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In this work we present deep learning implementations of two popular theoretical constrained optimization algorithms in infinite dimensional Hilbert spaces, namely, the penalty and the augmented Lagrangian methods. We test these algorithms on some toy problems originating in either calculus of variations or physics. We demonstrate that both methods are able to produce decent approximations for the test problems and are comparable in terms of different errors. Leveraging the common occurrence of the Lagrange multiplier update rule being computationally less expensive than solving subproblems in the penalty method, we achieve significant speedups in cases when the output of the constraint function is itself a function.

In this paper, we propose an iterative convolution-thresholding method (ICTM) based on prediction-correction for solving the topology optimization problem in steady-state heat transfer equations. The problem is formulated as a constrained minimization problem of the complementary energy, incorporating a perimeter/surface-area regularization term, while satisfying a steady-state heat transfer equation. The decision variables of the optimization problem represent the domains of different materials and are represented by indicator functions. The perimeter/surface-area term of the domain is approximated using Gaussian kernel convolution with indicator functions. In each iteration, the indicator function is updated using a prediction-correction approach. The prediction step is based on the variation of the objective functional by imposing the constraints, while the correction step ensures the monotonically decreasing behavior of the objective functional. Numerical results demonstrate the efficiency and robustness of our proposed method, particularly when compared to classical approaches based on the ICTM.

In this paper, our objective is to present a constraining principle governing the spectral properties of the sample covariance matrix. This principle exhibits harmonious behavior across diverse limiting frameworks, eliminating the need for constraints on the rates of dimension $p$ and sample size $n$, as long as they both tend to infinity. We accomplish this by employing a suitable normalization technique on the original sample covariance matrix. Following this, we establish a harmonic central limit theorem for linear spectral statistics within this expansive framework. This achievement effectively eliminates the necessity for a bounded spectral norm on the population covariance matrix and relaxes constraints on the rates of dimension $p$ and sample size $n$, thereby significantly broadening the applicability of these results in the field of high-dimensional statistics. We illustrate the power of the established results by considering the test for covariance structure under high dimensionality, freeing both $p$ and $n$.

The causal inference literature frequently focuses on estimating the mean of the potential outcome, whereas the quantiles of the potential outcome may carry important additional information. We propose a universal approach, based on the inverse estimating equations, to generalize a wide class of causal inference solutions from estimating the mean of the potential outcome to its quantiles. We assume that an identifying moment function is available to identify the mean of the threshold-transformed potential outcome, based on which a convenient construction of the estimating equation of quantiles of potential outcome is proposed. In addition, we also give a general construction of the efficient influence functions of the mean and quantiles of potential outcomes, and identify their connection. We motivate estimators for the quantile estimands with the efficient influence function, and develop their asymptotic properties when either parametric models or data-adaptive machine learners are used to estimate the nuisance functions. A broad implication of our results is that one can rework the existing result for mean causal estimands to facilitate causal inference on quantiles, rather than starting from scratch. Our results are illustrated by several examples.

In this paper, we propose the application of shrinkage strategies to estimate coefficients in the Bell regression models when prior information about the coefficients is available. The Bell regression models are well-suited for modeling count data with multiple covariates. Furthermore, we provide a detailed explanation of the asymptotic properties of the proposed estimators, including asymptotic biases and mean squared errors. To assess the performance of the estimators, we conduct numerical studies using Monte Carlo simulations and evaluate their simulated relative efficiency. The results demonstrate that the suggested estimators outperform the unrestricted estimator when prior information is taken into account. Additionally, we present an empirical application to demonstrate the practical utility of the suggested estimators.

In this paper, to the best of our knowledge, we make the first attempt at studying the parametric semilinear elliptic eigenvalue problems with the parametric coefficient and some power-type nonlinearities. The parametric coefficient is assumed to have an affine dependence on the countably many parameters with an appropriate class of sequences of functions. In this paper, we obtain the upper bound estimation for the mixed derivatives of the ground eigenpairs that has the same form obtained recently for the linear eigenvalue problem. The three most essential ingredients for this estimation are the parametric analyticity of the ground eigenpairs, the uniform boundedness of the ground eigenpairs, and the uniform positive differences between ground eigenvalues of linear operators. All these three ingredients need new techniques and a careful investigation of the nonlinear eigenvalue problem that will be presented in this paper. As an application, considering each parameter as a uniformly distributed random variable, we estimate the expectation of the eigenpairs using a randomly shifted quasi-Monte Carlo lattice rule and show the dimension-independent error bound.

Within the ViSE (Voting in Stochastic Environment) model, we study the effectiveness of majority voting in various environments. By the pit of losses paradox, majority decisions in apparently hostile environments systematically reduce the capital of society. In such cases, the basic action of ``rejecting all proposals without voting'' outperforms simple majority. We reveal another pit of losses appearing in favorable environments. Here, the simple action of ``accepting all proposals without voting'' is superior to simple majority, which thus causes a loss compared to total acceptance. We show that the second pit of losses is a mirror image of the pit of losses in hostile environments and explain this phenomenon. Technically, we consider a voting society consisting of individual agents whose strategy is supporting all proposals that increase their capital and a group whose members vote for the increase of the total group capital. According to the main result, the expected capital gain of each agent in the environment whose proposal generator $\xi$ has mean $\mu>0$ exceeds by $\mu$ their expected capital gain with generator $-\xi$. This result extends to the shift-based families of generators with symmetric distributions. The difference by $\mu$ causes symmetry relative to the basic action that rejects/accepts all proposals in unfavorable/favorable environments.

In this study, we introduce the first-of-its-kind class of tests for detecting change points in the distribution of a sequence of independent matrix-valued random variables. The tests are constructed using the weighted square integral difference of the empirical orthogonal Hankel transforms. The test statistics have a convenient closed-form expression, making them easy to implement in practice. We present their limiting properties and demonstrate their quality through an extensive simulation study. We utilize these tests for change point detection in cryptocurrency markets to showcase their practical use. The detection of change points in this context can have various applications in constructing and analyzing novel trading systems.

Laguerre spectral approximations play an important role in the development of efficient algorithms for problems in unbounded domains. In this paper, we present a comprehensive convergence rate analysis of Laguerre spectral approximations for analytic functions. By exploiting contour integral techniques from complex analysis, we prove that Laguerre projection and interpolation methods of degree $n$ converge at the root-exponential rate $O(\exp(-2\rho\sqrt{n}))$ with $\rho>0$ when the underlying function is analytic inside and on a parabola with focus at the origin and vertex at $z=-\rho^2$. As far as we know, this is the first rigorous proof of root-exponential convergence of Laguerre approximations for analytic functions. Several important applications of our analysis are also discussed, including Laguerre spectral differentiations, Gauss-Laguerre quadrature rules, the scaling factor and the Weeks method for the inversion of Laplace transform, and some sharp convergence rate estimates are derived. Numerical experiments are presented to verify the theoretical results.

When and why can a neural network be successfully trained? This article provides an overview of optimization algorithms and theory for training neural networks. First, we discuss the issue of gradient explosion/vanishing and the more general issue of undesirable spectrum, and then discuss practical solutions including careful initialization and normalization methods. Second, we review generic optimization methods used in training neural networks, such as SGD, adaptive gradient methods and distributed methods, and theoretical results for these algorithms. Third, we review existing research on the global issues of neural network training, including results on bad local minima, mode connectivity, lottery ticket hypothesis and infinite-width analysis.