Papers

Reconstructing continuous terrain manifolds from massive, unstructured airborne LiDAR point clouds remains challenging in complex Wildland-Urban Interface (WUI) environments, where deep neural networks require costly point-wise annotations and nonparametric surface reconstruction methods often lack structural interpretability. This paper introduces the Gaussian Linear Functional Manifold (GLFM), a physics-informed statistical framework that represents continuous surface topography using deterministic linear functional bases while modeling microscale diffuse laser backscatter as an isotropic Gaussian process. To avoid the quadratic computational cost of exact constrained maximum likelihood estimation, we develop an algebraic singular value decomposition (SVD) rank-reduction algorithm that enables linear-time parameter estimation and closed-form quadric classification. Evaluated on 35.2 km^2 of real-world aerial LiDAR data, GLFM automatically filters ground points and extracts morphological features, achieving an adjusted Rand index (ARI) of 0.9933 against field-verified ground truth and outperforming four leading baselines while maintaining an out-of-core memory footprint. The framework provides a rigorous, interpretable, and scalable foundation for large-scale point cloud analytics.

We study variance-preserving diffusion of the response in mixed linear regression (MLR) with unknown mixing weights. Our analysis separates the statistical guarantees of score matching from the loss geometry and optimization signal at a fixed diffusion noise level. The KL divergence links the denoising score matching objective integrated over the diffusion path with the likelihood and a terminal discrepancy. Under mild regularity conditions and terminal schedule, the resulting estimator converges up to the ground truth parameters of MLR, and its scaled error converges to the Gaussian limit of the maximum-likelihood estimator. At a fixed scale of the diffusion noise level, we derive a decomposition linking the score matching loss to cross-entropy and Expectation-Maximization (EM) operators. This decomposition yields an EM-related low-noise gradient expansion with additional correction terms of latent variance. In the high-noise limit, we further characterize gradient descent on this limiting loss under isotropic covariance. Along fixed high signal-to-noise ratio rays, the score matching imbalance gradient and the latent-variance term tend to zero pointwise. Numerical experiments illustrate our theoretical findings and statistical guarantees.

Single-cell RNA sequencing studies routinely use clustering to define putative cell types and cell states, yet the observed separation may arise from sampling variability rather than genuine biological heterogeneity. This paper studies formal significance testing of such clustering structure in high-dimensional data. Existing SigClust methods assess clustering significance through Monte Carlo simulation under a Gaussian single-cluster null, but this assumption can be unreliable for normalized gene expression data and other non-Gaussian settings. We propose SigClust-LCP, a nonparametric extension that models a single cluster by a log-concave distribution. To make this approach computationally feasible in moderate to high dimensions, we develop a score-matching estimator for log-concave projection inspired by recent generative modeling ideas. We establish theoretical guarantees for the estimator and for its use in clustering significance testing. Simulations show that SigClust-LCP controls Type-I error more reliably than existing methods across a range of unimodal and mixture distributions while retaining competitive power. In a single-cell RNA sequencing analysis of Hydra cells, the method avoids spurious subclusters within annotated cell populations and supports biologically meaningful separation across lineages and body-axis regions.

Estimating a low-dimensional subspace shared across noisy data matrices is a fundamental problem in multi-view matrix estimation. We study this problem under the two-view JIVE model, where each data matrix contains shared and view-specific low-rank components. We demonstrate that standard plug-in subspace intersection, including AJIVE, suffers from a second-order bias caused by direction-dependent leakage of the empirical singular vectors. We propose bias-corrected subspace intersection (BCSI), which removes this bias before estimating the shared subspace. We establish finite-sample risk bounds for BCSI that accommodate unequal view dimensions, signal strengths, and view-specific ranks and require no condition-number assumptions on the signal matrices. When the shared and view-specific ranks are comparable, these bounds match our minimax lower bounds up to universal constants. The resulting minimax rate contains a new second-order term, arising from quadratic leakage perturbations relative to the shrinking spectral gap when the view-specific subspaces are nearly aligned. This term is absent from previous JIVE minimax lower bounds. Numerical experiments demonstrate the advantage of BCSI over AJIVE when the leakage bias is pronounced. Along the way, we establish a nonasymptotic concentration result for the bias-corrected leakage Gram matrix of a rectangular spiked matrix, which may be of independent interest.

