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Sums of squares certificates for polynomial moment inequalities
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Abstract
This paper introduces and develops the algebraic framework of moment polynomials, which are polynomial expressions in commuting variables and their formal mixed moments. Their positivity and optimization over probability measures supported on semialgebraic sets and subject to moment polynomial constraints is investigated. A positive solution to Hilbert's 17th problem for pseudo-moments is given. On the other hand, moment polynomials positive on actual measures are shown to be sums of squares and formal moments of squares up to arbitrarily small perturbation of their coefficients. When only measures supported on a bounded semialgebraic set are considered, a stronger algebraic certificate for moment polynomial positivity is derived. This result gives rise to a converging hierarchy of semidefinite programs for moment polynomial optimization. Finally, as an application, two nonlinear Bell inequalities from quantum physics are settled.
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We introduce Bell inequalities based on covariance, one of the most common measures of correlation. Explicit examples are discussed, and violations in quantum theory are demonstrated. A crucial feature of these covariance Bell inequalities is their nonlinearity; this has nontrivial consequences for the derivation of their local bound, which is not reached by deterministic local correlations. For our simplest inequality, we derive analytically tight bounds for both local and quantum correlations. An interesting application of covariance Bell inequalities is that they can act as "shared randomness witnesses": specifically, the value of the Bell expression gives device-independent lower bounds on both the dimension and the entropy of the shared random variable in a local model.
A classical result of Tchakalo on the existence of exact quadra- ture formulae up to a given degree is extended to positive measures without compact support. A criterion for the existence of Gaussian quadratures for a class of such measures is also derived from the main proof.
A survey of the theory of k-hyponormal operators starts with the construction of a polynomially hyponorrnal operator which is not subnormal. This is
achieved via a natural dictionary between positive functionals on specific convex cones of polynomials and linear bounded
operators acting on a Hilbert space, with a distinguished cyclic vector. The class of unilateral weighted shifts provides
an optimal framework for studying k-hyponormality. Non-trivial links with the theory of Toeplitz operators on Hardy space are also exposed in detail. A good
selection of intriguing open problems, with precise references to prior works and partial solutions, is offered.
Mathematics Subject Classification (2000)Primary 47B20–Secondary 47B35–47B37–46A55–30E05
We show that every real polynomial f nonnegative on [−1,1]
n
can be approximated in the l
1-norm of coefficients, by a sequence of polynomials {fer}{\{f_{\epsilon r}\}} that are sums of squares (s.o.s). This complements the existence of s.o.s. approximations in the denseness result of Berg,
Christensen and Ressel, as we provide a very simple and explicit approximation sequence. Then we show that if the moment problem holds for a basic closed semi-algebraic set
KS Ì \mathbbRn{K_{S} \subset \mathbb{R}^n} with nonempty interior, then every polynomial nonnegative on K
S
can be approximated in a similar fashion by elements from the corresponding preordering. Finally, we show that the degree
of the perturbation in the approximating sequence depends on e{\epsilon} as well as the degree and the size of coefficients of the nonnegative polynomial f, but not on the specific values of its coefficients.
This paper derives exponential concentration inequalities and polynomial
moment inequalities for the spectral norm of a random matrix. The analysis
requires a matrix extension of the scalar concentration theory developed by
Sourav Chatterjee using Stein's method of exchangeable pairs. When applied to a
sum of independent random matrices, this approach yields matrix generalizations
of the classical inequalities due to Hoeffding, Bernstein, Khintchine, and
Rosenthal. The same technique delivers bounds for sums of dependent random
matrices and more general matrix-valued functions of dependent random
variables.
This paper presents a study of the large-sample behavior of the posterior distribution of a structural parameter which is partially identified by moment inequalities. The posterior density is derived based on the limited information likelihood. The posterior distribution converges to zero exponentially fast on any -contraction outside the identified region. Inside, it is bounded below by a positive constant if the identified region is assumed to have a nonempty interior. Our simulation evidence indicates that the Bayesian approach has advantages over frequentist methods, in the sense that, with a proper choice of the prior, the posterior provides more information about the true parameter inside the identified region. We also address the problem of moment and model selection. Our optimality criterion is the maximum posterior procedure and we show that, asymptotically, it selects the true moment/model combination with the most moment inequalities and the simplest model. Comment: Published in at http://dx.doi.org/10.1214/09-AOS714 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)
A general method for obtaining moment inequalities for functions of independent random variables is presented. It is a generalization of the entropy method which has been used to derive concentration inequalities for such functions [Boucheron, Lugosi and Massart Ann. Probab. 31 (2003) 1583–1614], and is based on a generalized tensorization inequality due to Latała and Oleszkiewicz [Lecture Notes in Math. 1745 (2000) 147–168]. The new inequalities prove to be a versatile tool in a wide range of applications. We illustrate the power of the method by showing how it can be used to effortlessly re-derive classical inequalities including Rosenthal and Kahane–Khinchine-type inequalities for sums of independent random variables, moment inequalities for suprema of empirical processes and moment inequalities for Rademacher chaos and U-statistics. Some of these corollaries are apparently new. In particular, we generalize Talagrand’s exponential inequality for Rademacher chaos of order 2 to any order. We also discuss applications for other complex functions of independent random variables, such as suprema of Boolean polynomials which include, as special cases, subgraph counting problems in random graphs.
