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Simply Exponential Approximation of the Permanent of Positive Semidefinite Matrices
[article]

2017
*
arXiv
*
pre-print

We design a deterministic polynomial time c^n

arXiv:1704.03486v1
fatcat:bspbnw7jlrdoznuqyiqnoasi7u
*approximation*algorithm for*the**permanent**of**positive**semidefinite**matrices*where c=e^γ+1≃ 4.84. ... We write a natural convex relaxation and show that its optimum solution gives a c^n*approximation**of**the**permanent*. ... Prior to our paper, no efficient algorithm (deterministic, randomized, or quantum) was known for*simply**exponential**approximation**of**the**permanent**of*general*positive**semidefinite**matrices*. ...##
###
Quantum-inspired algorithm for estimating the permanent of positive semidefinite matrices

2017
*
Physical Review A
*

Our algorithm then

doi:10.1103/physreva.96.022329
fatcat:dcwwa7ybtvap5da4hej4lvfj2e
*approximates**the*matrix*permanent*from*the*corresponding sample mean and is shown to run in polynomial time for various sets*of*Hermitian*positive**semidefinite**matrices*, achieving a ... We construct a quantum-inspired classical algorithm for computing*the**permanent**of*Hermitian*positive**semidefinite**matrices*, by exploiting a connection between these mathematical structures and*the*boson ... to*approximate**the**permanent**of*Hermitian*positive**semidefinite**matrices*. ...##
###
A simple polynomial time algorithm to approximate the permanent within a simply exponential factor
[article]

1997
*
arXiv
*
pre-print

We present a simple randomized polynomial time algorithm to

arXiv:math/9704218v1
fatcat:4az67x6idffenlueils2fvrkkq
*approximate**the*mixed discriminant*of*n*positive**semidefinite*n × n*matrices*within a factor 2^O(n). ... When applied to*approximating**the**permanent*,*the*algorithm turns out to be a simple modification*of**the*well-known Godsil-Gutman estimator. ... Hence we get a randomized polynomial time algorithm*approximating**the*mixed discriminant*of**positive**semidefinite**matrices*(and hence*the**permanent**of*a non-negative matrix) within a*simply**exponential*...##
###
Maximizing Products of Linear Forms, and The Permanent of Positive Semidefinite Matrices
[article]

2021
*
arXiv
*
pre-print

We show that this convex program is also a relaxation

arXiv:2002.04149v2
fatcat:ra37myimgrfn3e5ro5iqqqonrq
*of**the**permanent**of*Hermitian*positive**semidefinite*(HPSD)*matrices*. ... By analyzing a constructive randomized rounding algorithm, we obtain an improved multiplicative*approximation*factor to*the**permanent**of*HPSD*matrices*, as well as computationally efficient certificates ... [AGGS17] gave*the*first polynomial-time algorithm for*approximating**the**permanent**of*HPSD*matrices*with a*simply**exponential*multiplicative*approximation*factor*of*n! ...##
###
Relative entropy optimization and its applications

2016
*
Mathematical programming
*

We provide solutions based on REPs to a range

doi:10.1007/s10107-016-0998-2
fatcat:5xjfffz5yjcxrpsh5fslvq24ce
*of*problems such as*permanent*maximization, robust optimization formulations*of*GPs, and hitting-time estimation in dynamical systems. ... We conclude with a discussion*of*quantum analogs*of**the*relative entropy function, including a review*of**the*similarities and distinctions with respect to*the*classical case. ... Acknowledgements*The*authors would like to thank Pablo Parrilo and Yong-Sheng Soh for helpful conversations, and Leonard Schulman for pointers to*the*literature on Von-Neumann entropy. ...##
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A Deterministic Algorithm for Approximating the Mixed Discriminant and Mixed Volume, and a Combinatorial Corollary

2002
*
Discrete & Computational Geometry
*

We present a deterministic polynomial-time algorithm that computes

doi:10.1007/s00454-001-0083-2
fatcat:kkaajcxvabfv7fjwjscpegsqjy
*the*mixed discriminant*of*an n-tuple*of**positive**semidefinite**matrices*to within an*exponential*multiplicative factor. ... To this end we extend*the*notion*of*doubly stochastic matrix scaling to a larger class*of*n-tuples*of**positive**semidefinite**matrices*, and provide a polynomial-time algorithm for this scaling. ...*The*realistic goal, then, is to try and*the**permanent*efficiently*approximate*as well as possible, for large classes*of**matrices*. How well can*the**permanent*be*approximated*in polynomial time? ...##
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When is the Stability and Complexity of a mixed Discriminant is described
[article]

2019
*
Zenodo
*

We show that

doi:10.5281/zenodo.5835395
fatcat:xatvk6u3cjh5zlf75wybsiy3da
*the*mixed discriminant*of*n*positive**semidefinite*n×n real symmetric*matrices*can be*approximated*within a relative error > 0 in quasipolynomial time, provided*the*distance*of*each matrix ... As is shown by Gurvits, for m = 2*the*problem is #P-hard and covers*the*problem*of*computing*the*mixed discriminant*of**positive**semidefinite**matrices**of*rank 2 ... for many inspiring conversations about mixed discriminants during*the*"Geometry*of*Polynomials" program at*the*Simons Institute for*the*Theory*of*Computing. ...##
###
Approximation of the joint spectral radius using sum of squares

