositions

2018
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Algorithmica
*

We introduce the non-commutative

doi:10.1007/s00453-018-0489-3
pmid:30872883
pmcid:PMC6386049
fatcat:nv3dazhwrngczjn4bcr5hkxdqu
*subset**convolution*-a*convolution*of functions useful when working with determinant-based algorithms. ... I would also like to thank Paul Leopardi for helping me understand the*fast**Fourier*-like transform for Clifford algebras. ... This is actually the case in the*Fast**Subset**Convolution*[2] , where the isomorphism is given by the*Möbius*transform. ...##
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There are 1,132,835,421,602,062,347 nonisomorphic one-factorizations ofK14

2009
*
Journal of combinatorial designs (Print)
*

Koivisto,

doi:10.1002/jcd.20188
fatcat:gqreaywcwnacjhodu5oebzzn64
*Fourier**meets**Möbius*:*fast**subset**convolution*, Proceedings of the 39th Annual ACM Symposium on Theory of Computing (San Diego, CA, June 11-13, 2007), Association for Computing Machinery, New ... For α < 1/4 a positive answer is obtained by combining a trimmed*fast**subset**convolution*of f 1 , f 2 with the*fast*intersection transform of f 3 , where f 1 , f 2 , f 3 are indicator functions of F 1 ...##
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Fast Fourier transforms for finite inverse semigroups

2010
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Journal of Algebra
*

We extend the theory of

doi:10.1016/j.jalgebra.2009.11.031
fatcat:ifzrvo6i45fxpn77jlu3r2b5ru
*fast**Fourier*transforms on finite groups to finite inverse semigroups. ... on its maximal subgroups and a*fast*zeta transform on its poset structure. ... In this case, the isomorphism (1) is the usual discrete*Fourier*transform: Here is the general*convolution*theorem. Proof. ...##
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Homomorphic Hashing for Sparse Coefficient Extraction
[chapter]

2012
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Lecture Notes in Computer Science
*

investigate the systematic use of homomorphic hash functions to combine the best of these methods and obtain improved space-efficient algorithms for problems including LINEAR SAT, SET PARTITION, and

doi:10.1007/978-3-642-33293-7_15
fatcat:fc3uzfmcsraj7k7f4bkw3t2r4i
*SUBSET*... Algorithm*mobius*clearly runs in O ⋆ (|P | 2 ) time, so this procedure*meets*the claimed time bound. ⊓ ⊔ Proof (of Theorem 3, self-contained). Recall that we already know that supp(φ 0 ) = {∅}. ... Then by Theorem 26 and Lemma 30, the zeta-transform of the output of C ′ can be computed*fast*using point-wise multiplication, and then using*Möbius*inversion the original output can be computed. ...##
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Homomorphic Hashing for Sparse Coefficient Extraction
[article]

2012
*
arXiv
*
pre-print

investigate the systematic use of homomorphic hash functions to combine the best of these methods and obtain improved space-efficient algorithms for problems including LINEAR SAT, SET PARTITION, and

arXiv:1203.4063v1
fatcat:in5o4y77zrc2boiljz777ybbiq
*SUBSET*... Algorithm*mobius*clearly runs in O ⋆ (|P | 2 ) time, so this procedure*meets*the claimed time bound. ⊓ ⊔ Proof (of Theorem 3, self-contained). Recall that we already know that supp(φ 0 ) = {∅}. ... Then by Theorem 26 and Lemma 30, the zeta-transform of the output of C ′ can be computed*fast*using point-wise multiplication, and then using*Möbius*inversion the original output can be computed. ...##
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Constant Curvature Graph Convolutional Networks
[article]

2020
*
arXiv
*
pre-print

Here, we bridge this gap by proposing mathematically grounded generalizations of graph

arXiv:1911.05076v3
fatcat:qfvdahbvxbakpjacnq3jgtcfpq
*convolutional*networks (GCN) to (products of) constant curvature spaces. ... the element-wise product since*convolutions*become products in the*Fourier*domain. ... In order to define a*convolution*for graphs, we shift from the vertex domain to the*Fourier*domain: x G y = U U T x U T y Note thatx = U T x andŷ = U T y are the graph*Fourier*representations and we use ...##
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Diffraction from visible lattice points and k-th power free integers
[article]

2000
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arXiv
*
pre-print

The older numerical calculations in [29] , using the

arXiv:math/9906132v2
fatcat:y7bqyd2pjveszg7aesl43ny3ni
*fast**Fourier*transform, suffer from an insufficient resolution and are misleading. ... In this article, we will only*meet*the simple case that ν is a Dirac comb ω S . ... which weak*-converges to the diffraction spectrum of V , since the*Fourier*transform operator is weak*-continuous. ...##
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Constrained L^2-approximation by polynomials on subsets of the circle
[article]

2017
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arXiv
*
pre-print

We study best approximation to a given function, in the least square sense on a

arXiv:1710.10808v1
fatcat:3sxemyd77zbu7ipdxvatecw5wi
*subset*of the unit circle, by polynomials of given degree which are pointwise bounded on the complementary*subset*. ... More precisely, recall that a linear time-invariant dynamical system is just a*convolution*operator, hence the*Fourier*-Laplace transform of its output is that of its input times the*Fourier*-Laplace transform ... Clearly E n ⊂ F are convex and nonempty*subsets*of L 2 (I), as they contain 0. ...##
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Constrained L2-Approximation by Polynomials on Subsets of the Circle
[chapter]

2018
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Fields Institute Communications
*

We study best approximation to a given function, in the least square sense on a

doi:10.1007/978-1-4939-7543-3_8
fatcat:mkpzrikfj5gkdi3tjjroyiwatm
*subset*of the unit circle, by polynomials of given degree which are pointwise bounded on the complementary*subset*. ... More precisely, recall that a linear time-invariant dynamical system is just a*convolution*operator, hence the*Fourier*-Laplace transform of its output is that of its input times the*Fourier*-Laplace transform ... Clearly E n ⊂ F are convex and nonempty*subsets*of L 2 (I), as they contain 0. ...##
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From EMI to AI: a brief history of commercial CT reconstruction algorithms

2021
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Journal of Medical Imaging
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Bracewell was a leading expert on

doi:10.1117/1.jmi.8.5.052111
pmid:34660842
pmcid:PMC8492478
fatcat:kgw6hlnptnfr5nsmn2kajhu22i
*Fourier*transforms, and published a classic text on them in 1965, 7 but he also understood that they were computationally expensive at the time, as the*fast**Fourier*... the use of*Fourier*transforms. ...##
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Table of Contents

2020
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IEEE Transactions on Signal Processing
*

Tang 1197 Generalized

doi:10.1109/tsp.2020.3042287
fatcat:nh7viihaozhd7li3txtadnx5ui
*Fast*-*Convolution*-Based Filtered-OFDM: Techniques and Application to 5G New Radio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... Pesavento 3194 Dynamic Sensor*Subset*Selection for Centralized Tracking of an IID Process .. . . . . . . . . A. Chattopadhyay and U. ...
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