We propose a semidefinite optimization approach to the problem of deriving tight moment inequalities for P(X is an element of S), for a set S defined by polynomial inequalities and a random vector X de. ned on Omega subset of R-n that has a given collection of up to kth- order moments. In the univariate case, we provide optimal bounds on P(X is an element of S), when the first k moments of X are given, as the solution of a semidefinite optimization problem in k + 1 dimensions. In the multivariate case, if the sets S and. are given by polynomial inequalities, we obtain an improving sequence of bounds by solving semidefinite optimization problems of polynomial size in n, for fixed k. We characterize the complexity of the problem of deriving tight moment inequalities. We show that it is NP-hard to find tight bounds for k >= 4 and Omega = R-n and for k >= 2 and Omega = R-+(n), when the data in the problem is rational. For k = 1 and Omega = R-+(n) we show that we can find tight upper bounds by solving n convex optimization problems when the set S is convex, and we provide a polynomial time algorithm when S and Omega are unions of convex sets, over which linear functions can be optimized efficiently. For the case k = 2 and Omega = R-n, we present an efficient algorithm for finding tight bounds when S is a union of convex sets, over which convex quadratic functions can be optimized efficiently.
This paper provides conditions under which the inequality constraints generated by either single agent optimizing behavior, or by the Nash equilibria of multiple agent problems, can be used as a basis for estimation and inference. We also add to the econometric literature on inference in models defined by inequality constraints by providing a new specification test and methods of inference for the boundaries of the model's identified set. Two applications illustrate how the use of inequality constraints can simplify the problem of obtaining estimators from complex behavioral models of substantial applied interest.
We consider the problem of finding the unconstrained global minimum of a realvalued polynomial p(x) : R R, as well as the global minimum of p(x), in a compact set K defined by polynomial inequalities. It is shown that this problem reduces to solving an (often finite) sequence of convex linear matrix inequality (LMI) problems. A notion of Karush--Kuhn--Tucker polynomials is introduced in a global optimality condition. Some illustrative examples are provided. Key words. global optimization, theory of moments and positive polynomials, semidefinite programming AMS subject classifications. 90C22, 90C25 PII. S1052623400366802 1.
We are interested in the problem of characterizing the correlations that arise when performing local measurements on separate quantum systems. In a previous work [Phys. Rev. Lett. 98, 010401 (2007)], we introduced an infinite hierarchy of conditions necessarily satisfied by any set of quantum correlations. Each of these conditions could be tested using semidefinite programming. We present here new results concerning this hierarchy. We prove in particular that it is complete, in the sense that any set of correlations satisfying every condition in the hierarchy has a quantum representation in terms of commuting measurements. Although our tests are conceived to rule out non-quantum correlations, and can in principle certify that a set of correlations is quantum only in the asymptotic limit where all tests are satisfied, we show that in some cases it is possible to conclude that a given set of correlations is quantum after performing only a finite number of tests. We provide a criterion to detect when such a situation arises, and we explain how to reconstruct the quantum states and measurement operators reproducing the given correlations. Finally, we present several applications of our approach. We use it in particular to bound the quantum violation of various Bell inequalities.
We review approaches to identification and inference on models in Industrial Organization with partial identification and/or moment inequalities. Often, such approaches are intentionally built directly on assumptions of optimizing behavior that are credible in Industrial Organization settings, while avoiding the use of strong modeling and measurement assumptions that may not be warranted. The result is an identified set for the object of interest, reflecting what the econometrician can learn from the data and assumptions. The chapter formally defines identification, reviews the assumptions underlying the identification argument, and provides examples of their use in Industrial Organization settings. We then discuss the corresponding statistical inference problem paying particular attention to practical implementation issues.
