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In this paper we derive sharp lower and upper bounds for the covariance of two bounded random variables when knowledge about their expected values, variances or both is available. When only the expected values are known, our result can be viewed as an extension of the Bhatia-Davis Inequality for variances. We also provide a number of different ways to standardize covariance. For a binary pair random variables, one of these standardized measures of covariation agrees with a frequently used measure of dependence between genetic variants.

The Bayesian persuasion paradigm of strategic communication models interaction between a privately-informed agent, called the sender, and an ignorant but rational agent, called the receiver. The goal is typically to design a (near-)optimal communication (or signaling) scheme for the sender. It enables the sender to disclose information to the receiver in a way as to incentivize her to take an action that is preferred by the sender. Finding the optimal signaling scheme is known to be computationally difficult in general. This hardness is further exacerbated when there is also a constraint on the size of the message space, leading to NP-hardness of approximating the optimal sender utility within any constant factor. In this paper, we show that in several natural and prominent cases the optimization problem is tractable even when the message space is limited. In particular, we study signaling under a symmetry or an independence assumption on the distribution of utility values for the actions. For symmetric distributions, we provide a novel characterization of the optimal signaling scheme. It results in a polynomial-time algorithm to compute an optimal scheme for many compactly represented symmetric distributions. In the independent case, we design a constant-factor approximation algorithm, which stands in marked contrast to the hardness of approximation in the general case.

Generalized sampling consists in the recovery of a function $f$, from the samples of the responses of a collection of linear shift-invariant systems to the input $f$. The reconstructed function is typically a member of a finitely generated integer-shift-invariant space that can reproduce polynomials up to a given degree $M$. While this property allows for an approximation power of order $(M+1)$, it comes with a tradeoff on the length of the support of the basis functions. Specifically, we prove that the sum of the length of the support of the generators is at least $(M+1)$. Following this result, we introduce the notion of shortest basis of degree $M$, which is motivated by our desire to minimize the computational costs. We then demonstrate that any basis of shortest support generates a Riesz basis. Finally, we introduce a recursive algorithm to construct the shortest-support basis for any multi-spline space. It provides a generalization of both polynomial and Hermite B-splines. This framework paves the way for novel applications such as fast derivative sampling with arbitrarily high approximation power.

This work proposes a method of wind farm scenario generation to support real-time optimization tools and presents key findings therein. This work draws upon work from the literature and presents an efficient and scalable method for producing an adequate number of scenarios for a large fleet of wind farms while capturing both spatial and temporal dependencies. The method makes probabilistic forecasts using conditional heteroscedastic regression for each wind farm and time horizon. Past training data is transformed (using the probabilistic forecasting models) into standard normal samples. A Gaussian copula is estimated from the normalized samples and used in real-time to enforce proper spatial and temporal dependencies. The method is evaluated using historical data from MISO and performance within the MISO real-time look-ahead framework is discussed.

We present a stationary iteration method, namely Alternating Symmetric positive definite and Scaled symmetric positive semidefinite Splitting (ASSS), for solving the system arisen from finite element discretization of a distributed optimal control problem with time-periodic parabolic equations. An upper bound for the spectral radius of the iteration method is given which is always less than 1. So convergence of the ASSS iteration method is guaranteed. The induced ASSS preconditioner is applied to accelerate the convergence speed of the GMRES method for solving the system. Numerical results are presented to demonstrate the effectiveness of both the ASSS iteration method and the ASSS preconditioner.

We analyze (stochastic) gradient descent (SGD) with delayed updates on smooth quasi-convex and non-convex functions and derive concise, non-asymptotic, convergence rates. We show that the rate of convergence in all cases consists of two terms: (i) a stochastic term which is not affected by the delay, and (ii) a higher order deterministic term which is only linearly slowed down by the delay. Thus, in the presence of noise, the effects of the delay become negligible after a few iterations and the algorithm converges at the same optimal rate as standard SGD. This result extends a line of research that showed similar results in the asymptotic regime or for strongly-convex quadratic functions only. We further show similar results for SGD with more intricate form of delayed gradients---compressed gradients under error compensation and for local~SGD where multiple workers perform local steps before communicating with each other. In all of these settings, we improve upon the best known rates. These results show that SGD is robust to compressed and/or delayed stochastic gradient updates. This is in particular important for distributed parallel implementations, where asynchronous and communication efficient methods are the key to achieve linear speedups for optimization with multiple devices.

