Low-rank adaptation (LoRA) has emerged as a powerful tool for parameter-efficient fine-tuning of large language models (LLMs). This paper studies LoRA under a federated learning setting, enabling collaborative fine-tuning across clients while preserving parameter efficiency. We focus on a highly heterogeneous regime in which clients share only partial structure and a substantial subset may be contaminated. We propose Collaborative Low-rank Alignment and Identifiable Recovery (CLAIR), a contamination-aware framework that relies only on preliminary local estimators. Its formulation applies broadly, from linear regression to neural network and LLM modules, whenever local adaptation can be represented by matrix-valued updates. CLAIR recovers the shared LoRA subspace and detects contaminated clients via a structured low-rank plus block-sparse decomposition. We prove exact recovery of the shared LoRA subspace in the noiseless case, stable recovery under preliminary estimation error, and consistent collaborative-set recovery under mild separation conditions. We further quantify the gain from CLAIR refinement: it reduces off-subspace estimation error through cross-client averaging while preserving client-specific variation within the shared LoRA subspace, thus improves over local fine-tuning whenever this oracle gain outweighs the costs of subspace estimation and benign-client heterogeneity. Empirically, we demonstrate the benefits of CLAIR by fine-tuning a Transformer architecture on a text-copying task. The results show accurate contamination detection and improved benign-client performance compared with local fine-tuning and non-robust federated averaging.

Compositional score-based approaches to simulation-based inference (SBI) approximate the posterior over a shared parameter given $n$ independent observations by aggregating individually learned posterior scores: currently, there are two main propositions of such methods (Geffner et al. (2023), Linhart et al. (2026)). As the resulting composite score does not correspond to the score of any distribution along the forward diffusion path of the true multi-observation posterior, sampling from it via a reverse SDE leads to an irreducible bias. Annealed Langevin dynamics provides a principled alternative: it treats the composite score as the genuine score of a sequence of tractable bridging densities and samples from them in succession. When properly tuned, it could lead to a controllable bias. However, its hyperparameters, namely step sizes, the number of steps per level, and the number of annealing levels, have so far been chosen empirically. We derive Wasserstein bounds for annealed Langevin with approximate scores and translate them into explicit decision rules for these hyperparameters that guarantee a prescribed sampling accuracy, while highlighting different theoretical aspects of each composite score formulation. In the Gaussian setting, we obtain closed-form expressions for all relevant quantities and prove that the bridging densities of Linhart et al. (2026) consistently admit larger step sizes and require fewer total Langevin steps than those of Geffner et al. (2023). Furthermore, we show empirically that the tuning obtained in the Gaussian setting generalizes to more complex problems, thus providing a well-understood and theoretically grounded starting point for practitioners using compositional score-based approaches.

Gradient-flow analyses show that simplified linear transformers can learn the in-context linear-regression algorithm, but they do not explain the finite-step behavior of gradient descent at large learning rates. Motivated by empirical work on high-learning-rate transformer instabilities and by the cubic-map phase diagram for quadratic regression, we study an exactly reducible one-prompt linear-transformer training problem. After normalization, the dynamics reduce to a two-factor product map with an effective step-size parameter \(μ\). On the balanced slice, this map recovers the known scalar cubic transition from monotone convergence to catapult convergence, periodic and chaotic bounded nonconvergence, and divergence. We then analyze the full two-dimensional system and show that, for \(0<μ<2\), it has an explicit invariant Chebyshev ellipse separating forward-invariant regions; this ellipse carries off-balanced chaotic dynamics but is transversely repelling, while balanced scalar attractors can be transversely attracting. These results show that large constant learning rates can change the training attractor of the learned transformer rather than merely accelerating convergence: beyond sharp stability thresholds, finite-step training may settle into cycles, bounded chaos, or divergence instead of a single in-context linear-regression solution. We also discuss the consequences for mini-batch gradient descent based training methods.

