We utilize a discrete version of the notion of degree of freedom to prove a sharp min-entropy-variance inequality for integer valued log-concave random variables. More specifically, we show that the geometric distribution minimizes the min-entropy within the class of log-concave probability sequences with fixed variance. As an application, we obtain a discrete R\'enyi entropy power inequality in the log-concave case, which improves a result of Bobkov, Marsiglietti and Melbourne (2022).
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Most of the literature on causality considers the structural framework of Pearl and the potential-outcome framework of Neyman and Rubin to be formally equivalent, and therefore interchangeably uses the do-notation and the potential-outcome subscript notation to write counterfactual outcomes. In this paper, we superimpose the two causal frameworks to prove that structural counterfactual outcomes and potential outcomes do not coincide in general -- not even in law. More precisely, we express the law of the potential outcomes in terms of the latent structural causal model under the fundamental assumptions of causal inference. This enables us to precisely identify when counterfactual inference is or is not equivalent between approaches, and to clarify the meaning of each kind of counterfactuals.
Optimizing multiple competing objectives is a common problem across science and industry. The inherent inextricable trade-off between those objectives leads one to the task of exploring their Pareto front. A meaningful quantity for the purpose of the latter is the hypervolume indicator, which is used in Bayesian Optimization (BO) and Evolutionary Algorithms (EAs). However, the computational complexity for the calculation of the hypervolume scales unfavorably with increasing number of objectives and data points, which restricts its use in those common multi-objective optimization frameworks. To overcome these restrictions we propose to approximate the hypervolume function with a deep neural network, which we call DeepHV. For better sample efficiency and generalization, we exploit the fact that the hypervolume is scale-equivariant in each of the objectives as well as permutation invariant w.r.t. both the objectives and the samples, by using a deep neural network that is equivariant w.r.t. the combined group of scalings and permutations. We evaluate our method against exact, and approximate hypervolume methods in terms of accuracy, computation time, and generalization. We also apply and compare our methods to state-of-the-art multi-objective BO methods and EAs on a range of synthetic benchmark test cases. The results show that our methods are promising for such multi-objective optimization tasks.
In this paper, we obtain sufficient and necessary conditions for quasi-cyclic codes with index even to be symplectic self-orthogonal. Then, we propose a method for constructing symplectic self-orthogonal quasi-cyclic codes, which allows arbitrary polynomials that coprime $x^{n}-1$ to construct symplectic self-orthogonal codes. Moreover, by decomposing the space of quasi-cyclic codes, we provide lower and upper bounds on the minimum symplectic distances of a class of 1-generator quasi-cyclic codes and their symplectic dual codes. Finally, we construct many binary symplectic self-orthogonal codes with excellent parameters, corresponding to 117 record-breaking quantum codes, improving Grassl's table (Bounds on the Minimum Distance of Quantum Codes. //www.codetables.de).
We develop a provably efficient importance sampling scheme that estimates exit probabilities of solutions to small-noise stochastic reaction-diffusion equations from scaled neighborhoods of a stable equilibrium. The moderate deviation scaling allows for a local approximation of the nonlinear dynamics by their linearized version. In addition, we identify a finite-dimensional subspace where exits take place with high probability. Using stochastic control and variational methods we show that our scheme performs well both in the zero noise limit and pre-asymptotically. Simulation studies for stochastically perturbed bistable dynamics illustrate the theoretical results.
Most identification methods of unknown parameters of linear regression equations (LRE) ensure only boundedness of a parametric error in the presence of additive perturbations, which is almost always unacceptable for practical scenarios. In this paper, a new identification law is proposed to overcome this drawback and guarantee asymptotic convergence of the unknown parameters estimation error to zero in case the mentioned additive perturbation meets special averaging conditions. Theoretical results are illustrated by numerical simulations.
We consider goodness-of-fit methods for multivariate symmetric and asymmetric stable Paretian random vectors in arbitrary dimension. The methods are based on the empirical characteristic function and are implemented both in the i.i.d. context as well as for innovations in GARCH models. Asymptotic properties of the proposed procedures are discussed, while the finite-sample properties are illustrated by means of an extensive Monte Carlo study. The procedures are also applied to real data from the financial markets.
This article investigates a local discontinuous Galerkin (LDG) method for one-dimensional and two-dimensional singularly perturbed reaction-diffusion problems on a Shishkin mesh. During this process, due to the inability of the energy norm to fully capture the behavior of the boundary layers appearing in the solutions, a balanced norm is introduced. By designing novel numerical fluxes and constructing special interpolations, optimal convergences under the balanced norm are achieved in both 1D and 2D cases. Numerical experiments support the main theoretical conclusions.
We present a relational representation of odd Sugihara chains. The elements of the algebra are represented as weakening relations over a particular poset which consists of two densely embedded copies of the rationals. Our construction mimics that of Maddux (2010) where a relational representation of the even Sugihara chains is given. An order automorphism between the two copies of the rationals is the key to ensuring that the identity element of the monoid is fixed by the involution.
A new mechanical model on noncircular shallow tunnelling considering initial stress field is proposed in this paper by constraining far-field ground surface to eliminate displacement singularity at infinity, and the originally unbalanced tunnel excavation problem in existing solutions is turned to an equilibrium one of mixed boundaries. By applying analytic continuation, the mixed boundaries are transformed to a homogenerous Riemann-Hilbert problem, which is subsequently solved via an efficient and accurate iterative method with boundary conditions of static equilibrium, displacement single-valuedness, and traction along tunnel periphery. The Lanczos filtering technique is used in the final stress and displacement solution to reduce the Gibbs phenomena caused by the constrained far-field ground surface for more accurte results. Several numerical cases are conducted to intensively verify the proposed solution by examining boundary conditions and comparing with existing solutions, and all the results are in good agreements. Then more numerical cases are conducted to investigate the stress and deformation distribution along ground surface and tunnel periphery, and several engineering advices are given. Further discussions on the defects of the proposed solution are also conducted for objectivity.
We develop randomized matrix-free algorithms for estimating partial traces. Our algorithm improves on the typicality-based approach used in [T. Chen and Y-C. Cheng, Numerical computation of the equilibrium-reduced density matrix for strongly coupled open quantum systems, J. Chem. Phys. 157, 064106 (2022)] by deflating important subspaces (e.g. corresponding to the low-energy eigenstates) explicitly. This results in a significant variance reduction for matrices with quickly decaying singular values. We then apply our algorithm to study the thermodynamics of several Heisenberg spin systems, particularly the entanglement spectrum and ergotropy.
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