Late Gadolinium Enhancement (LGE) on cardiac magnetic resonance is a key marker of myocardial scar, but its limited accessibility motivates routine ECG-based screening. We evaluated whether $\beta$-variational autoencoder (VAE)-derived ECG representations can discriminate LGE+ from LGE- cardiomyopathic patients in a local cohort of 300 subjects. We compared 32-dimensional features from the foundation ECGx.AI model with those from a shallower $\beta$-VAE trained on normal PTB-XL ECGs, evaluating downstream classification and Dynamic Time Warping (DTW)-based reconstruction errors. ECGx.AI reached an area under ROC of 0.686 with Random Forest, while the proposed $\beta$-VAE reached 0.577 with sensitivity of 0.775 with Gradient Boosting. Notably, DTW-reconstruction errors significantly differed between classes in 10 out of 12 leads according to Mann-Whitney U test and help in classification, leading to an area under ROC of 0.643 with Logistic Regression, supporting their potential as markers of scar-related ECG alterations.

Forecasting time series accurately is critical for applications with complex data ranging from energy systems to healthcare and finance. Among current state of the art models, generative latent variable models are increasingly implemented; yet principled generalisation guarantees for modern latent variable models remain limited. In particular, while Variational AutoEncoders are widely used for sequential data, their theoretical analysis is largely restricted to i.i.d. settings. In this work, we develop a PAC-Bayesian framework for latent variables models applied to time series. Building on reconstruction-based bounds, we extend PAC-Bayesian guarantees to Markovian latent structures, capturing temporal dependencies through a sequential generative process. These guarantees do not grow with the length of the trajectory. Our bounds depend on assumptions which are common in the literature; we provide an example framework where they would be verified to show that they are not as restrictive as they may seem.

Dynamical symbolic regression methods identify governing differential equations from noisy data, balancing interpretability and predictive accuracy. However, standard methods often produce expressions that violate known physical laws. To address this, we propose FluxDisco, a physics-informed framework tailored for flux-based, stoichiometric ODE systems. By leveraging a known stoichiometry, we reduce the expression search space and ensure physical adherence. Our framework adapts the Monte Carlo Graph Search algorithm for the unique challenges associated with joint flux discovery of stoichiometric systems. We evaluate our method across a range of physical and biological systems, demonstrating its ability to accurately recover governing dynamics through interpretable equations.

Conformal counterfactual inference enables network operators to use logged telemetry to reliably answer 'what-if' questions about network operation. These answers typically take the form of prediction sets that contain, with a user-defined probability, the key performance indicators (KPIs) that would have been observed under alternative control actions. A key challenge is that logged telemetry may omit variables used by the controller, resulting in hidden confounding and invalidating the statistical guarantees of counterfactual analysis. In principle, this issue can be addressed using randomized telemetry, collected by assigning control actions independently of the network state. However, because such randomization may disrupt normal operation, randomized telemetry is typically scarce, causing counterfactual analysis based solely on it to produce uninformative prediction sets. To address these challenges, we propose Confounding-Valid Counterfactual Conformal Inference (CV-CCI), which combines abundant, potentially confounded observational telemetry with limited randomized data through the General Synthetic-Powered Inference (GESPI) principle. CV-CCI leverages observational data to improve efficiency while using randomized data to retain finite-sample coverage guarantees under arbitrary hidden confounding. Experiments on two representative radio access network (RAN) control tasks show that CV-CCI remains valid under hidden confounding while producing more efficient prediction sets than state-of-the-art confounding-valid baselines.

We study regression under bounded responses in terms of excess mean squared error. When the comparator class is finite, this setting is known as model selection aggregation, and achieving minimax excess risk requires improper learning algorithms. Contrary to this, in the universal learning framework no improperness is needed, as simple empirical risk minimization achieves the best-possible exponential learning rate. Hence, the two frameworks suggest different optimal algorithmic principles. This poses the question of best-of-both-worlds guarantees: Are minimax and universal exponential rates achievable by the same algorithm? For finite hypothesis classes, we answer this question in the affirmative by showing that the $Q$-aggregation estimator - which is known to achieve minimax optimal tails - achieves exponential universal rates. A wide range of other estimators and algorithmic principles (ERM, sequential averaging, pruning, and star estimation) do not achieve both. For countably infinite hypothesis classes, we answer the question in the negative by showing that there is an inherent trade-off between achieving exponential universal and minimax uniform rates. This trade-off is exactly traced by combining optimal algorithms from each world using $Q$-aggregation. Besides these results, we prove several additional structural results about universal rates in learning with squared loss.