Motivated by recent progress in quantum information theory, this article aims at optimizing trace polynomials, i.e., polynomials in noncommuting variables and traces of their products. A novel Positivstellensatz certifying positivity of trace polynomials subject to trace constraints is presented, and a hierarchy of semidefinite relaxations converging monotonically to the optimum of a trace polynomial subject to tracial constraints is provided. This hierarchy can be seen as a tracial analog of the Pironio, Navascués and Acín scheme (Pironio et al. in New J. Phys. 10(7):073013, 2008) for optimization of noncommutative polynomials. The Gelfand–Naimark–Segal (GNS) construction is applied to extract optimizers of the trace optimization problem if flatness and extremality conditions are satisfied. These conditions are sufficient to obtain finite convergence of our hierarchy. The results obtained are applied to violations of polynomial Bell inequalities in quantum information theory. The main techniques used in this paper are inspired by real algebraic geometry, operator theory, and noncommutative algebra.
Network Bell experiments give rise to a form of quantum nonlocality that conceptually goes beyond Bell’s theorem. We investigate here the simplest network, known as the bilocality scenario. We depart from the typical use of the Bell state measurement in the network central node and instead introduce a family of symmetric isoentangled measurement bases that generalize the so-called “elegant joint measurement.” This leads us to report noise-tolerant quantum correlations that elude bilocal variable models. Inspired by these quantum correlations, we introduce network Bell inequalities for the bilocality scenario and show that they admit noise-tolerant quantum violations. In contrast to many previous studies of network Bell inequalities, neither our inequalities nor their quantum violations are based on standard Bell inequalities and standard quantum nonlocality. Moreover, we pave the way for an experimental realization by presenting a simple two-qubit quantum circuit for the implementation of the elegant joint measurement and our generalization.
A univariate trace polynomial is a polynomial in a variable x and formal trace symbols Tr(xj). Such an expression can be naturally evaluated on matrices, where the trace symbols are evaluated as normalized traces. This paper addresses global and constrained positivity of univariate trace polynomials on symmetric matrices of all finite sizes. A tracial analog of Artin's solution to Hilbert's 17th problem is given: a positive semidefinite univariate trace polynomial is a quotient of sums of products of squares and traces of squares of trace polynomials.
A non-commutative polynomial which is positive on a bounded semi-algebraic set of operators has a weighted sum of squares representation. This Positivstellensatz parallels similar results in the commutative case. A broader issue is, to what extent does real semi-algebraic geometry extend to non-commutative polynomials? Our “strict" Positivstellensatz is positive news, on the opposite extreme from strict positivity would be a Real Nullstellensatz. We give an example which shows that there is no non-commutative Real Nullstellensatz along certain lines. However, we include a successful type of non-commutative Nullstellensatz proved by George Bergman.
This book presents recent results on positivity and optimization of polynomials in non-commuting variables. Researchers in non-commutative algebraic geometry, control theory, system engineering, optimization, quantum physics and information science will find the unified notation and mixture of algebraic geometry and mathematical programming useful. Theoretical results are matched with algorithmic considerations; several examples and information on how to use NCSOStools open source package to obtain the results provided. Results are presented on detecting the eigenvalue and trace positivity of polynomials in non-commuting variables using Newton chip method and Newton cyclic chip method, relaxations for constrained and unconstrained optimization problems, semidefinite programming formulations of the relaxations and finite convergence of the hierarchies of these relaxations, and the practical efficiency of algorithms.
We show that every real nonnegative polynomial f can be approximated as closely as desired (in the l₁-norm of its coefficient vector) by a sequence of polynomials that are sums of squares. The novelty is that each has a simple and explicit form in terms of f and ε.
We consider a generic basic semi-algebraic subset of the space
of generalized functions, that is a set given by (not necessarily countably
many) polynomial constraints. We derive necessary and sufficient conditions for
an infinite sequence of generalized functions to be realizable on
, namely to be the moment sequence of a finite measure
concentrated on . Our approach combines the classical results
about the moment problem on nuclear spaces with the techniques recently
developed to treat the moment problem on basic semi-algebraic sets of
. In this way, we determine realizability conditions that can be
more easily verified than the well-known Haviland type conditions. Our result
completely characterizes the support of the realizing measure in terms of its
moments. As concrete examples of semi-algebraic sets of generalized functions,
we consider the set of all Radon measures and the set of all the measures
having bounded Radon-Nikodym density w.r.t. the Lebesgue measure.
Moment problems were originally studied in the one-variable case. In 1894, Stieltjes [25] showed that for K ? ?0; y#, the K-moment problem is solvable for L if and only if L ? f, # tg,# X 0 for all f ; g A R?t?. The most famous example is the case K ? R, which was solved by Hamburger in 1921 [9]. He showed that a necessary and su?-cient condition in this case is L ? f,# X 0 for all f A R?t?. In 1923, Hausdor篓 [11] studied the case K ? ?0; 1? and showed that a necessary and su?cient condition is L ? f, # tg, # ? 1 ? t#h,# X 0 for all f ; g; h A R?t?. These results can be viewed as par-
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