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Topological edge states of continuous Hamiltonians

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arXiv 2025年

作者： Frazier, Matthew Bal, Guillaume Committee on Computational and Applied Mathematics University of Chicago ChicagoIL60637 United States Departments of Mathematics and Statistics Committee on Computational and Applied Mathematics University of Chicago ChicagoIL60637 United States

This paper concerns the topological classification of continuous Hamiltonians that find applications in biased cold plasmas and photonics. Besides a magnetic bias, the Hamiltonians are parametrized by a plasma frequency and a fixed vertical wavenumber. Eight distinct phases of matter are identified as these parameters vary. When insulating gaps are shared by two such phases, asymmetric edge modes propagate along interfaces separating the two phases. Here we apply the notion of a bulk difference invariant (BDI) to this Hamiltonian, and show by numerical diagonalizations of interface Hamiltonians that after an appropriate regularization our BDI correctly predicts edge transport as described by a bulk edge correspondence. We also derive theoretical tools to compute the BDI and show the limitations of the bulk edge correspondence (BEC) when the phase transition is too singular. Copyright © 2025, The Authors. All rights reserved.

关键词： Topology

Batched Nonparametric Contextual Bandits

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IEEE Transactions on Information Theory 2025年

作者： Jiang, Rong Ma, Cong University of Chicago Committee on Computational and Applied Mathematics IL60637 United States University of Chicago Department of Statistics ChicagoIL60637 United States

We study nonparametric contextual bandits under batch constraints, where the expected reward for each action is modeled as a smooth function of covariates, and the policy updates are made at the end of each batch of observations. We establish a minimax regret lower bound for this setting and propose a novel batch learning algorithm that achieves the optimal regret (up to logarithmic factors). In essence, our procedure dynamically splits the covariate space into smaller bins, carefully aligning their widths with the batch size. Our theoretical results suggest that for nonparametric contextual bandits, a nearly constant number of policy updates can attain optimal regret in the fully online setting. © 1963-2012 IEEE.

关键词： batch learning contextual bandits dynamic binning minimax rate regret bounds

A flow-kick model of dryland vegetation patterns: the impact of rainfall variability on resilience

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arXiv 2025年

作者： Gandhi, Punit Oline, Matthew Silber, Mary Department of Mathematics and Applied Mathematics Virginia Commonwealth University RichmondVA23284 United States Computational and Applied Mathematics University of Chicago ChicagoIL60637 United States Department of Statistics and Committee on Computational and Applied Mathematics University of Chicago ChicagoIL60637 United States

In many drylands around the globe, vegetation self-organizes into regular spatial patterns in response to aridity stress. We consider the regularly-spaced vegetation bands, on gentle hill-slopes, that survive low rainfall conditions by harvesting additional stormwater from upslope low-infiltration bare zones. We are interested in the robustness of this pattern formation survival mechanism to changes in rainfall variability. For this, we use a flow-kick modeling framework that treats storms as instantaneous kicks to the soil water. The positive feedbacks in the storm-level hydrology, that act to concentrate water within the vegetation bands, are captured through the spatial profiles of the soil water kicks. Between storms, the soil water and vegetation, modeled by a two-component reaction-diffusion system, evolve together. We use a combination of linear stability analysis and numerical simulation to compare predictions of idealized periodic rainfall, with no variability, to predictions when there is randomness in the timing and magnitude of water input from storms. We show that including these random elements leads to a decrease in the parameter range over which patterns appear. This suggests that an increase in storm variability, even with the same mean annual rainfall, may negatively impact the resilience of these pattern-forming dryland ecosystems. Copyright © 2025, The Authors. All rights reserved.

关键词： Vegetation

Sharp concentration of simple random tensors

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arXiv 2025年

作者： Al-Ghattas, Omar Chen, Jiaheng Sanz-Alonso, Daniel Department of Statistics University of Chicago IL60637 United States Committee on Computational and Applied Mathematics University of Chicago IL60637 United States

This paper establishes sharp dimension-free concentration inequalities and expectation bounds for the deviation of the sum of simple random tensors from its expectation. As part of our analysis, we use generic chaining techniques to obtain a sharp high-probability upper bound on the suprema of multi-product empirical processes. In so doing, we generalize classical results for quadratic and product empirical processes to higher-order settings. © 2025, CC BY.