We consider density estimation under the relaxed local differential privacy condition that the privatized distributions are $\alpha$-close in total variation distance. We show that adding independent noise with a convenient symmetrized Gamma distribution to each sensitive observation attains the $\alpha$-TV-LDP. We prove that the deconvolution estimator of $r$-Sobolev smooth functions attains the pointwise rate $(n\alpha)^{-\frac{2r-1}{2r}}$ up to log factors which is faster than $(n\alpha^2)^{-\frac{2r-1}{2r+1}}$ under the classical $\alpha$-LDP and closer to the nonprivate minimax rate $n^{-\frac{2r-1}{2r}}$. Next, we use a Goldenshluger-Lepski procedure to build a free of the smoothness adaptive procedure and show optimality of our rates in the convolution model of our privatisation scheme. We illustrate the benefits of this simple privacy mechanism by implementing a neural network estimator which does not need to add more noise in the optimization steps. Numerical results show significant improvement of the estimation rate over the Laplace and the private-SGD mechanisms.

Self-supervised learning relies on so-called data augmentations $\phi(x)$ of unlabeled datapoints $x$ — for example, masking random pixels in an image $x$ — that should leave the label of $x$ invariant and are often used to learn a lower-complexity invariant subspace $\cal V$ for downstream tasks. In practice, such augmentations $\{ \phi_l(x_i) \}$ are pooled together to learn $\cal V$, despite obvious inter-dependencies between different augmentations $\phi_l(x), \phi_k(x)$ of the same datapoint $x$. However, theoretical works on the subject typically consider procedures that avoid such dependencies, and are therefore limited to operate on smaller subsets of independent data. We show in this work that pooling augmentations together, despite inter-dependencies, is a better alternative than the baseline of partitioning the data into subsets of independent data. More precisely, in the context of estimating $\cal V$, the statistical estimation error bounds for pooling are never worse than the partitioning baseline, and in some cases — such as masking or noise injection-based augmentations over a shallow neural network — naive pooling leads to faster rates in terms of the number of augmentations. The benefits of pooling are particularly prominent when the correlations between different augmentations $\phi_l(x), \phi_k(x)$ have mild effects on estimation or help decrease the estimation variance. The analysis, therefore, yields new insights into the success of pooling augmented samples in self-supervised pre-training, and provides an intuition behind the practical preference towards using many augmentations.

While diffusion-based methods have recently emerged as effective tools for probing the intrinsic geometry of high-dimensional data, their statistical difficulty remains largely unexplored. We study estimation of the finite-scale population functional underlying FLIPD (Kamkari et al., 2024; arXiv:2406.03537), a diffusion-based local intrinsic dimension (LID) quantity defined through the logarithmic scale derivative of a Gaussian-smoothed density. Intuitively, Gaussian smoothing turns local dimension into a scale law: near a $d$-dimensional manifold, the kernel mass grows like $\sigma^d$, so differentiating with respect to the noise scale reveals the intrinsic exponent. Under a regular manifold model, we show uniformly over the model class that the finite-scale field differs from the manifold dimension $d$ by at most $O(\sigma^2)$. We then establish a minimax lower bound of order $(n\sigma^d)^{-1}$ for estimating this finite-scale field from $n$ observations, for $n^{-1/(2\alpha+d)}\lesssim\sigma\le\sigma_0$. At the smallest scale covered by our lower-bound construction, the bound becomes the nonparametric rate $n^{-2\alpha/(2\alpha+d)}$.

