Comparing two probability distributions is a basic building block of statistics and machine learning, and the right family is well understood: the Rényi divergences of order $α\in[0,\infty]$ are the unique family monotone under data processing and additive on independent products. Many problems instead compare more than two distributions at once -- multi-population fairness, multi-prior PAC-Bayes bounds, multi-hypothesis testing -- and the right multi-distribution generalization of the Rényi family has been an open question. We characterize it. Every functional of $W$-tuples of distributions that is monotone under data processing and additive on independent products is a positive integral of multi-way coincidence divergences $C_α(π_1,\dots,π_W) := -\log\int π_1^{α_1}\cdotsπ_W^{α_W}$ (with $\sum_k α_k = 1$) over a parameter space with four strata: the simplex interior; mixed-sign exponent cones (the analogue of Rényi orders $>1$); a tropical boundary at infinity carrying max-divergences; and pairwise Kullback-Leibler edges at the simplex vertices. Each stratum is necessary -- the destination of an explicit data-processing-monotone, product-additive divergence the others cannot reproduce -- and each is a clean limit of simplex-interior atoms. The same family arises from several independent routes -- the structural axioms, Kolmogorov-Nagumo means with Rényi's entropy axiomatics, classical entropy characterizations, multi-hypothesis testing error exponents, and a multi-lottery betting interpretation -- structural evidence that this is the canonical multi-distribution Rényi calculus rather than an artefact of any one axiomatic input. The two-prior case recovers the standard Rényi result; a worked $W=3$ instance, numerical verification, and a conditional extension round out the treatment.

We connect stochastic resetting from non-equilibrium statistical physics with ridge regularization in statistical learning. For linear gradient flow, resetting to the origin at rate $r$ produces stationary mean $(X^\top X+rI)^{-1}X^\top y$, exactly the ridge estimator with penalty $λ=r$. This uses the known Laplace-transform relationship between ridge regression and exponential-time averaging of gradient flow, with the exponential time now interpreted as the stationary age associated with Poisson resetting. We then extend this identity to general renewal reset laws: the exponential reset time distribution is the unique renewal law whose stationary mean reproduces scalar ridge in every eigendirection as an exact filter identity for every positive curvature, while non-exponential renewal laws generate alternative spectral filters. At the fluctuation level, we study a separate additive Ornstein-Uhlenbeck extension with constant diffusion, interpreted as a stylized SGD approximation. In this setting, the equality holds only at the level of the mean, since the reset process has a nonzero stationary covariance from accumulated OU noise and reset-timing variance, whereas deterministic ridge is a fixed estimator with the same center. Stylized experiments compare the deterministic renewal-induced filters directly and illustrate when filters induced by non-exponential reset-time laws can differ predictively from ridge. The results for the stationary mean and the induced spectral filters are established for continuous-time gradient flow with isotropic resetting on quadratic objectives; the covariance and risk formulas additionally assume additive noise with state-independent covariance.

Aligning large language models (LLMs) to human preferences typically relies on aggregating pooled feedback into a single reward model. However, this standard approach assumes that all labelers share the same underlying preferences, ignoring the fact that real-world labelers are highly heterogeneous and usually anonymous. Consequently, relying solely on binary choice data fundamentally distorts the learned policy, making the true population-average preference unidentifiable. To overcome this critical limitation, we demonstrate that augmenting preference datasets with a simple, secondary signal -- the user's response time -- can restore the identifiability of the population's average preference. By modeling each decision as a Drift-Diffusion Model (DDM), we introduce a novel, consistent estimator of heterogeneous preferences that successfully corrects the distortions of standard choice-only labels. We prove that our estimator asymptotically converges to the true average preference even in extreme cases where each anonymous labeler contributes only a single choice. Empirically, across both synthetic and real-world datasets, our method consistently outperforms standard baselines that otherwise fail and plateau at a bias floor. Because response times are essentially free to record and require zero user tracking or identification, our results bring promises and open up new opportunities for future data-collection pipelines to improve the social benefit without requiring user-level identifiers or repeated elicitations.

