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Fourier meets möbius: fast subset convolution

Andreas Björklund, Thore Husfeldt, Petteri Kaski, Mikko Koivisto
2007 Proceedings of the thirty-ninth annual ACM symposium on Theory of computing - STOC '07  
We present a 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.  ... 
doi:10.1145/1250790.1250801 dblp:conf/stoc/BjorklundHKK07 fatcat:hvtaeoesqfdo5hnekpxequsb5e

Fourier meets Möbius: fast subset convolution [article]

Andreas Björklund, Thore Husfeldt, Petteri Kaski, Mikko Koivisto
2006 arXiv   pre-print
We present a 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  ... 
arXiv:cs/0611101v1 fatcat:zvrm7bgjwbew7mw5pocjriqsku

Fast Algorithms for Join Operations on Tree Decompositions [article]

Johan M. M. van Rooij
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 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] .  ... 
arXiv:2006.01588v1 fatcat:36zw37waejhrhjaib4tpquhbyu

Fourier inversion for finite inverse semigroups [article]

Martin E. Malandro
2013 arXiv   pre-print
Finally, we give 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  ... 
arXiv:1212.6462v2 fatcat:akf63gi2gbenbexzqt6ema2a4i

Clifford Algebras Meet Tree Decompositions

Michał Włodarczyk
2018 Algorithmica  
We introduce the non-commutative 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.  ... 
doi:10.1007/s00453-018-0489-3 pmid:30872883 pmcid:PMC6386049 fatcat:nv3dazhwrngczjn4bcr5hkxdqu

There are 1,132,835,421,602,062,347 nonisomorphic one-factorizations ofK14

Petteri Kaski, Patric R. J. Östergård
2009 Journal of combinatorial designs (Print)  
Koivisto, 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  ... 
doi:10.1002/jcd.20188 fatcat:gqreaywcwnacjhodu5oebzzn64

Fast Fourier transforms for finite inverse semigroups

Martin E. Malandro
2010 Journal of Algebra  
We extend the theory of 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.  ... 
doi:10.1016/j.jalgebra.2009.11.031 fatcat:ifzrvo6i45fxpn77jlu3r2b5ru

Homomorphic Hashing for Sparse Coefficient Extraction [chapter]

Petteri Kaski, Mikko Koivisto, Jesper Nederlof
2012 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 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.  ... 
doi:10.1007/978-3-642-33293-7_15 fatcat:fc3uzfmcsraj7k7f4bkw3t2r4i

Homomorphic Hashing for Sparse Coefficient Extraction [article]

Petteri Kaski, Mikko Koivisto, Jesper Nederlof
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 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.  ... 
arXiv:1203.4063v1 fatcat:in5o4y77zrc2boiljz777ybbiq

Constant Curvature Graph Convolutional Networks [article]

Gregor Bachmann, Gary Bécigneul, Octavian-Eugen Ganea
2020 arXiv   pre-print
Here, we bridge this gap by proposing mathematically grounded generalizations of graph 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  ... 
arXiv:1911.05076v3 fatcat:qfvdahbvxbakpjacnq3jgtcfpq

Diffraction from visible lattice points and k-th power free integers [article]

Michael Baake, Peter A. B. Pleasants
2000 arXiv   pre-print
The older numerical calculations in [29] , using the 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.  ... 
arXiv:math/9906132v2 fatcat:y7bqyd2pjveszg7aesl43ny3ni

Constrained L^2-approximation by polynomials on subsets of the circle [article]

L Baratchart
2017 arXiv   pre-print
We study best approximation to a given function, in the least square sense on a 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.  ... 
arXiv:1710.10808v1 fatcat:3sxemyd77zbu7ipdxvatecw5wi

Constrained L2-Approximation by Polynomials on Subsets of the Circle [chapter]

Laurent Baratchart, Juliette Leblond, Fabien Seyfert
2018 Fields Institute Communications  
We study best approximation to a given function, in the least square sense on a 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.  ... 
doi:10.1007/978-1-4939-7543-3_8 fatcat:mkpzrikfj5gkdi3tjjroyiwatm

From EMI to AI: a brief history of commercial CT reconstruction algorithms

Patrick J. La Riviere, Carl R. Crawford
2021 Journal of Medical Imaging  
Bracewell was a leading expert on 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.  ... 
doi:10.1117/1.jmi.8.5.052111 pmid:34660842 pmcid:PMC8492478 fatcat:kgw6hlnptnfr5nsmn2kajhu22i

Table of Contents

2020 IEEE Transactions on Signal Processing  
Tang 1197 Generalized 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.  ... 
doi:10.1109/tsp.2020.3042287 fatcat:nh7viihaozhd7li3txtadnx5ui
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