We show that susceptibilities, an interpretability technique developed for neural networks, can identify the presence of algorithmic structure in Turing machines by probing the local loss landscape of a learning problem for noisy Turing machines introduced by Murfet and Troiani (arXiv:2504.08075). We prove that symmetries and path separation in the algorithm implemented by a Turing machine induce permutation symmetries and low-rank blocks in its susceptibility matrix. We study this empirically on a set of deterministic finite automata (DFAs) and demonstrate that algorithmic features can be recovered by principal component analysis and clustering methods in susceptibility space.

AI-workflow governance cannot be reduced to checking the final answer: an apparently safe answer may rest on a fragile evidence path that ordinary evaluation cannot see, localize or govern. We call this hidden fragility a "structural iceberg": hallucinations and unsupported claims may form its visible tip, while consequential weakness remains submerged. Stochastic semantic evidence graphs (SSEGs) expose these icebergs by preserving workflow channels, propagating local uncertainty and identifying the hidden paths on which an apparently safe output depends. ALCE and RAGTruth show that visible failures at the tip -unsupported citations and hallucinated spans -rest on distinct submerged weaknesses and therefore require different interventions. Across retrieval, tool-use and controlled stress tests, SSEG localizes those weaknesses, supports targeted repair, produces no false automatic passes in 35,000 known-truth cases and reduces ToolSandbox review by 28.8% across 96 executions from two agent models. The same structural view carries into end-to-end governance: in a separately sealed 1,200-case FinGovBench study, adding SSEG to GPT-OSS-20B reduces unsafe releases from 452/660 to 8/660 while releasing all 540 safe cases and correctly distinguishing 592/600 matched workflow pairs. An unchanged-gate transfer to Qwen3-8B releases all 540 safe cases and none of 660 unsafe cases, whereas flat-UQ releases 520 unsafe cases. SSEG therefore moves governance below surface-level output checking, turning hidden evidence dependencies into auditable, path-specific decisions about intervention, revalidation and release.

Millions worldwide require Renal Replacement Therapy (RRT) as a treatment essential for survival. However, optimizing RRT strategies via AI is challenging due to heterogeneous patient dynamics, missing data, and the absence of an AI-oriented health assessment criterion. We propose an AI-Oriented Comprehensive Normalized Assessment (CNA) for healthy status and apply it to optimize RRT strategies by using offline reinforcement learning (RL). The key idea of CNA is transforming vital-sign distributions into a standard normal space, enabling a unified, data-driven health-status score defined by deviations from referent intervals, which also provides an AI-oriented criterion to assess strategy quality and supports RL termination. We further design a structured 23-dimensional state representation that integrates 19 indicators with 4 RRT descriptors, and employ matrix decomposition to reconstruct missing vital signs, improving data completeness for learning. These components are incorporated into multiple offline RL algorithms and validated via systematic ablation studies on RRT feature subsets. Compared with physicians' observed treatments, the best learned strategy reduces mortality from 13.2% to 5.0% (reducing 62.24%) and shortens average in-hospital stay from 308.5 to 250.1 hours (reducing 18.93%), demonstrating both methodological innovation and the potential of CNA-guided RL to improve RRT outcomes in nephrology.

Large language models increasingly rely on long chain-of-thought (CoT) trajectories for complex reasoning, but autoregressive generation brings substantial memory and inference costs. Latent reasoning models (LRMs) offer a more efficient alternative by compressing intermediate reasoning into a small number of continuous latent vectors. Despite their efficiency, however, the adversarial robustness of LRMs remains largely underexplored. In this work, we systematically evaluate the robustness of latent reasoning across textual and multimodal settings, covering eight models and six benchmarks. We find that, across our evaluated settings, LRMs are generally less robust than explicit CoT baselines under adversarial perturbations, with particularly severe degradation under white-box attacks. Further analysis reveals distinct failure modes across modalities: textual latent states exhibit brittle dynamics and high sensitivity to specific input patterns, while latent states in multimodal models can remain largely invariant to input perturbations and have limited influence on final predictions. These findings expose robustness limitations of current latent reasoning approaches and highlight the need to jointly consider efficiency and robustness when designing implicit reasoning systems. We have open-sourced our code to facilitate reproduction of our research https://github.com/PKU-ML/latent-reasoning-model-assessment.

