Search

This paper addresses the challenges of high-dimensional non-convex optimization, particularly the inefficiencies caused by saddle points. The authors propose several techniques for detecting, evading, and optimizing in the presence of these saddle points. We begin by analyzing saddle point detection through the Hessian spectrum, showing that the likelihood of encountering saddle points increases with dimensionality. We introduce stochastic gradient perturbation, which adds noise to escape saddle points and avoid premature convergence, and emphasize the importance of gradient flow dynamics and adaptive learning rates in ensuring convergence to local minima. The paper validates these methods within constrained optimization problems and explores randomized subspace optimization, reducing search space dimensionality while maintaining global convergence efficiency. These findings offer a comprehensive framework for enhancing the reliability and efficiency of high-dimensional non-convex optimization.

This paper addresses critical challenges in machine learning, particularly the stability, consistency, and convergence of neural networks under non-IID data, distribution shifts, and high-dimensional settings. We provide new theoretical results on uniform stability for neural networks with dynamic learning rates in non-convex settings. Further, we establish consistency bounds for federated learning models in non-Euclidean spaces, accounting for distribution shifts and curvature effects. For Physics-Informed Neural Networks (PINNs), we derive stability, consistency, and convergence guarantees for solving Partial Differential Equations (PDEs) in noisy environments. These results fill significant gaps in understanding model behavior in complex, non-ideal conditions, paving the way for more robust and reliable machine learning applications.

In this paper, a multi-class Aw-Rascle \textrm{(AR)} model with time fractional order derivative is presented. The conservative form of the proposed model is considered for the natural extension and generalization of equations involved. The fractional order derivative involved in the model equations is computed by applying the Caputo fractional derivative definition. An explicit difference scheme is obtained through finite difference method of discretization. The scheme is shown to be consistent, conditionally stable and convergent. Numerical flux is computed by original Roe decomposition and an entropy condition applied to the Roe decomposition. From numerical results, the effect of fractional-order derivative of time, on the traffic flow of vehicle classes is determined. Results obtained from the proposed model remain within limits therefore, they are realistic., Comment: Looking forward to receiving comments. arXiv admin note: text overlap with arXiv:2306.00692

An extended multi-class Aw-Rascle (AR) model with pressure term described as a function of area occupancy defined in form of proportional densities is presented. Two vehicle classes that is; cars and motorcycles are considered based on an assumption that proportions of these form total traffic density. Qualitative properties of the proposed equilibrium velocity is established. Conditions under which the proposed model is stable are determine by linear stability analysis. To compute numerical flux, the model is discretized by the original Roe decomposition scheme, where Roe matrix, averaged data variables and wave strengths are explicitly derived. The Roe matrix is shown to be hyperbolic, consistent and conservative. From the numerical results, the effect of motorcycles proportion on the flow of vehicle classes is determined. Results obtained remain within limits therefore, the proposed model is realistic., Comment: 25 pages, comments are welcome

Eigenvector centrality is one of the outstanding measures of central tendency in graph theory. In this paper we consider the problem of calculating eigenvector centrality of graph partitioned into components and how this partitioning can be used. Two cases are considered; first where the a single component in the graph has the dominant eigenvalue, secondly when there are at least two components that share the dominant eigenvalue for the graph. In the first case we implement and compare the method to the usual approach (power method) for calculating eigenvector centrality while in the second case with shared dominant eigenvalues we show some theoretical and numerical results. Keywords: Eigenvector centrality, power iteration, graph, strongly connected component., Comment: 21 pages