Page 89 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 89
COMPUTATIONAL ASPECTS OF COMMUTATIVE AND NONCOMMUTATIVE
POSITIVE POLYNOMIALS (MS-77)
semidefinite programming relaxation for the logarithmic Sobolev constant of a finite Markov
chain. This relaxation gives certified lower bounds on the logarithmic Sobolev constant. Nu-
merical experiments show that the solution to this relaxation is often very close to the true
constant. Finally, we use this relaxation to obtain a sum-of-squares proof that the logarithmic
Sobolev constant is equal to half the Poincaré constant for the specific case of a simple random
walk on the odd n-cycle, with n in 5,7,. . . ,21. Previously this was known only for n=5 and even
n.
Optimizing conditional entropies for quantum correlations
Omar Fawzi, omar.fawzi@ens-lyon.fr
Inria, France
Coauthors: Peter Brown, Hamza Fawzi
The rates of quantum cryptographic protocols are usually expressed in terms of a conditional
entropy minimized over a certain set of quantum states. In the so-called device-independent
setting, the minimization is over all the quantum states of arbitrary dimension jointly held by
the adversary and the parties that are consistent with the statistics that are seen by the parties.
We introduce new quantum divergences and use techniques from noncommutative polynomial
optimization to approximate such entropic quantities.
Based on https://arxiv.org/abs/2007.12575
Fast Semidefinite Optimization with Latent Basis Learning
Georgina Hall, georgina.hall@insead.edu
INSEAD, France
When faced with a semidefinite program (SDP), it is often the case that we do not need to
solve just one specific SDP, but rather a family of very similar problems with varying data, for
example, when solving matrix completion problems to obtain movie recommendations for users
with similar preferences. In this talk, I will present Fast Semidefinite Optimization (FSDO), a
data-driven method to quickly solve SDPs coming from the same family. Our method learns
a shared latent basis representation across the family, which is then used as input to a second-
order cone program, whose solution constitutes an approximate solution to the original SDP.
The learning is done using neural networks and leverages recent advances in differentiable
convex optimization.
Positive maps and trace polynomials from the symmetric group
Felix Huber, felix.huber@uj.edu.pl
Jagiellonian University, Poland
With techniques borrowed from quantum information theory, we develop a method to systemat-
ically obtain operator inequalities and identities in several matrix variables. These take the form
of trace polynomials: polynomial-like expressions that involve matrix monomials Xα1 · · · Xαr
and their traces tr(Xα1 · · · Xαr ). Our method rests on translating the action of the symmetric
87
POSITIVE POLYNOMIALS (MS-77)
semidefinite programming relaxation for the logarithmic Sobolev constant of a finite Markov
chain. This relaxation gives certified lower bounds on the logarithmic Sobolev constant. Nu-
merical experiments show that the solution to this relaxation is often very close to the true
constant. Finally, we use this relaxation to obtain a sum-of-squares proof that the logarithmic
Sobolev constant is equal to half the Poincaré constant for the specific case of a simple random
walk on the odd n-cycle, with n in 5,7,. . . ,21. Previously this was known only for n=5 and even
n.
Optimizing conditional entropies for quantum correlations
Omar Fawzi, omar.fawzi@ens-lyon.fr
Inria, France
Coauthors: Peter Brown, Hamza Fawzi
The rates of quantum cryptographic protocols are usually expressed in terms of a conditional
entropy minimized over a certain set of quantum states. In the so-called device-independent
setting, the minimization is over all the quantum states of arbitrary dimension jointly held by
the adversary and the parties that are consistent with the statistics that are seen by the parties.
We introduce new quantum divergences and use techniques from noncommutative polynomial
optimization to approximate such entropic quantities.
Based on https://arxiv.org/abs/2007.12575
Fast Semidefinite Optimization with Latent Basis Learning
Georgina Hall, georgina.hall@insead.edu
INSEAD, France
When faced with a semidefinite program (SDP), it is often the case that we do not need to
solve just one specific SDP, but rather a family of very similar problems with varying data, for
example, when solving matrix completion problems to obtain movie recommendations for users
with similar preferences. In this talk, I will present Fast Semidefinite Optimization (FSDO), a
data-driven method to quickly solve SDPs coming from the same family. Our method learns
a shared latent basis representation across the family, which is then used as input to a second-
order cone program, whose solution constitutes an approximate solution to the original SDP.
The learning is done using neural networks and leverages recent advances in differentiable
convex optimization.
Positive maps and trace polynomials from the symmetric group
Felix Huber, felix.huber@uj.edu.pl
Jagiellonian University, Poland
With techniques borrowed from quantum information theory, we develop a method to systemat-
ically obtain operator inequalities and identities in several matrix variables. These take the form
of trace polynomials: polynomial-like expressions that involve matrix monomials Xα1 · · · Xαr
and their traces tr(Xα1 · · · Xαr ). Our method rests on translating the action of the symmetric
87