In this paper, the deployment of federated learning (FL) is investigated in an energy harvesting wireless network in which the base station (BS) employs massive multiple-input multiple-output (MIMO) to serve a set of users powered by independent energy harvesting sources. Since a certain number of users may not be able to participate in FL due to the interference and energy constraints, a joint energy management and user scheduling problem in FL over wireless systems is formulated. This problem is formulated as an optimization problem whose goal is to minimize the FL training loss via optimizing user scheduling. To find how the factors such as transmit power and number of scheduled users affect the training loss, the convergence rate of the FL algorithm is first analyzed. Given this analytical result, the user scheduling and energy management optimization problem can be decomposed, simplified, and solved. Further, the system model is extended by considering multiple BSs. Hence, a joint user association and scheduling problem in FL over wireless systems is studied. The optimal user association problem is solved using the branch-and-bound technique. Simulation results show that the proposed user scheduling and user association algorithm can reduce training loss compared to a standard FL algorithm.

We consider multilevel decompositions of piecewise constants on simplicial meshes that are stable in $H^{-s}$ for $s\in (0,1)$. Proofs are given in the case of uniformly and locally refined meshes. Our findings can be applied to define local multilevel diagonal preconditioners that lead to bounded condition numbers (independent of the mesh-sizes and levels) and have optimal computational complexity. Furthermore, we discuss multilevel norms based on local (quasi-)projection operators that allow the efficient evaluation of negative order Sobolev norms. Numerical examples and a discussion on several extensions and applications conclude this article.

This work presents a suitable mathematical analysis to understand the properties of convergence and bounded variation of a new { fully discrete locally conservative} Lagrangian--Eulerian {explicit} numerical scheme to the entropy solution in the sense of Kruzhkov via weak asymptotic method. We also make use of the weak asymptotic method to connect the theoretical developments with the computational approach within the practical framework of a solid numerical analysis. This method also serves to address the issue of notions of solutions, and its resulting algorithms have been proven to be effective to study nonlinear wave formations and rarefaction interactions in intricate applications. The weak asymptotic solutions we compute in this study with our novel Lagrangian--Eulerian framework are shown to coincide with classical solutions and Kruzhkov entropy solutions in the scalar case. Moreover, we present and discuss significant computational aspects by means of numerical experiments related to nontrivial problems: a nonlocal traffic model, the $2 \times 2$ symmetric Keyfitz--Kranzer system, and numerical studies via Wasserstein distance to explain shock interaction with the fundamental inviscid Burgers' model for fluids. Therefore, the proposed weak asymptotic analysis, when applied to the Lagrangian--Eulerian framework, fits in properly with the classical theory while optimizing the mathematical computations for the construction of new accurate numerical schemes.

Many factors influence speech yielding different renditions of a given sentence. Generative models, such as variational autoencoders (VAEs), capture this variability and allow multiple renditions of the same sentence via sampling. The degree of prosodic variability depends heavily on the prior that is used when sampling. In this paper, we propose a novel method to compute an informative prior for the VAE latent space of a neural text-to-speech (TTS) system. By doing so, we aim to sample with more prosodic variability, while gaining controllability over the latent space's structure. By using as prior the posterior distribution of a secondary VAE, which we condition on a speaker vector, we can sample from the primary VAE taking explicitly the conditioning into account and resulting in samples from a specific region of the latent space for each condition (i.e. speaker). A formal preference test demonstrates significant preference of the proposed approach over standard Conditional VAE. We also provide visualisations of the latent space where well-separated condition-specific clusters appear, as well as ablation studies to better understand the behaviour of the system.