Decode-time alignment methods steer a frozen language model by scoring candidate continuations with an external reward and selecting the maximiser. We argue that this shared design is a single degenerate point in a much larger space. We introduce Swiss-Knife, a framework for externalised multi-objective alignment in which the alignment specification is a first-class runtime object: hot-swappable scoring blades, a batch normaliser, a pairwise aggregation operator, and a selection rule. Six axioms characterise the admissible aggregation operators, and we prove a representation theorem: every operator satisfying them has the form $R_i = \sum_{j \neq i} g((\mu_i - \mu_j)/s(\sigma_i,\sigma_j))$, a two-parameter family containing probit and logistic comparison rules and pointwise argmax as named coordinates. Within it, pairwise aggregation is Lipschitz-stable under adversarial reward contamination while argmax is not, and Candidate-Batch Normalization (CBN) makes the weight simplex invariant to the rescalings under which reward models are only ever identified. Our reference instantiation pairs DPO-LoRA blades with an uncertainty-aware pairwise tournament. Sweeping the helpfulness/honesty/harmlessness simplex, it attains the best balanced frontier of six decode-time methods (harmonic $F_1$ 0.797 vs. 0.750 for the strongest baseline, $p < 10^{-14}$) with the lowest refusal rate and highest helpfulness of any arm, and reconfigures its objectives in 0.05 ms with no gradient computation. Ablating CBN costs 0.217 $F_1$, and the metadata confirms the predicted mechanism: the lowest-variance blade retains 9% of its nominal 33% influence, and the collapse follows the predicted ordering.

Rollcast is a probabilistic forecasting method for univariate time series that combines a compact set of rolling statistical anchors rather than relying on a single global model. Rolling means, medians, extrema, regression endpoints, and quantiles define candidate forecast locations and a representation of the current state. A state-dependent softmax gate learns anchor probabilities by minimizing negative log predictive density, while residual distributions retrieved from similar historical states provide local uncertainty. Recursive simulation propagates the resulting mixture over multiple forecast horizons. The method is evaluated in a Monte Carlo study covering eight data-generating processes, including autoregressive, random-walk, local-trend, threshold, regime-switching, stochastic-volatility, heavy-tailed, and variance-break dynamics. Across 2,000 independent fitted series, Rollcast is compared with the true conditional predictive distribution generated by an oracle simulator. Overall empirical coverage is 86.2% for nominal 90% intervals and 91.5% for nominal 95% intervals. Predictive intervals are on average 13.6% wider than the oracle at the 90% level, while CRPS is 14.4% higher than oracle CRPS. Performance is closest to the oracle under autoregressive, threshold, stochastic-volatility, heavy-tailed, and variance-break dynamics, while local trends and regime switching are more challenging. The results indicate that Rollcast can construct competitive probabilistic forecasts from simple, interpretable local summaries, while also identifying limitations in calibration and recursive uncertainty propagation.

Let a finite population of n labelled examples carry a class-weighted loss, with pi*n in a rare positive class weighted by N0/N1. We study estimation of total risk from a subsample K << n under designs allocating K0 and K1 draws to the two strata. We derive the exact finite-population variance of the weighted risk estimator under class-conditional sampling without replacement and solve for the optimal allocation. The class multiplier inflates positive-stratum dispersion by the imbalance ratio, causing that ratio to cancel from the optimal allocation and making equal, rather than proportional, allocation the natural default. Simple random sampling is dominated by an explicit between-stratum term; an exact bias identity shows that cluster-representative selection has no general unbiasedness guarantee; and a Serfling bound transfers the allocation result to selection error over a finite candidate set. Under the implemented truncation, the realised allocation ratio is gamma=min(2*pi/f,1), independent of n, yielding the parameter-free efficiency prediction A(pi,f)=gamma/[pi(1-pi)(1+gamma)^2]. A separate measurability result bounds the record occupied by a labelled example, determining the required train-test separation and controlling departure from block independence under absolute regularity. We test these predictions on forecasting the onset of statistically explosive price regimes, dated ex post by the Phillips-Shi-Yu procedure, using 350 U.S. equities from 2004-2011 with under 1% positive rows and five purged forward blocks. The predicted ordering of the four designs holds, and at the 10-day horizon the five design points are ordered exactly as predicted by A (Spearman rho=1, exact p=0.0167). The predicted dependence on pi across horizons does not hold; we identify the channels lying outside the design-based argument.

