March 2026

arXiv:2603.24594v1 Announce Type: cross Abstract: We introduce the Multilevel Euler-Maruyama (ML-EM) method compute solutions of SDEs and ODEs using a range of approximators $f^1,dots,f^k$ to the drift $f$ with increasing accuracy and computational cost, only requiring a few evaluations of the most accurate $f^k$ and many evaluations of the less costly $f^1,dots,f^{k-1}$. If the drift lies in the so-called Harder than Monte Carlo (HTMC) regime, i.e. it requires $epsilon^{-gamma}$ compute to be $epsilon$-approximated for some $gamma>2$, then ML-EM […]

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arXiv:2603.24545v1 Announce Type: cross Abstract: We study the problem of detecting local geometry in random graphs. We introduce a model $mathcal{G}(n, p, d, k)$, where a hidden community of average size $k$ has edges drawn as a random geometric graph on $mathbb{S}^{d-1}$, while all remaining edges follow the ErdH{o}s–R’enyi model $mathcal{G}(n, p)$. The random geometric graph is generated by thresholding inner products of latent vectors on $mathbb{S}^{d-1}$, with each edge having marginal probability equal to $p$. This implies […]

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arXiv:2603.24495v1 Announce Type: cross Abstract: While the mathematical foundations of score-based generative models are increasingly well understood for unconstrained Euclidean spaces, many practical applications involve data restricted to bounded domains. This paper provides a statistical analysis of reflected diffusion models on the hypercube $[0,1]^D$ for target distributions supported on $d$-dimensional linear subspaces. A primary challenge in this setting is the absence of Gaussian transition kernels, which play a central role in standard theory in $mathbb{R}^D$. By employing an […]

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arXiv:2603.24384v1 Announce Type: cross Abstract: The theory of Local Intrinsic Dimensionality (LID) has become a valuable tool for characterizing local complexity within and across data manifolds, supporting a range of data mining and machine learning tasks. Accurate LID estimation requires samples drawn from small neighborhoods around each query to avoid biases from nonlocal effects and potential manifold mixing, yet limited data within such neighborhoods tends to cause high estimation variance. As a variance reduction strategy, we propose an […]

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arXiv:2603.24333v1 Announce Type: cross Abstract: Recently, Forr’e (arXiv:2104.11547, 2021) introduced transitional conditional independence, a notion of conditional independence that provides a unified framework for both random and non-stochastic variables. The original paper establishes a strong global Markov property connecting transitional conditional independencies with suitable graphical separation criteria for directed mixed graphs with input nodes (iDMGs), together with a version of causal calculus for iDMGs in a general measure-theoretic setting. These notes aim to further illustrate the motivations behind […]

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arXiv:2603.23926v1 Announce Type: cross Abstract: Online reinforcement learning in infinite-horizon Markov decision processes (MDPs) remains less theoretically and algorithmically developed than its episodic counterpart, with many algorithms suffering from high “burn-in” costs and failing to adapt to benign instance-specific complexity. In this work, we address these shortcomings for two infinite-horizon objectives: the classical average-reward regret and the $gamma$-regret. We develop a single tractable UCB-style algorithm applicable to both settings, which achieves the first optimal variance-dependent regret guarantees. Our […]

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arXiv:2603.23923v1 Announce Type: cross Abstract: Predictive inference is a fundamental task in statistics, traditionally addressed using parametric assumptions about the data distribution and detailed analyses of how models learn from data. In recent years, conformal prediction has emerged as a rapidly growing alternative framework that is particularly well suited to modern applications involving high-dimensional data and complex machine learning models. Its appeal stems from being both distribution-free — relying mainly on symmetry assumptions such as exchangeability — and […]

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arXiv:2603.23831v1 Announce Type: cross Abstract: Deep neural networks (DNNs), particularly those using Rectified Linear Unit (ReLU) activation functions, have achieved remarkable success across diverse machine learning tasks, including image recognition, audio processing, and language modeling. Despite this success, the non-convex nature of DNN loss functions complicates optimization and limits theoretical understanding. In this paper, we highlight how recently developed convex equivalences of ReLU NNs and their connections to sparse signal processing models can address the challenges of training […]

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arXiv:2603.23805v1 Announce Type: cross Abstract: Neural Collapse is a phenomenon that helps identify sparse and low rank structures in deep classifiers. Recent work has extended the definition of neural collapse to regression problems, albeit only measuring the phenomenon at the last layer. In this paper, we establish that Neural Regression Collapse (NRC) also occurs below the last layer across different types of models. We show that in the collapsed layers of neural regression models, features lie in a […]

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arXiv:2603.23792v1 Announce Type: cross Abstract: Diffusion models often generate novel samples even when the learned score is only emph{coarse} — a phenomenon not accounted for by the standard view of diffusion training as density estimation. In this paper, we show that, under the emph{manifold hypothesis}, this behavior can instead be explained by coarse scores capturing the emph{geometry} of the data while discarding the fine-scale distributional structure of the population measure~$mu_{scriptscriptstylemathrm{data}}$. Concretely, whereas estimating the full data distribution $mu_{scriptscriptstylemathrm{data}}$ […]

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