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research
Sparse sum-of-squares certificates on finite abelian groups
Authors
Hamza Fawzi
Pablo A. Parrilo
James Saunderson
Publication date
3 March 2015
Publisher
'Springer Science and Business Media LLC'
Doi
Cite
View
on
arXiv
Abstract
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
^
\hat{G}
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. We apply our general result to two examples. First, in the case where
G
=
Z
2
n
G = \mathbb{Z}_2^n
G
=
Z
2
n
β
, 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
β
n
/
2
β
\lceil n/2 \rceil
β
n
/2
β
, establishing a conjecture of Laurent. Second, we consider nonnegative functions of degree d on
Z
N
\mathbb{Z}_N
Z
N
β
(when d divides N). By constructing a particular chordal cover of the d'th power of the N-cycle, we prove that any such function is a sum of squares of functions with at most
3
d
log
β‘
(
N
/
d
)
3d\log(N/d)
3
d
lo
g
(
N
/
d
)
nonzero Fourier coefficients. Dually this shows that a certain cyclic polytope in
R
2
d
\mathbb{R}^{2d}
R
2
d
with N vertices can be expressed as a projection of a section of the cone of psd matrices of size
3
d
log
β‘
(
N
/
d
)
3d\log(N/d)
3
d
lo
g
(
N
/
d
)
. Putting
N
=
d
2
N=d^2
N
=
d
2
gives a family of polytopes
P
d
β
R
2
d
P_d \subset \mathbb{R}^{2d}
P
d
β
β
R
2
d
with LP extension complexity
xc
L
P
(
P
d
)
=
Ξ©
(
d
2
)
\text{xc}_{LP}(P_d) = \Omega(d^2)
xc
L
P
β
(
P
d
β
)
=
Ξ©
(
d
2
)
and SDP extension complexity
xc
P
S
D
(
P
d
)
=
O
(
d
log
β‘
(
d
)
)
\text{xc}_{PSD}(P_d) = O(d\log(d))
xc
PS
D
β
(
P
d
β
)
=
O
(
d
lo
g
(
d
))
. To the best of our knowledge, this is the first explicit family of polytopes in increasing dimensions where
xc
P
S
D
(
P
d
)
=
o
(
xc
L
P
(
P
d
)
)
\text{xc}_{PSD}(P_d) = o(\text{xc}_{LP}(P_d))
xc
PS
D
β
(
P
d
β
)
=
o
(
xc
L
P
β
(
P
d
β
))
.Comment: 34 page
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Crossref
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info:doi/10.1109%2Fcdc.2015.74...
Last time updated on 18/02/2019