Large-scale logistics networks require synthetic data generation capabilities to support scenario-based planning under novel conditions-such as network reconfiguration and demand shocks. Existing approaches, which rely primarily on historical observations, lack the ability to generate demand patterns that adapt to changes in network topology while respecting operational constraints. We propose a constraint-aware conditional generative framework for synthetic origin-destination demand generation in hierarchical logistics networks. The framework models demand as a conditional distribution over destinations given each origin, enabling topology-aware synthesis that is both topologically realistic and operationally feasible. Operational guidance is incorporated directly into the generative objective via differentiable constraints, while a flexible conditioning mechanism supports various operational contexts and adaptation to evolving network configurations. We instantiate the proposed framework based on a conditional generative model. Experimental validation on industrial real fulfillment and transportation network demonstrates 16% improvement over graph neural network baselines, 87% operational compliance, and efficient cold-start adaptation, enabling applications in capacity planning, network design evaluation, and routing optimization.

Nano self-assembly organizes molecular components into bioactive nanoscale structures. Self-assembled nanoparticles (NAPs) derived from Chinese herbal formulas and applications such as anti-lung-cancer therapy demonstrate the substantial potential of self-assembly for nanomedicine discovery. Yet discovery still relies on costly wet-lab screening, while existing machine learning approaches lack standardized tasks, effective pairwise compatibility modeling, and public benchmarks with unified evaluation. To address these limitations, we formalize NSA prediction as a binary classification task for predicting self-assembly between molecular pairs and then establish NSA-Bench, the first public benchmark with curated molecular combinations, experimental conditions, self-assembly labels, and standardized evaluation protocols. We further develop NSA-Net, an interaction-aware multimodal framework that integrates complementary molecular evidence from graph topology, sequence semantics, and physicochemical descriptors to learn molecular-pair representations for self-assembly prediction. Extensive experiments on NSA-Bench show that NSA-Net achieves a ROC-AUC of $0.9470\pm0.0112$ (Small) and $0.9492\pm0.0062$ (Large). On the Small track, it surpasses the strongest machine-learning and graph-based baselines by 3.9 and 17.1 percentage points, respectively. Representation analyses reveal interpretable molecular characteristics associated with self-assembly prediction captured by the learned representations. Moreover, an NSA-Agent case study further demonstrates how NSA-Net predictions can support formulation refinement through experimental-condition-aware reasoning. Our code is available at https://github.com/developer-hq/NSA-Net.

Multi-Task semantic communication (SemCom) prioritizes simultaneous execution of multiple tasks over bit-accurate reconstruction in future intelligent networks. In our prior work [1], we introduced the cooperative multi-task SemCom (CMT-SemCom) framework, in which the semantic encoder is divided into a common unit (CU) and multiple specific units (SUs) to facilitate cooperative multi-task processing. However, CMT-SemCom has been evaluated on homogeneous classification tasks on simplistic datasets, limiting its applicability to real-world perception systems. In this paper, we extend our CMT-SemCom to jointly handle heterogeneous classification and regression tasks on the complex Cityscapes dataset. We adopt the information maximization (InfoMax) principle so that it accommodates mixed discrete and continuous semantic variables. In particular, we benchmark the proposed framework against independent single-task training, a conventional task-agnostic digital transmission, and single-encoder multi-decoder SemCom. Additionally, we investigate the impact of CU capacity on joint task performance, providing design insights. Extensive evaluations demonstrate that CMT-SemCom significantly outperforms the benchmarks.