关键词： Tensors

PRECISE: PRivacy-loss-Efficient and Consistent Inference based on poSterior quantilEs

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arXiv 2025年

作者： Zhou, Ruyu Liu, Fang Department of Applied and Computational Mathematics and Statistics University of Notre Dame IN46556 United States

Differential privacy (DP) is a mathematical framework for releasing information with formal privacy guarantees. Despite the existence of various DP procedures for performing a wide range of statistical analysis and machine learning tasks, methods of good utility are still lacking for obtaining valid statistical inference with DP guarantees. We address this gap by introducing the notion of valid Privacy-Preserving Interval Estimation (PPIE) and proposing PRivacy-loss-Efficient and Consistent Inference based on poSterior quantilEs (PRECISE). PRECISE is a general-purpose Bayesian approach for constructing privacy-preserving posterior intervals. We establish the Mean-Squared-Error (MSE) consistency for our proposed private posterior quantiles converging to the population posterior quantile as sample size or privacy loss increases. We conduct extensive experiments to compare the utilities of PRECISE with common existing privacy-preserving inferential approaches in various inferential tasks, data types and sizes, privacy loss levels. The results demonstrated a significant advantage of PRECISE with its nominal coverage and substantially narrower intervals than the existing methods, which are prone to either under-coverage or impractically wide intervals. © 2025, CC BY.

关键词： Differential privacy

Statistical Inference for Low-Rank Tensor Models

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arXiv 2025年

作者： Xu, Ke Chen, Elynn Han, Yuefeng Department of Applied and Computational Mathematics and Statistics University of Notre Dame United States Department of Technology Operations and Statistics New York University United States

Statistical inference for tensors has emerged as a critical challenge in analyzing high-dimensional data in modern data science. This paper introduces a unified framework for inferring general and low-Tucker-rank linear functionals of low-Tucker-rank signal tensors for several low-rank tensor models. Our methodology tackles two primary goals: achieving asymptotic normality and constructing minimax-optimal confidence intervals. By leveraging a debiasing strategy and projecting onto the tangent space of the low-Tucker-rank manifold, we enable inference for general and structured linear functionals, extending far beyond the scope of traditional entrywise inference. Specifically, in the low-Tucker-rank tensor regression or PCA model, we establish the computational and statistical efficiency of our approach, achieving near-optimal sample size requirements (in regression model) and signal-to-noise ratio (SNR) conditions (in PCA model) for general linear functionals without requiring sparsity in the loading tensor. Our framework also attains both computationally and statistically optimal sample size and SNR thresholds for low-Tucker-rank linear functionals. Numerical experiments validate our theoretical results, showcasing the framework’s utility in diverse applications. This work addresses significant methodological gaps in statistical inference, advancing tensor analysis for complex and high-dimensional data *** Codes 62H10 (Primary) 62H25, 62F12 (Secondary) © 2025, CC BY-NC-SA.

关键词： Tensors

Riemannian Proximal Sampler for High-accuracy Sampling on Manifolds

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arXiv 2025年

作者： Guan, Yunrui Balasubramanian, Krishnakumar Ma, Shiqian Department of Computational Applied Mathematics and Operations Research Rice University United States Department of Statistics University of California Davis United States

We introduce the Riemannian Proximal Sampler, a method for sampling from densities defined on Riemannian manifolds. The performance of this sampler critically depends on two key oracles: the Manifold Brownian Increments (MBI) oracle and the Riemannian Heat-kernel (RHK) oracle. We establish high-accuracy sampling guarantees for the Riemannian Proximal Sampler, showing that generating samples with Ε-accuracy requires O(log(1/Ε)) iterations in Kullback-Leibler divergence assuming access to exact oracles and O(log2(1/Ε)) iterations in the total variation metric assuming access to sufficiently accurate inexact oracles. Furthermore, we present practical implementations of these oracles by leveraging heat-kernel truncation and Varadhan’s asymptotics. In the latter case, we interpret the Riemannian Proximal Sampler as a discretization of the entropy-regularized Riemannian Proximal Point Method on the associated Wasserstein space. We provide preliminary numerical results that illustrate the effectiveness of the proposed methodology. © 2025, CC BY.

