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In this paper we prove upper and lower bounds on the minimal spherical dispersion. In particular, we see that the inverse $N(\varepsilon,d)$ of the minimal spherical dispersion is, for fixed $\varepsilon>0$, up to logarithmic terms linear in the dimension $d$. We also derive upper and lower bounds on the expected dispersion for points chosen independently and uniformly at random from the Euclidean unit sphere.

Uncertainty in physical parameters can make the solution of forward or inverse light scattering problems in astrophysical, biological, and atmospheric sensing applications, cost prohibitive for real-time applications. For example, given a probability density in the parametric space of dimensions, refractive index and wavelength, the number of required evaluations for the expected scattering increases dramatically. In the case of dielectric and weakly absorbing spherical particles (both homogeneous and layered), we begin with a Fraunhofer approximation of the scattering coefficients consisting of Riccati-Bessel functions, and reduce it into simpler nested trigonometric approximations. They provide further computational advantages when parameterized on lines of constant optical path lengths. This can reduce the cost of evaluations by large factors $\approx$ 50, without a loss of accuracy in the integrals of these scattering coefficients. We analyze the errors of the proposed approximation, and present numerical results for a set of forward problems as a demonstration.

The generalized Biot-Brinkman equations describe the displacement, pressures and fluxes in an elastic medium permeated by multiple viscous fluid networks and can be used to study complex poromechanical interactions in geophysics, biophysics and other engineering sciences. These equations extend on the Biot and multiple-network poroelasticity equations on the one hand and Brinkman flow models on the other hand, and as such embody a range of singular perturbation problems in realistic parameter regimes. In this paper, we introduce, theoretically analyze and numerically investigate a class of three-field finite element formulations of the generalized Biot-Brinkman equations. By introducing appropriate norms, we demonstrate that the proposed finite element discretization, as well as an associated preconditioning strategy, is robust with respect to the relevant parameter regimes. The theoretical analysis is complemented by numerical examples.

In this paper, we proposed a multi-objective approach for the EEG Inverse Problem. This formulation does not need unknown parameters that involve empirical procedures. Due to the combinatorial characteristics of the problem, this alternative included evolutionary strategies to resolve it. The result is a Multi-objective Evolutionary Algorithm based on Anatomical Restrictions (MOEAAR) to estimate distributed solutions. The comparative tests were between this approach and 3 classic methods of regularization: LASSO, Ridge-L and ENET-L. In the experimental phase, regression models were selected to obtain sparse and distributed solutions. The analysis involved simulated data with different signal-to-noise ratio (SNR). The indicators for quality control were Localization Error, Spatial Resolution and Visibility. The MOEAAR evidenced better stability than the classic methods in the reconstruction and localization of the maximum activation. The norm L0 was used to estimate sparse solutions with the evolutionary approach and its results were relevant.

Under-approximations of reachable sets and tubes have received recent research attention due to their important roles in control synthesis and verification. Available under-approximation methods designed for continuous-time linear systems typically assume the ability to compute transition matrices and their integrals exactly, which is not feasible in general. In this note, we attempt to overcome this drawback for a class of linear time-invariant (LTI) systems, where we propose a novel method to under-approximate finite-time forward reachable sets and tubes utilizing approximations of the matrix exponential. In particular, we consider the class of continuous-time LTI systems with an identity input matrix and uncertain initial and input values belonging to full dimensional sets that are affine transformations of closed unit balls. The proposed method yields computationally efficient under-approximations of reachable sets and tubes with first order convergence guarantees in the sense of the Hausdorff distance. To illustrate its performance, the proposed method is implemented in three numerical examples, where linear systems of dimensions ranging between 2 and 200 are considered.

This paper discusses the estimation of the generalization gap, the difference between a generalization error and an empirical error, for overparameterized models (e.g., neural networks). We first show that a functional variance, a key concept in defining a widely-applicable information criterion, characterizes the generalization gap even in overparameterized settings where a conventional theory cannot be applied. We also propose a computationally efficient approximation of the function variance, the Langevin approximation of the functional variance (Langevin FV). This method leverages only the $1$st-order gradient of the squared loss function, without referencing the $2$nd-order gradient; this ensures that the computation is efficient and the implementation is consistent with gradient-based optimization algorithms. We demonstrate the Langevin FV numerically by estimating the generalization gaps of overparameterized linear regression and non-linear neural network models.

We study the problem of policy evaluation with linear function approximation and present efficient and practical algorithms that come with strong optimality guarantees. We begin by proving lower bounds that establish baselines on both the deterministic error and stochastic error in this problem. In particular, we prove an oracle complexity lower bound on the deterministic error in an instance-dependent norm associated with the stationary distribution of the transition kernel, and use the local asymptotic minimax machinery to prove an instance-dependent lower bound on the stochastic error in the i.i.d. observation model. Existing algorithms fail to match at least one of these lower bounds: To illustrate, we analyze a variance-reduced variant of temporal difference learning, showing in particular that it fails to achieve the oracle complexity lower bound. To remedy this issue, we develop an accelerated, variance-reduced fast temporal difference algorithm (VRFTD) that simultaneously matches both lower bounds and attains a strong notion of instance-optimality. Finally, we extend the VRFTD algorithm to the setting with Markovian observations, and provide instance-dependent convergence results that match those in the i.i.d. setting up to a multiplicative factor that is proportional to the mixing time of the chain. Our theoretical guarantees of optimality are corroborated by numerical experiments.

We study the problem of learning classification functions from noiseless training samples, under the assumption that the decision boundary is of a certain regularity. We establish universal lower bounds for this estimation problem, for general classes of continuous decision boundaries. For the class of locally Barron-regular decision boundaries, we find that the optimal estimation rates are essentially independent of the underlying dimension and can be realized by empirical risk minimization methods over a suitable class of deep neural networks. These results are based on novel estimates of the $L^1$ and $L^\infty$ entropies of the class of Barron-regular functions.

Covariance matrix estimation is a fundamental statistical task in many applications, but the sample covariance matrix is sub-optimal when the sample size is comparable to or less than the number of features. Such high-dimensional settings are common in modern genomics, where covariance matrix estimation is frequently employed as a method for inferring gene networks. To achieve estimation accuracy in these settings, existing methods typically either assume that the population covariance matrix has some particular structure, for example sparsity, or apply shrinkage to better estimate the population eigenvalues. In this paper, we study a new approach to estimating high-dimensional covariance matrices. We first frame covariance matrix estimation as a compound decision problem. This motivates defining a class of decision rules and using a nonparametric empirical Bayes g-modeling approach to estimate the optimal rule in the class. Simulation results and gene network inference in an RNA-seq experiment in mouse show that our approach is comparable to or can outperform a number of state-of-the-art proposals, particularly when the sample eigenvectors are poor estimates of the population eigenvectors.

A fundamental problem in numerical analysis and approximation theory is approximating smooth functions by polynomials. A much harder version under recent consideration is to enforce bounds constraints on the approximating polynomial. In this paper, we consider the problem of approximating functions by polynomials whose Bernstein coefficients with respect to a given degree satisfy such bounds, which implies such bounds on the approximant. We frame the problem as an inequality-constrained optimization problem and give an algorithm for finding the Bernstein coefficients of the exact solution. Additionally, our method can be modified slightly to include equality constraints such as mass preservation. It also extends naturally to multivariate polynomials over a simplex.