Attention mechanisms are central to modern foundation models, yet their training dynamics remain poorly understood, especially when the attention matrices have extensive rank. In this work, we study attention-indexed models, a broad framework that can represent multi-layer and multi-head attention architectures. First, we show that, in a suitable high-dimensional limit, the population-loss landscape is characterized by a finite set of trace order parameters. In contrast, online stochastic gradient descent (SGD) is governed by an infinite hierarchy of matrix moments, which we show can be exponentially well-approximated by a finite truncated system. Second, this framework reveals that attention parameterization itself can act as an architectural implicit bias. Direct optimization of an attention matrix $S\in\mathbb{R}^{d\times d}$ can remain trapped in an uninformative state. Tied attention ($S=WW^\top$) induces an automatic symmetry-breaking mechanism and yields weak recovery in $\Theta(d^2\log d)$ samples. For untied attention, $S=UV^\top$, we uncover a fast-slow mechanism: the pre-activation mean first evolves on a fast timescale, while the overlaps evolve on a slower one. Weak recovery on the $\Theta(d^2\log d)$ scale occurs when the state selected by the fast dynamics breaks the initial symmetry.

High-dimensional simulation of multivariate extremes is fundamentally limited by the combinatorial complexity of dependence, often more than by the scarcity of extreme observations. We show that symmetry admits a lossless orbit-space representation that preserves structured extreme dependence while replacing an exponentially large dependence space with a compact set of symmetry classes. Based on this principle, we develop Q-Edge (Quantum Extreme Dependence Engine), a symmetry-reduced quantum framework that operates directly in orbit space, enabling scalable simulation and digital twins of structured extreme systems. By transferring symmetry into the data representation rather than the quantum circuit, Q-Edge allows unconstrained quantum generative models to exploit dramatically reduced state spaces. For a 30-dimensional problem, approximately 1.6 million angular states collapse to 256 orbit states, reducing the required quantum representation from about 21 qubits to 8. Our results establish a general computational principle for scalable quantum simulation of structured extreme dependence.

We study the large-depth behavior of residual networks whose weights are correlated across layers at initialization. Our results confirm and extend a conjecture of Marion et al. [2025], according to which correlated initializations should interpolate continuously between the Brownian stochastic differential equation arising from independent initialization and the ordinary differential equation arising from perfectly correlated initialization. When the initialization is obtained from the application of a feature function to a stationary Gaussian sequence with regularly varying correlation, we prove that there exists a unique critical scaling such that the infinite-depth limit is the solution of a Young differential equation driven by a Hermite process. Hermite processes reduce to the fractional Brownian motion if the feature function generating the initialization has Hermite rank one, which is the case for the identity function, for example. We show that the critical scaling and asymptotic limit are uniquely determined by the decay of correlations together with the Hermite rank of the feature function. Consequently, the correlation structure and Hermite rank of the initialization represent meaningful hyperparameters in the asymptotic regime. By contrast, under finite-variance iid initialization, the asymptotic driver is universally Brownian up to normalization regardless of the choice of distribution. Our proofs rely on a collection of novel results establishing a robust stability theory for Young differential equations in Banach spaces.

Mixture-of-experts (MoE) architectures increase model capacity by combining a collection of expert predictors through input-dependent routing, while often activating only a small subset of experts for each input. Despite their growing importance in modern large-scale models, the statistical roles of their design choices, especially routing, sparse activation, and shared experts, remain only partially understood, as existing theory has largely focused on parametric or correctly specified MoE models. In this paper, we view MoE as a form of localized aggregation and show how this localization reshapes the approximation-estimation-computation tradeoff. We derive oracle risk bounds for learning dense and sparse routing with evolving experts, separating approximation, expert-learning, and router-estimation errors, and characterize how sparse Top-K routing can retain the benefits of localized aggregation while controlling per-input computation. We also interpret gating through the geometry of input space, relating routing performance to regions of local expert advantage, and show how shared experts, as adopted in architectures such as DeepSeekMoE, can extract common predictive structure so that routed experts focus on residual local variation. Together, these results provide a unified statistical framework for understanding MoE through input-dependent expert aggregation, in which expert specialization and computational tradeoffs are governed by local predictive structure.

This paper examines the relationship between the parameters of autoencoder models and the statistical properties of the data on which they are trained. Autoencoders are defined as models with an encoder-decoder architecture, trained to reconstruct input data through a compressed latent representation. It is proposed that the model parameters can be viewed as a dense vector representation of the corresponding sample. To test this hypothesis, a theoretical and experimental study is conducted in which a vector representation is formed based on the spectral characteristics of the autoencoder parameter matrices. Theoretical analysis shows that the singular values of the model parameter matrices are related to the eigenvalues of the covariance matrix of the training data, ensuring the transfer of information between the data space and the parameter space. Experimental results on the CIFAR-10 and FashionMNIST datasets confirm that the resulting vector representations allow for a high degree of accuracy in distinguishing between models trained on different data subsets, without resorting to complex vector generation algorithms or using the original samples. These results suggest that the parameters of trained autoencoders can be viewed as sample representations.

