Positivstellensätze for noncommutative rational expressions

Pascoe, J. E.

doi:10.1090/proc/13773

Cited by 11 publications

(10 citation statements)

References 11 publications

(12 reference statements)

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“…Subsequent developments ultimately led to a noncommutative version of the Positivstellensatz for semialgebraic sets. We refer interested readers to Pascoe (2018) for an overview of this topic.…”

Section: Noncommutative Positivstellensatzmentioning

confidence: 99%

Recht-Ré Noncommutative Arithmetic-Geometric Mean Conjecture is False

Lai,

Lim

2020

Preprint

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Stochastic optimization algorithms have become indispensable in modern machine learning. An unresolved foundational question in this area is the difference between with-replacement sampling and without-replacement sampling -does the latter have superior convergence rate compared to the former? A groundbreaking result of Recht and Ré reduces the problem to a noncommutative analogue of the arithmeticgeometric mean inequality where n positive numbers are replaced by n positive definite matrices. If this inequality holds for all n, then withoutreplacement sampling indeed outperforms withreplacement sampling. The conjectured Recht-Ré inequality has so far only been established for n = 2 and a special case of n = 3. We will show that the Recht-Ré conjecture is false for general n. Our approach relies on the noncommutative Positivstellensatz, which allows us to reduce the conjectured inequality to a semidefinite program and the validity of the conjecture to certain bounds for the optimum values, which we show are false as soon as n = 5.

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Section: Noncommutative Positivstellensatzmentioning

confidence: 99%

Recht-Ré Noncommutative Arithmetic-Geometric Mean Conjecture is False

Lai,

Lim

2020

Preprint

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“…It is clear that extension of f to the new domain must still be given by the same formula as before. Either using the algorithms in [10,21] or by brute force, one can see that…”

Section: The Noncommutative Contextmentioning

confidence: 99%

“…We also point out that the case where R 1 = R d as a diagonal algebra and R 2 = R was explored in [22,19], and that the current work simplifies the proof of the main result of those works if we are willing to use the commutative Löwner theorem from [1] as a black box. Moreover, if we are given a rational expression, such as the Schur complement, on a nice finite dimensional operator system, such as a matrix algebra, one can apply the algorithms in [10] which make the rational convex Positivstellensatz [21] effective to check that a function is matrix monotone in our sense.…”

Section: The Noncommutative Contextmentioning

confidence: 99%

The noncommutative Löwner theorem for matrix monotone functions over operator systems

Pascoe¹

2018

Linear Algebra and its Applications

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Given a function f : (a, b) → R, Löwner's theorem states f is monotone when extended to self-adjoint matrices via the functional calculus, if and only if f extends to a self-map of the complex upper half plane. In recent years, several generalizations of Löwner's theorem have been proven in several variables. We use the relaxed Agler, M c Carthy, and Young theorem on locally matrix monotone functions in several commuting variables to generalize results in the noncommutative case. Specifically, we show that a real free function defined over an operator system must analytically continue to a noncommutative upper half plane as map into another noncommutative upper half plane.

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“…Given a monic Hermitian pencil = + 1 1 + • • • + , the associated free spectrahedron D( ) is the set of Hermitian tuples X satisfying the linear matrix inequality ( ) 0. Since every convex solution set of a noncommutative polynomial is a free spectrahedron [HM12], the following statement is called a rational convex Positivstellensatz, and it generalises its analogues in the polynomial context [HKM12] and regular rational context [Pas18].…”

Section: Introductionmentioning

confidence: 99%

Hilbert’s 17th problem in free skew fields

Volčič

2020

Forum of Mathematics, Sigma

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This paper solves the rational noncommutative analogue of Hilbert’s 17th problem: if a noncommutative rational function is positive semidefinite on all tuples of Hermitian matrices in its domain, then it is a sum of Hermitian squares of noncommutative rational functions. This result is a generalisation and culmination of earlier positivity certificates for noncommutative polynomials or rational functions without Hermitian singularities. More generally, a rational Positivstellensatz for free spectrahedra is given: a noncommutative rational function is positive semidefinite or undefined at every matricial solution of a linear matrix inequality $L\succeq 0$ if and only if it belongs to the rational quadratic module generated by L. The essential intermediate step toward this Positivstellensatz for functions with singularities is an extension theorem for invertible evaluations of linear matrix pencils.

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Positivstellensätze for noncommutative rational expressions

Cited by 11 publications

References 11 publications

Recht-Ré Noncommutative Arithmetic-Geometric Mean Conjecture is False

Recht-Ré Noncommutative Arithmetic-Geometric Mean Conjecture is False

The noncommutative Löwner theorem for matrix monotone functions over operator systems

Hilbert’s 17th problem in free skew fields

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