A new class of large-sample covariance and spectral density matrix estimators is proposed based on the notion of flat-top kernels. The new estimators are shown to be higher-order accurate when higher-order accuracy is...
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A new class of large-sample covariance and spectral density matrix estimators is proposed based on the notion of flat-top kernels. The new estimators are shown to be higher-order accurate when higher-order accuracy is possible. A discussion on kernel choice is presented as well as a supporting finite-sample simulation. The problem of spectral estimation under a potential lack of finite fourth moments is also addressed. The higher-order accuracy of flat-top kernel estimators typically comes at the sacrifice of the positive semidefinite property. Nevertheless, we show how a flattop estimator can be modified to become positive semidefinite (even strictly positive definite) while maintaining its higher-order accuracy. In addition, an easy (and consistent) procedure for optimal bandwidth choice is given;this procedure estimates the optimal bandwidth associated with each individual element of the target matrix, automatically sensing (and adapting to) the underlying correlation structure.
We solve a 20-year old problem posed by Yannakakis and prove that no polynomial-size linear program (LP) exists whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to ...
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We solve a 20-year old problem posed by Yannakakis and prove that no polynomial-size linear program (LP) exists whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the cut polytope and the stable set polytope. These results were discovered through a new connection that we make between one-way quantum communication protocols and semidefinite programming reformulations of LPs.
In this article, we disprove a conjecture of Goemans and Linial;namely, that every negative type metric embeds into l(1) with constant distortion. We show that for an arbitrarily small constant delta > 0, for all l...
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In this article, we disprove a conjecture of Goemans and Linial;namely, that every negative type metric embeds into l(1) with constant distortion. We show that for an arbitrarily small constant delta > 0, for all large enough n, there is an n-point negative type metric which requires distortion at least (log log n)(1/6-delta) to embed into l(1). Surprisingly, our construction is inspired by the Unique Games Conjecture (UGC), establishing a previously unsuspected connection between probabilistically checkable proof systems (PCPs) and the theory of metric embeddings. We first prove that the UGC implies a super-constant hardness result for the (nonuniform) SPARSESTCUT problem. Though this hardness result relies on the UGC, we demonstrate, nevertheless, that the corresponding PCP reduction can be used to construct an "integrality gap instance" for SPARSESTCUT. Towards this, we first construct an integrality gap instance for a natural SDP relaxation of UNIQUEGAMES. Then we "simulate" the PCP reduction and "translate" the integrality gap instance of UNIQUEGAMES to an integrality gap instance of SPARSESTCUT. This enables us to prove a (log log n)(1/6-delta) integrality gap for SPARSESTCUT, which is known to be equivalent to the metric embedding lower bound.
The background method is a widely used technique to bound mean properties of turbulent flows rigorously. This work reviews recent advances in the theoretical formulation and numerical implementation of the method. Fir...
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The background method is a widely used technique to bound mean properties of turbulent flows rigorously. This work reviews recent advances in the theoretical formulation and numerical implementation of the method. First, we describe how the background method can be formulated systematically within a broader 'auxiliary function' framework for bounding mean quantities, and explain how symmetries of the flow and constraints such as maximum principles can be exploited. All ideas are presented in a general setting and are illustrated on Rayleigh-Benard convection between stress-free isothermal plates. Second, we review a semidefinite programming approach and a timestepping approach to optimizing bounds computationally, revealing that they are related to each other through convex duality and low-rank matrix factorization. Open questions and promising directions for further numerical analysis of the background method are also *** article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 1)'.
We consider a recently proposed optimization formulation of multi-task learning based on trace norm regularized least squares. While this problem may be formulated as a semidefinite program (SDP), its size is beyond g...
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We consider a recently proposed optimization formulation of multi-task learning based on trace norm regularized least squares. While this problem may be formulated as a semidefinite program (SDP), its size is beyond general SDP solvers. Previous solution approaches apply proximal gradient methods to solve the primal problem. We derive new primal and dual reformulations of this problem, including a reduced dual formulation that involves minimizing a convex quadratic function over an operator-norm ball in matrix space. This reduced dual problem may be solved by gradient-projection methods, with each projection involving a singular value decomposition. The dual approach is compared with existing approaches and its practical effectiveness is illustrated on simulations and an application to gene expression pattern analysis.
