Fourier Analysis on Finite Groups and Applications

Terras, Audrey

doi:10.1017/cbo9780511626265

Cited by 356 publications

(363 citation statements)

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“…In this section we present some background material needed for the paper: we first recall some of the basic results and terminology concerning Fourier analysis on finite abelian groups [Rud90,Ter99], then we review the definition of chordal graph and the main results concerning sparse positive semidefinite matrices and matrix completion. We also review some of the terminology concerning lifts of polytopes/extended formulations.…”

Section: Preliminariesmentioning

confidence: 99%

Sparse sums of squares on finite abelian groups and improved semidefinite lifts

2016

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Let G be a finite abelian group. This paper is concerned with nonnegative functions on G that are sparse with respect to the Fourier basis. We establish combinatorial conditions on subsets S and T of Fourier basis elements under which nonnegative functions with Fourier support S are sums of squares of functions with Fourier support T . Our combinatorial condition involves constructing a chordal cover of a graph related to G and S (the Cayley graph Cay( G, S)) with maximal cliques related to T . Our result relies on two main ingredients: the decomposition of sparse positive semidefinite matrices with a chordal sparsity pattern, as well as a simple but key observation exploiting the structure of the Fourier basis elements of G (the characters of G).We apply our general result to two examples. First, in the case where G = Z n 2 , by constructing a particular chordal cover of the half-cube graph, we prove that any nonnegative quadratic form in n binary variables is a sum of squares of functions of degree at most n/2 , establishing a conjecture of Laurent. Second, we consider nonnegative functions of degree d on ZN (when d divides N ). By constructing a particular chordal cover of the dth power of the N -cycle, we prove that any such function is a sum of squares of functions with at most 3d log(N/d) nonzero Fourier coefficients. Dually this shows that a certain cyclic polytope in R 2d with N vertices can be expressed as a projection of a section of the cone of positive semidefinite matrices of size 3d log(N/d). Putting N = d 2 gives a family of polytopes in R 2d with linear programming extension complexity Ω(d 2 ) and semidefinite programming extension complexity O(d log(d)). To the best of our knowledge, this is the first explicit family of polytopes (P d ) in increasing dimensions where xcPSD(P d ) = o(xcLP(P d )) (where xcPSD and xcLP are respectively the SDP and LP extension complexity).The authors are with the

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Section: Preliminariesmentioning

confidence: 99%

Sparse sums of squares on finite abelian groups and improved semidefinite lifts

2016

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“…, and [L(G)] u,v = 0, otherwise (see [9,33]). If a graph has no isolated vertex then Tr(L(G)) = n. Therefore, we can define the density matrix…”

Section: Resultsmentioning

confidence: 99%

Quantifying Disorder in Networks

Passerini

Severini

2011

Developments in Intelligent Agent Technologies and Multi-Agent Systems

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We normalize the combinatorial Laplacian of a graph by the degree sum, look at its eigenvalues as a probability distribution and then study its Shannon entropy. Equivalently, we represent a graph with a quantum mechanical state and study its von Neumann entropy. At the graph-theoretic level, this quantity may be interpreted as a measure of regularity; it tends to be larger in relation to the number of connected components, long paths and nontrivial symmetries. When the set of vertices is asymptotically large, we prove that regular graphs and the complete graph have equal entropy, and specifically it turns out to be maximum. On the other hand, when the number of edges is fixed, graphs with large cliques appear to minimize the entropy.

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“…. , i k } and for any Discrete Fourier Transforms: For background on discrete Fourier transforms in computer science, the reader is referred to [39,40]. Let f : Σ 1 × · · · × Σ n → C be any function defined over the discrete product space, we define the Fourier transform of D as, for all a ∈ Σ 1 × · · · × Σ n ,…”

Section: Preliminariesmentioning

confidence: 99%

Testing Non-uniform k-Wise Independent Distributions over Product Spaces

Rubinfeld

Xie

2010

Automata, Languages and Programming

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Abstract. A distribution D over Σ1 × · · · × Σn is called (non-uniform) k-wise independent if for any set of k indices {i1, . . . , i k } and for any z1We study the problem of testing (non-uniform) k-wise independent distributions over product spaces. For the uniform case we show an upper bound on the distance between a distribution D from the set of k-wise independent distributions in terms of the sum of Fourier coefficients of D at vectors of weight at most k. Such a bound was previously known only for the binary field. For the non-uniform case, we give a new characterization of distributions being k-wise independent and further show that such a characterization is robust. These greatly generalize the results of Alon et al.[1] on uniform k-wise independence over the binary field to non-uniform k-wise independence over product spaces. Our results yield natural testing algorithms for k-wise independence with time and sample complexity sublinear in terms of the support size when k is a constant. The main technical tools employed include discrete Fourier transforms and the theory of linear systems of congruences.A full version of this paper is available at

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Fourier Analysis on Finite Groups and Applications

Cited by 356 publications

References 0 publications

Sparse sums of squares on finite abelian groups and improved semidefinite lifts

Sparse sums of squares on finite abelian groups and improved semidefinite lifts

Quantifying Disorder in Networks

Testing Non-uniform k-Wise Independent Distributions over Product Spaces

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