Bilevel optimization provides a natural framework for problems in which one learning task is constrained by the solution of another. This hierarchical structure appears across machine learning, including hyperparameter optimization [43, 39, 36], meta-learning [20, 18, 45], inverse problems and optimal control [31, 1], reinforcement learning [25], domain adaptation [35], and instrumental variable regression [42, 50, 49]. In these applications, the outer parameter is typically updated using gradient-based methods, so the quality of the resulting bilevel gradient directly affects both optimization and statistical performance. Most existing theory for bilevel optimization has been developed in finite-dimensional parametric settings, often under strong convexity of the lower-level problem [21, 27, 29, 61]. This assumption gives a unique inner solution and makes implicit differentiation stable [43, 36]. It is also convenient for algorithmic convergence and stability analyses [9, 23, 40].

Optimal transport provides an inherently geometric and highly structured framework for studying spaces of probability measures, supplying a rich theoretical toolkit for contemporary statistics, machine learning, and generative modelling. In applications, however, the measures of interest are almost never known precisely, calling for a theory of optimal transport that accounts for statistical uncertainty. We construct such a framework, lifting the classical theory to the setting of random probability measures. We introduce the $L^2$ over Wasserstein space establishing that it inherits the formal Riemannian structure of the Wasserstein space by characterising distances and geodesic geometry. The structure induces random flows with Wasserstein gradient flow sample paths, making it the natural extension of the Wasserstein space which allows for random gradient flow dynamics. We ensemble statistical convergence results of the optimal transport machinery using the empirical measure within the $L^2$ over Wasserstein framework. Moreover, in the setting of Bayesian non-parametrics, we refine Schwartz's consistency theorem to the Wasserstein topology and deduce posterior convergence of the same machinery in the $L^2$ over Wasserstein space. We demonstrate that the growing theory of random token sampling for transformer models using self-attention flow paths can be embedded into the our framework. The results provide a unified treatment of random optimal transport and its consequences for principled inference and generative modelling under the statistical uncertainty of random sampling.

Despite the remarkable empirical success of generative models, the available theory on their statistical accuracy in scientific computing remains largely pessimistic. This paper develops a theoretical framework for understanding the regularity of transport maps and the generalization properties of one-step Wasserstein-guided generative models for PDE-induced probability measures. We consider normalized target densities associated with linear elliptic and parabolic equations on bounded domains, as well as diffusion and Fokker--Planck equations on the torus. Under standard structural assumptions, we prove that these target measures satisfy doubling conditions. By combining this fact with regularity theory for optimal transport between doubling measures, we show that the optimal transport map from a uniform source measure to the target measure is Hölder continuous. This regularity yields an approximation-theoretic justification for one-step generative models that learn PDE-induced distributions via a single pushforward map. As a representative instance, we study DeepParticle and derive excess-risk bounds characterizing the discrepancy between the learned map and the population-optimal map. We also establish a robustness estimate under target shift and illustrate the theory with experiments which support the derived rates.

Generative neural networks learn how to produce highly realistic images from a large, but finite number of examples - or do they simply memorise their training set? To settle this question, Kadkhodaie, Guth, Simoncelli and Mallat (ICLR '24) trained diffusion models independently on disjoint subsets of a dataset and showed that they converge to nearly the same density when the number of training images is large enough. This result raises two basic questions: how much data do you need for convergence, and what does convergence capture about learning the data distribution? Here, we address these questions by providing an exact analytical characterisation of the transition from memorisation to generalisation in linear generative models. We find that these models memorise at small load, while convergence emerges continuously when the number of samples is linear in the input dimension. Strikingly, we find that convergence is insensitive to recovery of the principal latent factors of the data, which are recovered in a sharp transition. After extending our approach to data with power-law spectra, we find the same distinction between convergence and latent recovery in our experiments with convolutional denoisers and in the data of Kadkhodaie et al. We thus show that generalisation in generative models decomposes into at least two distinct objectives: matching the bulk of the data distribution and recovering the principal latent factors. These objectives correspond to two different distances between true and learnt data distribution, and only the first one is captured by convergence.

