The Efficient Computation of Fourier Transforms on Semisimple Algebras

Maslen, David K.; Rockmore, Daniel N.; Wolff, Sarah

doi:10.1007/s00041-017-9555-5

Cited by 15 publications

(11 citation statements)

References 46 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…One can further extend these methods to analyze data on matchings, subsets, and set partitions by using the representation theory of the corresponding semisimple algebras with "group-like" structure, the Brauer, Rook monoid, and partition algebras. Fourier analysis methods on these algebras are initiated in the recent work [63].…”

Section: Tight Frames For Analyzing Data On Other Cayley Graphs Group...mentioning

confidence: 99%

Signal Processing on the Permutahedron: Tight Spectral Frames for Ranked Data Analysis

Chen,

DeJong,

Halverson

et al. 2021

Preprint

View full text Add to dashboard Cite

Ranked data sets, where m judges/voters specify a preference ranking of n objects/candidates, are increasingly prevalent in contexts such as political elections, computer vision, recommender systems, and bioinformatics. The vote counts for each ranking can be viewed as an n! data vector lying on the permutahedron, which is a Cayley graph of the symmetric group with vertices labeled by permutations and an edge when two permutations differ by an adjacent transposition. Leveraging combinatorial representation theory and recent progress in signal processing on graphs, we investigate a novel, scalable transform method to interpret and exploit structure in ranked data. We represent data on the permutahedron using an overcomplete dictionary of atoms, each of which captures both smoothness information about the data (typically the focus of spectral graph decomposition methods in graph signal processing) and structural information about the data (typically the focus of symmetry decomposition methods from representation theory). These atoms have a more naturally interpretable structure than any known basis for signals on the permutahedron, and they form a Parseval frame, ensuring beneficial numerical properties such as energy preservation. We develop specialized algorithms and open software that take advantage of the symmetry and structure of the permutahedron to improve the scalability of the proposed method, making it more applicable to the high-dimensional ranked data found in applications.

show abstract

Section: Tight Frames For Analyzing Data On Other Cayley Graphs Group...mentioning

confidence: 99%

Signal Processing on the Permutahedron: Tight Spectral Frames for Ranked Data Analysis

Chen,

DeJong,

Halverson

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…There is a rather large body of literature on this approach and it has been applied to a wide variety of group algebras and more general algebraic objects. For a fuller description of this approach and the results obtained, the reader is referred to the surveys [MR97b,Roc02], and the most recent paper in this line of work [MRW16].…”

Section: Past and Related Workmentioning

confidence: 99%

Fast Generalized DFTs for all Finite Groups

Umans

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

View full text Add to dashboard Cite

show abstract

“…A group that has resisted such exponent 1 algorithms despite a significant amount of work is SL 2 (F q ), where the best known algorithm achieves O(|G| 4/3 ) [LR92]. This group was described as a "particularly interesting and thorny special case" by Maslen, Rockmore, and Wolff in [MRW16a].…”

Section: ρ(G)mentioning

confidence: 99%

“…There is a rather large body of literature on this approach and it has been applied to a wide variety of group algebras and more general algebraic objects. For a fuller description of this approach and the results obtained, the reader is referred to the surveys [MR97b, MR97a,Roc02], and the most recent paper in this line of work [MRW16a].…”

Section: ρ(G)mentioning

confidence: 99%

A fast generalized DFT for finite groups of Lie type

Hsu¹,

Umans²

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

View full text Add to dashboard Cite

We give an arithmetic algorithm using O(|G| ω/2+o(1) ) operations to compute the generalized Discrete Fourier Transform (DFT) over group G for 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 ω is the exponent of matrix multiplication, so the exponent ω/2 is optimal if ω = 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 ω = 2) for families of linear groups of fixed dimension, and indeed the previous bestknown 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 ω = 2.We also show that ω = 2 implies a √ 2 exponent for general finite groups G, which beats the longstanding previous best upper bound (assuming ω = 2) of 3/2.

show abstract

The Efficient Computation of Fourier Transforms on Semisimple Algebras

Cited by 15 publications

References 46 publications

Signal Processing on the Permutahedron: Tight Spectral Frames for Ranked Data Analysis

Signal Processing on the Permutahedron: Tight Spectral Frames for Ranked Data Analysis

Fast Generalized DFTs for all Finite Groups

A fast generalized DFT for finite groups of Lie type

Contact Info

Product

Resources

About