Logit-based knowledge distillation for autoregressive language models usually aligns teacher and student next-token distributions over the entire vocabulary. However, this global objective overlooks relative preferences among likely token alternatives. Existing local approaches often select candidate tokens from either the teacher or the student alone. Teacher-only selection can miss tokens that the student considers likely, while student-only selection can rely on an inaccurate ranking early in training. We propose Adaptive Local Relational Alignment (ALRA), a position-specific framework combining student proposals with teacher guidance. At each valid prediction position, the student proposes likely tokens, while the teacher's most probable token is included as an anchor. ALRA adjusts the number of selected tokens according to how broadly the teacher distributes probability within this candidate set relative to the current batch. Adaptive Local Divergence retains the mass-matching term and separately matches the relative token distributions within the selected and remaining vocabulary regions. Unlike the exact full-vocabulary decomposition, it replaces the teacher-mass coefficients of the two conditional terms with unit coefficients, preventing either term from being downweighted solely because its region has low teacher probability. Student-Weighted Pairwise Relational Alignment emphasizes high-probability token pairs with small student probability gaps and gives less weight to unlikely or clearly separated pairs. Experiments on The Pile with randomly initialized 200M- and 500M-parameter students across nine zero-shot benchmarks yield average accuracies of 36.62% and 37.40%. ALRA exceeds the strongest competing distillation baseline by 0.94 and 0.83 percentage points and improves over pre-training without distillation by 2.31 and 2.91 points, respectively.

Language models encounter typos, corrupted text, altered words, and disrupted token order, yet robustness is usually evaluated only through output behavior. We study how six naturalistic and synthetic input perturbations propagate through decoder-only language models at three levels: output behavior, hidden-state geometry, and attention-head function. We evaluate behavioral effects across four GPT-2 and two Qwen2.5 checkpoints, analyze layerwise geometry using centered kernel alignment and intrinsic dimension, and examine attention-head responses in GPT-2. Perturbation types produce distinguishable metric profiles that are not fully captured by output measures and are only partly consistent across the tested checkpoints. Copying scores show the strongest pooled associations with activation-patching recovery under token substitution and shuffling, although these associations do not isolate copying-specific effects. Gradient-guided HotFlip perturbations also cause stronger behavioral and representational disruption than rate-matched random token substitutions in GPT-2; their behavioral effects are consistent across all six tested checkpoints. Our results show that robustness claims based on a single behavioral or representational metric can be misleading, and motivate multi-level evaluation of how perturbations alter language-model computation.

Validating a model on a time series asks for three things at once: each training run should use most of the sample (sufficiency), the test sets should together cover most of the sample (coverage), and training data should come before test data (causality). We prove that the three cannot be had together and price each one. Let $\alpha$ be the smallest training fraction over folds, $\beta$ the fraction of the sample covered by tests, $\Lambda$ the fraction of the sample used as training data from the future of a test point, and $\delta$ the distance from a test point to the nearest training point in its future. Every scheme on a sample of length $T$ satisfies $\alpha+\beta \le 1+\Lambda$ and $\alpha+\min\{\beta,\delta/T\} \le 1$, and under $\beta$-mixing the leakage bias at a test point is at most $2M\beta_{\mathrm{mix}}(\delta)$. In words: going beyond the causal frontier $\alpha+\beta=1$ requires training on the future; that future data must sit within $(1-\alpha)T$ of a test point; and its harm depends on its distance, not its amount. Hence expanding walk-forward is exactly the Pareto frontier of causal validation, $k$-fold cross-validation buys the most future data, and purged $k$-fold with an embargo pays in distance instead, which is cheap when the process forgets quickly but cannot repair the part of causality demanded by non-stationarity. On pure noise, shuffled 5-fold reports an information coefficient of $+0.32$, while contiguous 5-fold, using the same amount of future data, reports $+0.004$.