Earthquake forecasting is a critical task for natural risk management, infrastructure resilience planning, and emergency response operations. For Central Asia, and the Tian Shan mountain system in particular, this problem carries heightened importance due to high tectonic activity, complex geodynamics, and pronounced spatiotemporal heterogeneity of seismic processes. In the applied setting, the goal is not a deterministic forecast of individual events, but a macroscopic forecast of seismicity intensity: estimating the expected number of earthquakes with magnitude M 3.0 on a spatial grid at a weekly horizon. Historically, count data forecasting in fixed spatiotemporal cells has been formulated within the Poisson framework. However, its key assumption--equality of the conditional mean and conditional variance--is systematically violated in real seismological data. Earthquakes exhibit pronounced clustering associated with swarm activity, foreshock-aftershock sequences, and episodes of anomalous activity, resulting in overdispersion in which the variance substantially exceeds the mean. Under these conditions, uncritical application of the Poisson distribution leads to biased uncertainty estimates and, consequently, to underestimation of the risk of extreme scenarios. Despite the widespread adoption of machine learning methods in seismological problems, a substantial portion of existing work remains methodologically vulnerable. On one hand, several approaches apply continuous regression loss functions and metrics (e.g., MSE), ignoring the

Hyperparameter transfer allows extrapolating optimal optimization hyperparameters from small to large scales, making it critical for training large language models (LLMs). This is done either by fitting a scaling law to the hyperparameters or by a judicious choice of parameterization, such as Maximal Update ($μ$P), that renders optimal hyperparameters approximately scale invariant. In this paper, we first develop a framework to quantify hyperparameter transfer through three metrics: (1) the quality of the scaling law fit, (2) the robustness to extrapolation errors, and (3) the asymptotic loss penalty due to choice of parameterization. Next, we investigate through a comprehensive series of ablations why $μ$P appears to offer high-quality learning rate transfer relative to standard parameterization (SP), as existing theory is inadequate. We find that the overwhelming benefit of $μ$P relative to SP when training with AdamW arises simply from maximizing the learning rate of the embedding layer. In SP, the embedding layer learning rate acts as a bottleneck that induces training instabilities; increasing it by a factor of width to match $μ$P dramatically smooths out training while improving hyperparameter transfer. We also find that weight decay improves the scaling law fits, while, in the fixed token-per-parameter setting, it hurts the robustness of the extrapolation.

The main XAI attribution methods for deep neural networks -- GradCAM, SHAP, LIME, Integrated Gradients -- operate on separate theoretical foundations and are not formally comparable. We present GRALIS (Gradient-Riesz Averaged Locally-Integrated Shapley), a mathematical framework establishing a representation theory for attributions: every additive, linear, and continuous attribution functional on L^2(Q,mu) admits a unique canonical representation (Q, w, Delta), proved necessary by the Riesz Representation Theorem. This class encompasses SHAP, IG, LIME and linearized GradCAM, but excludes nonlinear functionals such as standard GradCAM or attention maps. Seven formal theorems provide simultaneous guarantees absent in any individual method: (T1) necessary canonical form; (T2) exact completeness; (T3) Monte Carlo convergence O(1/sqrt(m))+O(1/k); (T4) exact Shapley Interaction Values; (T5) Hoeffding ANOVA decomposition; (T6) Sobol sensitivity generalization; (T7) multi-scale extension (MS-GRALIS) with minimum-variance weights. An algebraic appendix justifies the GRALIS-SIV correspondence via the Mobius transform without circularity. GRALIS satisfies 13.5/14 axiomatic properties vs. 2.5-6/14 for individual methods, including completeness, sensitivity, locality, order-k interactions and optimal multi-scale aggregation simultaneously. Preliminary validation on BreaKHis (1,187 histology images, DenseNet-121) reports deletion faithfulness AUC +0.015 (malignant), 96% class-conditional consistency, SAL = 0.762+/-0.109 and sparsity index 0.39. Extended comparison with baseline XAI methods is planned for a companion paper.