We analyse how the sampling dynamics of distributions evolve in score-based diffusion models using cross-fluctuations, a centered-moment statistic from statistical physics. Specifically, we show that starting from an unbiased isotropic normal distribution, samples undergo sharp, discrete transitions, eventually forming distinct events of a desired distribution while progressively revealing finer structure. As this process is reversible, these transitions also occur in reverse, where intermediate states progressively merge, tracing a path back to the initial distribution. We demonstrate that these transitions can be detected as discontinuities in nth-order cross-fluctuations. For variance-preserving SDEs, we derive a closed-form for these cross-fluctuations that is efficiently computable for the reverse trajectory. We find that detecting these transitions directly boosts sampling efficiency, accelerates class-conditional and rare-class generation, and improves two zero-shot tasks-image classification and style transfer-without expensive grid search or retraining. We also show that this viewpoint unifies classical coupling and mixing from finite Markov chains with continuous dynamics while extending to stochastic SDEs and non Markovian samplers.

Topological descriptors have been increasingly utilized for capturing multiscale structural information in relational data. In this work, we consider various filtrations on the (box) product of graphs and the effect on their outputs on the topological descriptors - the Euler characteristic (EC) and persistent homology (PH). In particular, we establish a complete characterization of the expressive power of EC on general color-based filtrations. We also show that the PH descriptors of (virtual) graph products contain strictly more information than the computation on individual graphs, whereas EC does not. Additionally, we provide algorithms to compute the PH diagrams of the product of vertex-and edge-level filtrations on the graph product. We also substantiate our theoretical analysis with empirical investigations on runtime analysis, expressivity, and graph classification performance. Overall, this work paves way for powerful graph persistent descriptors via product filtrations.

By far the most common way to estimate an expected loss in machine learning is to draw samples, compute the loss on each one, and take the empirical average. However, sampling is not necessarily optimal. Given an MLP at initialization, we show how to estimate its expected output over Gaussian inputs without running samples through the network at all. Instead, we produce approximate representations of the distributions of activations at each layer, leveraging tools such as cumulants and Hermite expansions. We show both theoretically and empirically that for sufficiently wide networks, our estimator achieves a target mean squared error using substantially fewer FLOPs than Monte Carlo sampling. We find moreover that our methods perform particularly well at estimating the probabilities of rare events, and additionally demonstrate how they can be used for model training. Together, these findings suggest a path to producing models with a greatly reduced probability of catastrophic tail risks.

Calibration measures whether a model's predicted confidence aligns with its empirical accuracy, and is central to the reliable deployment of large language models (LLMs) in high-stakes domains such as medicine and law. While much recent work focuses on improving LLM calibration, the equally important question of how to evaluate it in realistic settings remains underdeveloped. Open-ended question answering (QA), the most common deployment setting for modern LLMs, is where existing evaluation methods fall short: logit-based metrics need restricted output formats and internal probabilities; verbalized confidence is self-reported and often overconfident; and sampling-based methods rely on task-specific extraction rules without a clear finite-sample target. We introduce Sem-ECE (Semantic-Sampling Expected Calibration Error), a calibration evaluation framework for open-ended QA that samples answers from the model, groups them into semantic classes, and uses the resulting frequencies as confidence. We study two estimators within this framework: Sem$_1$-ECE, the same-sample self-consistency score, and Sem$_2$-ECE, a held-out variant that separates answer selection from confidence evaluation. We prove both are asymptotically unbiased, and further show that they agree on easy questions but diverge on hard ones with Sem$_2$ achieving strictly smaller calibration error, so their gap also serves as a diagnostic for question difficulty. Experiments on three open-ended QA benchmarks across five leading commercial LLMs match our theoretical predictions and show that Sem-ECE outperforms verbalized confidence and existing sampling-based methods, while complementing logit-based evaluation when internal probabilities are unavailable.

Persistence diagrams provide stable, interpretable summaries of geometric and topological structure and are useful for simulation-based inference when low-order statistics miss key information. Yet persistence-based pipelines require hand-chosen filtrations, vectorizations, and compressors, typically without an objective tied to parameter uncertainty. We introduce \textbf{TopoFisher}, a differentiable persistent-homology pipeline that learns topological summaries by maximizing local Gaussian Fisher information. Using simulations near a fiducial parameter, TopoFisher optimizes trainable filtrations, diagram vectorizations, and compressors without posterior samples or supervised regression targets, while retaining stable topological inductive bias. We also give sufficient regularity conditions for the log-determinant Fisher loss to be locally Lipschitz in trainable parameters. Controlled experiments on noisy spirals and Gaussian random fields, where total Fisher information is known, show that TopoFisher recovers much of the available information and outperforms fixed topological vectorizations. Our main results are on weak gravitational lensing, a high-dimensional non-Gaussian cosmological field-inference problem. Learned topological summaries reach higher Fisher information than state-of-the-art cosmological summaries and approach an unconstrained Information Maximising Neural Network baseline with up to $\sim80\times$ fewer parameters. The learned filtrations also generalize better: under simulator shift from lognormal to LPT-based maps it retains most Fisher information, while the neural baseline drops, and in neural posterior estimation they give tighter constraints than the neural baseline, and of state-of-the-art cosmological summaries. These results support Fisher-based topological optimization as a robust, parameter-efficient front end for simulation-based inference.

G.1 Monoids Monoids are one of the simplest algebraic structure.

Since the random language model was proposed by E. DeGiuli [Phys. Rev. Lett. 122, 128301], language models have been investigated intensively from the viewpoint of statistical mechanics. Recently, the existence of a Berezinskii--Kosterlitz--Thouless transition was numerically demonstrated in models with long-range interactions between symbols. In statistical mechanics, it has long been known that long-range interactions can induce phase transitions. Therefore, it has remained unclear whether phase transitions observed in language models originate from genuinely linguistic properties that are absent in conventional spin models. In this study, we construct a random language model with short-range interactions and numerically investigate its statistical properties. Our model belongs to the class of context-sensitive grammars in the Chomsky hierarchy and allows explicit reference to contexts. We find that a phase transition occurs even when the model refers only to contexts whose length remains constant with respect to the sentence length. This result indicates that finite-temperature phase transitions in language models are genuinely induced by the intrinsic nature of language, rather than by long-range interactions.

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