arXiv subjects

Andrew M. Thomas

Publications and source records attributed to Andrew M. Thomas.

10 recordsLinked to original sources

arXiv2026

A generalized Bayesian approach to multiple changepoint analysis

We introduce a generalized Bayesian method for multiple changepoint analysis with a loss function inspired by multinomial logistic regression. The method does not require a specification of the data-generating process and avoids restrictive assumptions on the nature of changepoints. From the joint posterior distribution, we can make simultaneous inference on the locations of changepoints and the coefficients of a multinomial logistic regression model for distinguishing data across homogeneous segments. The multinomial logistic regression coefficients provide a familiar means of interpreting potentially complex changes. To select the number of changepoints, we leverage posterior summaries that measure whether the multinomial logistic classifier can distinguish data from either side of a potential changepoint. To simulate from the generalized posterior distribution, we present a Gibbs sampler based on Pólya-Gamma data augmentation. We assess the accuracy and flexibility of our method through simulation studies featuring different types of changes and demonstrate its interpretability through applications to financial network data and topological data derived from nanoparticle videos.

stat.ME↗

arXiv2025

Bayesian changepoint detection via logistic regression and the topological analysis of image series

We present a Bayesian method for multivariate changepoint detection that allows for simultaneous inference on the location of a changepoint and the coefficients of a logistic regression model for distinguishing pre-changepoint data from post-changepoint data. In contrast to many methods for multivariate changepoint detection, the proposed method is applicable to data of mixed type and avoids strict assumptions regarding the distribution of the data and the nature of the change. The regression coefficients provide an interpretable description of a potentially complex change. For posterior inference, the model admits a simple Gibbs sampling algorithm based on Pólya-gamma data augmentation. We establish conditions under which the proposed method is guaranteed to recover the true underlying changepoint. As a testing ground for our method, we consider the problem of detecting topological changes in time series of images. We demonstrate that our proposed method BCLR, combined with a topological feature embedding, performs well on both simulated and real image data. The method also successfully recovers the location and nature of changes in more traditional changepoint tasks.

stat.ME↗

arXiv2025

Convergence of persistence diagrams for discrete time stationary processes

In this article we establish two fundamental results for the sublevel set persistent homology for stationary processes indexed by the positive integers. The first is a strong law of large numbers for the persistence diagram (treated as a measure "above the diagonal" in the extended plane) evaluated on a large class of sets and functions, beyond continuous functions with compact support. We prove this result subject to only minor conditions that the sequence is ergodic and the tails of the marginals are not too heavy. The second result is a central limit theorem for the persistence diagram evaluated on the class of all step functions; this result holds as long as a $ρ$-mixing criterion is satisfied and the distributions of the partial maxima do not decay too slowly. Our results greatly expand those extant in the literature to allow for more fruitful use in statistical applications, beyond idealized settings. Examples of distributions and functions for which the limit theory holds are provided throughout.

math.PR↗

arXiv2024

Atomic Resolution Observations of Nanoparticle Surface Dynamics and Instabilities Enabled by Artificial Intelligence

Nanoparticle surface structural dynamics is believed to play a significant role in regulating functionalities such as diffusion, reactivity, and catalysis but the atomic-level processes are not well understood. Atomic resolution characterization of nanoparticle surface dynamics is challenging since it requires both high spatial and temporal resolution. Though ultrafast transmission electron microscopy (TEM) can achieve picosecond temporal resolution, it is limited to nanometer spatial resolution. On the other hand, with the high readout rate of new electron detectors, conventional TEM has the potential to visualize atomic structure with millisecond time resolutions. However, the need to limit electron dose rates to reduce beam damage yields millisecond images that are dominated by noise, obscuring structural details. Here we show that a newly developed unsupervised denoising framework based on artificial intelligence enables observations of metal nanoparticle surfaces with time resolutions down to 10 ms at moderate electron dose. On this timescale, we find that many nanoparticle surfaces continuously transition between ordered and disordered configurations. The associated stress fields can penetrate below the surface leading to defect formation and destabilization making the entire nanoparticle fluxional. Combining this unsupervised denoiser with electron microscopy greatly improves spatio-temporal characterization capabilities, opening a new window for future exploration of atomic-level structural dynamics in materials.

cond-mat.mtrl-sci↗

arXiv2023

Dynamic Atomic Column Detection in Transmission Electron Microscopy Videos via Ridge Estimation

