. Let F be a probability distribution on Rd which admits a bounded density. We investigate the Euler characteristic of the Cech complex on n points sampled from F i.i.d. as n -> infinity in the thermodynamic limit regime. As a main result, we identify a condition for two probability distributions to yield the same expected Euler characteristic under this construction. Namely, this happens if and only if their densities admit the same excess mass transform. Building on work of Bobrowski, we establish a connection between the limiting expected Euler characteristic of any such probability distribution F and the one of the uniform distribution on [0, 1]d through an integral transform. Our approach relies on constructive proofs, offering explicit calculations of expected Euler characteristics in lower dimensions as well as reconstruction of a distribution from its limiting Euler characteristic. In the context of topological data analysis, where the Euler characteristic serves as a summary of the shape of data, we address the inverse problem and determine what can be discriminated using this invariant. This research sheds light on the relationship between a probability distribution and topological properties of the Cech complex on its samples in the thermodynamic limit.
In this paper we investigate the efficacy of the score-based martingale posteriors (SMP) (Cui & Walker, 2025; Fong et al., 2023) in the context of modern and large-scale machine learning problems and its potential for meaningful uncertainty quantification. SMPs work with a stochastic gradient ascent-type recursion on the parameter space of stochastic models and construct a martingale on the parameter space. Under simple mathematical assumptions, the recursion can be built so that the parameters form a martingale sequence which possesses a limiting, in time, random variable, the latter of which can be simulated very quickly, in contrast to Monte Carlo-based methods such as Markov chain Monte Carlo. In this expository paper we explore the SMP for inferring the parameters of deep neural networks (DNNs) and, where feasible, compare our results to the state-of-the-art Monte Carlo methods aimed at inferring conventional Bayesian posteriors.
Algebraic topology has proved to be a very useful tool in addressing the computability of rendezvous problems in distributed computing. To date, the tools used include the fundamental group and ordinary homology. This paper introduces more sophisticated techniques from homotopy theory involving the Hilton-Milnor theorem in order to analyze whether rendezvous problems are solvable. A criterion is proved that ensures a solution given certain hypotheses. Families of examples are given when the hypotheses are satisfied and when they are not satisfied; these include new cases when homology alone suffices to ensure a solution, and cases when homology needs to be supplemented by additional properties to obtain a solution.
. Nonlinear dynamical systems are ubiquitous in various scientific areas. The Koopman operator theory provides a way to analyze nonlinear dynamical systems with a linear operator, whose spectra contain rich information in understanding the behaviors of dynamical systems. In this paper, we aim to use a data-driven method to construct the principal eigenfunctions of the Koopman operator. Our approach relies on the decomposition of the Ko opman eigenfunction where the linear part associates with the left eigenvectors of the linearization of the system at the equilibrium, and the nonlinear part satisfies a linear PDE. Then, we solve the linear PDE using a random features model, which is a data-driven machine learning technique for large-scale kernel approximation. In addition, the random features model can be viewed as a randomized shallow neural network where only the linear readout needs to be trained, which significantly reduces the computational complexity. We provide a computational framework for the approximation of Koopman eigenfunctions and an error analysis for our proposed method. Finally, we present simulation results for learning the principal eigenfunctions and constructing the Lyapunov function using learned principal eigenfunctions. Numerical experiments indicate that our method is fast and accurate. As such, it can serve as a benchmark for approximating Koopman eigenfunctions.