2008
*
Linear Algebra and its Applications
*

We provide a bound on

doi:10.1016/j.laa.2007.12.027
fatcat:p2rz22sl2jg3pbxiepwn6gbcjq
*the*quality*of**the**approximation*that unifies several earlier results and is independent*of**the*number*of**matrices*. ... We provide an asymptotically tight, computationally efficient*approximation**of**the*joint spectral radius*of*a set*of**matrices*using sum*of*squares (SOS) programming. ... Acknowledgements We thank*the*referees for their careful reading*of**the*manuscript, and their many useful suggestions. ...##
###
Stability and complexity of mixed discriminants
[article]

2019
*
arXiv
*
pre-print

We show that

arXiv:1806.05105v2
fatcat:gzlsceryc5ewtoim47gmmg5vge
*the*mixed discriminant*of*n*positive**semidefinite*n × n real symmetric*matrices*can be*approximated*within a relative error ϵ >0 in quasi-polynomial n^O( n -ϵ) time, provided*the*distance ... As is shown by Gurvits, for m=2*the*problem is #P-hard and covers*the*problem*of*computing*the*mixed discriminant*of**positive**semidefinite**matrices**of*rank 2. ... for many inspiring conversations about mixed discriminants during*the*"Geometry*of*Polynomials" program at*the*Simons Institute for*the*Theory*of*Computing. ...##
###
Inapproximability of Positive Semidefinite Permanents and Quantum State Tomography
[article]

2021
*
arXiv
*
pre-print

In

arXiv:2111.03142v1
fatcat:bltbloylj5ai5ikrdjynqofaxy
*the*process, we find that it reduces to*the*problem*of**approximately*computing*the**permanent**of*a Hermitian*positive**semidefinite*(HPSD) matrix. ... This implies that HPSD*permanents*are also NP-Hard to*approximate*, resolving a standing question with applications in quantum information and BosonSampling. ...*approximate**the**permanent**of**positive**semidefinite**matrices*within a factor*of*C. ...##
###
Page 643 of Mathematical Reviews Vol. , Issue 2000a
[page]

2000
*
Mathematical Reviews
*

*of*

*positive*

*semidefinite*

*matrices*. ... I.] (1-MI; Ann Arbor, MI) Polynomial time algorithms to

*approximate*

*permanents*and mixed discriminants within a

*simply*

*exponential*factor. ...

##
###
Classical deterministic complexity of Edmonds' Problem and quantum entanglement

2003
*
Proceedings of the thirty-fifth ACM symposium on Theory of computing - STOC '03
*

*The*notion

*of*

*Positive*operator, central in Quantum Theory, is a natural generalization

*of*

*matrices*with nonnegative entries. (Here operator refers to maps from

*matrices*to

*matrices*.) ... First, we reformulate

*the*Edmonds Problem in terms

*of*

*of*completely

*positive*operators, or equivalently, in terms

*of*bipartite density

*matrices*. ... time algorithm to

*approximate*within a

*simply*

*exponential*factor quantum

*permanents*

*of*separable unnormalized bipartite density

*matrices*(more details on this matter can be found in [7] A Proof

*of*...

##
###
Classical deterministic complexity of Edmonds' Problem and quantum entanglement

2003
*
Proceedings of the thirty-fifth ACM symposium on Theory of computing - STOC '03
*

*The*notion

*of*

*Positive*operator, central in Quantum Theory, is a natural generalization

*of*

*matrices*with nonnegative entries. (Here operator refers to maps from

*matrices*to

*matrices*.) ... First, we reformulate

*the*Edmonds Problem in terms

*of*

*of*completely

*positive*operators, or equivalently, in terms

*of*bipartite density

*matrices*. ... time algorithm to

*approximate*within a

*simply*

*exponential*factor quantum

*permanents*

*of*separable unnormalized bipartite density

*matrices*(more details on this matter can be found in [7] A Proof

*of*...

##
###
Classical deterministic complexity of Edmonds' problem and Quantum Entanglement
[article]

2003
*
arXiv
*
pre-print

*The*main subject here is

*the*so-called Edmonds' problem

*of*deciding if a given linear subspace

*of*square

*matrices*contains a nonsingular matrix . ... This property is shown to be very closely related to

*the*separability

*of*bipartite mixed states . One

*of*

*the*main tools used in

*the*paper is

*the*Quantum

*Permanent*introduced in quant-ph/0201022 . ... time algorithm to

*approximate*within a

*simply*

*exponential*factor quantum

*permanents*

*of*separable unnormalized bipartite density

*matrices*(more details on this matter can be found in [7] A Proof

*of*...

##
###
A polynomial time algorithm to approximate the mixed volume within a simply exponential factor
[article]

2009
*
arXiv
*
pre-print

for

arXiv:cs/0702013v4
fatcat:waymgofs3bcq3cdg2ubwjtv3f4
*the**approximation**of**the*volume*of*a convex set. ... We prove*the*mixed volume analogues*of**the*Van der Waerden and Schrijver-Valiant conjectures on*the**permanent*. ... And*the*mixed volume*of*ellipsoids is*approximated*by (D(A 1 , ..., A n )) 1 2*of**the*corresponding*positive**semidefinite**matrices*A i 0. ...
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