Ridge detection is a classical tool to extract curvilinear features in image processing. As such, it has great promise in applications to material science problems; specifically, for trend filtering relatively stable atom-shaped objects in image sequences, such as Transmission Electron Microscopy (TEM) videos. Standard analysis of TEM videos is limited to frame-by-frame object recognition. We instead harness temporal correlation across frames through simultaneous analysis of long image sequences, specified as a spatio-temporal image tensor. We define new ridge detection algorithms to non-parametrically estimate explicit trajectories of atomic-level object locations as a continuous function of time. Our approach is specially tailored to handle temporal analysis of objects that seemingly stochastically disappear and subsequently reappear throughout a sequence. We demonstrate that the proposed method is highly effective and efficient in simulation scenarios, and delivers notable performance improvements in TEM experiments compared to other material science benchmarks.

stat.AP↗

arXiv2023

Feature detection and hypothesis testing for extremely noisy nanoparticle images using topological data analysis

We propose a flexible algorithm for feature detection and hypothesis testing in images with ultra low signal-to-noise ratio using cubical persistent homology. Our main application is in the identification of atomic columns and other features in transmission electron microscopy (TEM). Cubical persistent homology is used to identify local minima and their size in subregions in the frames of nanoparticle videos, which are hypothesized to correspond to relevant atomic features. We compare the performance of our algorithm to other employed methods for the detection of columns and their intensity. Additionally, Monte Carlo goodness-of-fit testing using real valued summaries of persistence diagrams derived from smoothed images (generated from pixels residing in the vacuum region of an image) is developed and employed to identify whether or not the proposed atomic features generated by our algorithm are due to noise. Using these summaries derived from the generated persistence diagrams, one can produce univariate time series for the nanoparticle videos, thus providing a means for assessing fluxional behavior. A guarantee on the false discovery rate for multiple Monte Carlo testing of identical hypotheses is also established.

stat.AP↗

arXiv2022

The VC-dimension of a class of multiples of the primes, and a connection to AdaBoost

We discuss the VC-dimension of a class of multiples of integers and primes (equivalently indicator functions) and demonstrate connections to prime counting functions. Additionally, we prove limit theorems for the behavior of an empirical risk minimization rule as well as the weights assigned to the output hypothesis in AdaBoost for these "prime-identifying" indicator functions, when we sample $m_n$ i.i.d. points uniformly from the integers $\{2, \dots, n\}$.

math.NT↗

arXiv2022

Central limit theorems and asymptotic independence for local $U$-statistics on diverging halfspaces

We consider the stochastic behavior of a class of local $U$-statistics of Poisson processes$-$which include subgraph and simplex counts as special cases, and amounts to quantifying clustering behavior$-$for point clouds lying in diverging halfspaces. We provide limit theorems for distributions with light and heavy tails. In particular, we prove finite-dimensional central limit theorems. In the light tail case we investigate tails that decay at least as slow as exponential and at least as fast as Gaussian. These results also furnish as a corollary that $U$-statistics for halfspaces diverging at different angles are asymptotically independent, and that there is no asymptotic independence for heavy-tailed densities. Using state-of-the-art bounds derived from recent breakthroughs combining Stein's method and Malliavin calculus, we quantify the rate of this convergence in terms of Kolmogorov distance. We also investigate the behavior of local $U$-statistics of a Poisson Process conditioned to lie in diverging halfspace and show how the rate of convergence in the Kolmogorov distance is faster the lighter the tail of the density is.

math.PR↗

arXiv2021

Functional strong laws of large numbers for Euler characteristic processes of extreme sample clouds

To recover the topology of a manifold in the presence of heavy tailed or exponentially decaying noise, one must understand the behavior of geometric complexes whose points lie in the tail of these noise distributions. This study advances this line of inquiry, and demonstrates functional strong laws of large numbers for the Euler characteristic process of random geometric complexes formed by random points outside of an expanding ball in $\mathbb{R}^d$. When the points are drawn from a heavy tailed distribution with a regularly varying tail, the Euler characteristic process grows at a regularly varying rate, and the scaled process converges uniformly and almost surely to a smooth function. When the points are drawn from a distribution with an exponentially decaying tail, the Euler characteristic process grows logarithmically, and the scaled process converges to another smooth function in the same sense. All of the limit theorems take place when the points inside the expanding ball are densely distributed, so that the simplex counts outside of the ball of all dimensions contribute to the Euler characteristic process.

math.PR↗

arXiv2020

Functional limit theorems for the Euler characteristic process in the critical regime

This study presents functional limit theorems for the Euler characteristic of Vietoris-Rips complexes. The points are drawn from a non-homogeneous Poisson process on $\mathbb{R}^d$, and the connectivity radius governing the formation of simplices is taken as a function of time parameter $t$, which allows us to treat the Euler characteristic as a stochastic process. The setting in which this takes place is that of the critical regime, in which the simplicial complexes are highly connected and have non-trivial topology. We establish two "functional-level" limit theorems, a strong law of large numbers and a central limit theorem for the appropriately normalized Euler characteristic process.

math.PR↗