In this paper, we develop a numerically efficient scheme for set-membership prediction and filtering for discrete-time nonlinear systems, that takes into explicit account the effects of nonlinearities via local second...
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In this paper, we develop a numerically efficient scheme for set-membership prediction and filtering for discrete-time nonlinear systems, that takes into explicit account the effects of nonlinearities via local second-order information. The filtering scheme is based on a classical prediction/update recursion that requires at each step the solution of a convex semidefinite optimization problem. The technical results discussed in the paper build upon the recently developed paradigm of uncertain linear equations (ULE) and semidefinite relaxations.
We consider the problem of computing the global in. mum of a real polynomial f on R-n. Every global minimizer of f lies on its gradient variety, i.e., the algebraic subset of R-n where the gradient of f vanishes. If f...
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We consider the problem of computing the global in. mum of a real polynomial f on R-n. Every global minimizer of f lies on its gradient variety, i.e., the algebraic subset of R-n where the gradient of f vanishes. If f attains a minimum on R-n, it is therefore equivalent to look for the greatest lower bound of f on its gradient variety. Nie, Demmel, and Sturmfels proved recently a theorem about the existence of sums of squares certificates for such lower bounds. Based on these certificates, they find arbitrarily tight relaxations of the original problem that can be formulated as semidefinite programs and thus be solved efficiently. We deal here with the more general case when f is bounded from below but does not necessarily attain a minimum. In this case, the method of Nie, Demmel, and Sturmfels might yield completely wrong results. In order to overcome this problem, we replace the gradient variety by larger semialgebraic subsets of R-n which we call gradient tentacles. It now gets substantially harder to prove the existence of the necessary sums of squares certificates.
In this paper, we study the gradient descent-ascent method for convex-concave saddle-point problems. We derive a new non-asymptotic global convergence rate in terms of distance to the solution set by using the semidef...
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In this paper, we study the gradient descent-ascent method for convex-concave saddle-point problems. We derive a new non-asymptotic global convergence rate in terms of distance to the solution set by using the semidefinite programming performance estimation method. The given convergence rate incorporates most parameters of the problem and it is exact for a large class of strongly convex-strongly concave saddle-point problems for one iteration. We also investigate the algorithm without strong convexity and we provide some necessary and sufficient conditions under which the gradient descent-ascent enjoys linear convergence.
This paper proposes a reliability analysis framework that accounts for the error caused by characterizing a data set as a probabilistic model. To this end we model the uncertain parameters as a probability box (p-box)...
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This paper proposes a reliability analysis framework that accounts for the error caused by characterizing a data set as a probabilistic model. To this end we model the uncertain parameters as a probability box (p-box) of Sliced-Normal (SN) distributions. This class of distributions enables the analyst to characterize complex parameter dependencies with minimal modeling effort. The p-box, which spans the maximum likelihood and the moment-bounded maximum entropy estimates, yields a range of failure probability values. This range shrinks as the amount of data available increases. In addition, we leverage the semi-algebraic nature of the SNs to identify the most likely points of failure (MLPs). Such points allow the efficient estimation of failure probabilities using importance sampling. When the limit state functions are also semi-algebraic, semidefinite programming is used to guarantee that the computed MLPs are correct and complete, therefore ensuring that the resulting reliability analysis is accurate. This framework is applied to the reliability analysis of a truss structure subject to deflection and weight requirements.
Wedevelop a semidefinite programming method for the optimization of quantum networks, including both causal networks and networks with indefinite causal structure. Our method applies to a broad class of performance me...
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Wedevelop a semidefinite programming method for the optimization of quantum networks, including both causal networks and networks with indefinite causal structure. Our method applies to a broad class of performance measures, defined operationally in terms of interative tests set up by a verifier. We show that the optimal performance is equal to a max relative entropy, which quantifies the informativeness of the test. Building on this result, we extend the notion of conditional min-entropy from quantum states to quantum causal networks. The optimization method is illustrated in a number of applications, including the inversion, charge conjugation, and controlization of an unknown unitary dynamics. In the non-causal setting, we show a proof-of-principle application to the maximization of the winning probability in a non-causal quantum game.
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