A critical issue that affects engineers trying to assess the structural integrity of various infrastructures, such as metal rods or acoustic ducts, is the challenge of detecting internal fractures (defects). Traditionally, engineers depend on audible and visual aids to identify these fractures, as they do not physically dissect the object in question into multiple pieces to check for inconsistencies. This research introduces ideas towards the development of a robust strategy to image such defects using only a small set of minimal, non-invasive measurements. Assuming a one dimensional model (e.g. longitudinal waves in long and thin rods/acoustic ducts or transverse vibrations of strings), we make use of the continuous one-dimensional wave equation to model these physical phenomena and then employ specialized mathematical analysis tools (the Laplace transform and optimization) to introduce our defect imaging ideas. In particular, we will focus on the case of a long bar which is homogeneous throughout except in a small area where a defect in its Young's modulus is present. We will first demonstrate how the problem is equivalent to a spring-mass vibrational system, and then show how our imaging strategy makes use of the Laplace domain analytic map between the characteristics of the respective defect and the measurement data. More explicitly, we will utilize MATLAB (a platform for numerical computations) to collect synthetic data (computational alternative to real world measurements) for several scenarios with one defect of arbitrary location and stiffness. Subsequently, we will use this data along with our analytically developed map (between defect characteristics and measurements) to construct a residual function which, once optimized, will reveal the location and magnitude of the stiffness defect.

Modern optimization faces a fundamental challenge: local gradient-based methods provide no global information about the objective function $L$ landscape, often leading to suboptimal convergence and sensitivity to initialization. We introduce a novel optimization framework that leverages resurgence theory from complex analysis to extract global structural information from divergent asymptotic series. Our key insight is that the factorially divergent perturbative expansions of parameter space partition functions encode precise information about all critical objective function value in the landscape through their Borel transform singularities. The algorithm works by computing the statistical mechanical partition function $Z(g) = \int e^{-L(θ)/g} dθ$ for small coupling $g\ll 1$, extracting its asymptotic series coefficients, and identifying Borel plane singularities that correspond one-to-one with critical objective function values. These target values provide global guidance to local optimizers, enabling principled learning rate adaptation and escape from suboptimal regions. Unlike heuristic adaptive methods, targets are theoretically grounded in the geometry of the optimization landscape.

In this section, we derive the variational updates. The last term contains cross entropies between prior and variational PG distributions. The full VB model affects this limit in two ways. Its impact on the second parameter is indirect. How plausible is this "good fit" regime?

We will also expand Section 2 to make it more gradual, stating theorems and definitions explicitly.

We will first replicate an equation similar to (20) for the backward case. The derivation is similar to that of the forward equation, so that it uses a combination of equations (16), (18) and (19) while leaving out the observation likelihood function. The combination is again carried out using the Laplace transform.

-- Time series prediction, a crucial task across various domains, faces significant challenges due to the inherent complexities of time series data, including non-stationarity, multi-scale periodicity, and transient dynamics, particularly when tackling long-term predictions. While Transformer-based architectures have shown promise, their quadratic complexity with sequence length hinders their efficiency for long-term predictions. Meanwhile, they are susceptible to data noise issues in time series. This paper proposes a novel framework, FLDmamba (Fourier and Laplace Transform Decomposition Mamba), addressing these limitations. FLDmamba leverages the strengths of both Fourier and Laplace transforms to effectively capture both multi-scale periodicity, transient dynamics within time series data, and improve the robustness of the model to the data noise issue. Our extensive experiments demonstrate that FLDmamba achieves superior performance on time series prediction benchmarks, outperforming both Transformer-based and other Mamba-based architectures. IME series prediction, which forecasts the future values of a (multivariate) variable based on its historical values, finds its application across a wide range of fields. Examples include weather prediction [1, 2], power grid management [3], traffic prediction [4, 5, 6, 7, 8, 9, 10], and stock market [11, 12, 13, 14], to name just a few. Despite significant advancements in this domain, the inherent complexities of time series data, such as non-stationarity, multi-scale periodicity, intrinsic stochasticity, and noise, pose substantial challenges to existing predictive models in long-term prediction. Q. Zhang and S.M. Yiu are from the University of Hong Kong.