关键词： Numerical methods

Accuracy Versus Predominance: Reassessing the Validity of the Quasi-Steady-State Approximation

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Bulletin of Mathematical Biology 2025年第6期87卷 1-32页

作者： Srivastava, Kashvi Eilertsen, Justin Booth, Victoria Schnell, Santiago Department of Mathematics University of Michigan Ann Arbor USA Mathematical Reviews American Mathematical Society Ann Arbor USA Department of Biological Sciences and Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre Dame USA

The application of the standard quasi-steady-state approximation to the Michaelis-Menten reaction mechanism is a textbook example of biochemical model reduction, derived using singular perturbation theory. However, determining the specific biochemical conditions that dictate the validity of the standard quasi-steady-state approximation remains a challenging endeavor. Emerging research suggests that the accuracy of the standard quasi-steady-state approximation improves as the ratio of the initial enzyme concentration, $$e_0$$ , to the Michaelis constant, $$K_M$$ , decreases. In this work, we examine this ratio and its implications for the accuracy and validity of the standard quasi-steady-state approximation as compared to other quasi-steady-state reductions in its proximity. Using standard tools from the analysis of ordinary differential equations, we show that while $$e_0/K_M$$ provides an indication of the standard quasi-steady-state approximation’s asymptotic accuracy, the standard quasi-steady-state approximation’s predominance relies on a small ratio of $$e_0$$ to the Van Slyke-Cullen constant, K. Here, we define the predominance of a quasi-steady-state reduction when it offers the highest approximation accuracy among other well-known reductions with overlapping validity conditions. We conclude that the magnitude of $$e_0/K$$ offers the most accurate measure of the validity of the standard quasi-steady-state approximation.

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P-ORDER: A UNIFIED CONVERGENCE-ANALYSIS FRAMEWORK FOR MULTIVARIATE ITERATIVE METHODS

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arXiv 2025年

作者： Jiao, Xiangmin Gao, Hongji Department of Applied Mathematics & Statistics Institute for Advanced Computational Science Stony Brook University Stony BrookNY11794 United States Department of Applied Mathematics & Statistics Stony Brook University Stony BrookNY11794 United States

We propose P-order (Power-order), a unified, norm-independent framework for quantifying the convergence rates of iterative methods. Standard analyses based on Q-order are norm-dependent and require some uniformity of error reductions asymptotically. Although the Ortega–Rheinboldt R-order supports non-uniform (including non-monotonic sequences) and is norm-independent, it does not differentiate among various sublinear rates and has ambiguities for some superlinear rates. In contrast, our proposed framework parameterizes the convergence rate in direct analogy with asymptotic notation in combinatorial algorithm analysis (including little-o, big-Θ, and little-ω), thereby precisely distinguishing linear, sublinear (e.g., fractional-power), and superlinear (e.g., linearithmic) regimes. We also introduce two subclasses of P-order: Quasi-Uniform P-order (QUP) and Uniform P-order (UP), and show their relations with Q-order and R-order. We demonstrate the ease-of-use and flexibility of QUP-order by analyzing fixed-point iterations and show that P-order can provide tighter bounds on the convergence rate than R-order for both sublinearly and superlinearly convergent cases while avoiding some ambiguities in R-order. Furthermore, we present a refined analysis of Newton’s method for nonlinear equations with a wide spectrum of sublinear and superlinear convergence rates (including a newly defined anit-liearithmic rate). Copyright © 2025, The Authors. All rights reserved.

关键词： Combinatorial optimization