Papers

Deep operator networks can become statistically unstable when partial differential equation inputs are observed at thousands of strongly correlated sensors but only a small number of operator samples is available. We introduce FAST-DeepONet, a branch representation combining a fixed spectral path with a regularized projection of the orthogonal residual, in which the directional penalty acts on the effective residual map after each of its rows is normalized. On Navier–Stokes flow a plain DeepONet degrades from $0.0394$ to $0.1556$ mean relative $L_2$ error as the branch grows from $129$ to $8193$ coordinates, while FAST-DeepONet stays near $0.04$, so the sensor grid can be refined without a statistical penalty. Across independent test sets for Navier–Stokes flow, Darcy flow, and signed terminal wavefield prediction it lowers mean relative $L_2$ error by $4.7\%$ to $37.0\%$ with three to seven times fewer trainable parameters. A spectral-only branch sharing the same basis separates the two paths: the fixed spectral path carries the improvement on Navier–Stokes and Darcy, while terminal wave prediction requires the residual path together with its directional penalty. FAST-DeepONet targets coordinate-query architectures and trains on solution values alone.

Generative model alignment has received broad interest, and significant progress has been made in supervised fine-tuning and inference-time computation. Yet, alignment has remained poorly understood from a statistical learning perspective. We formulate inference-time alignment as a weak-to-strong learning problem, where a reference policy (weak model) is assumed to be fairly good and the goal is to produce a strong model that predicts a good response at test time with arbitrarily high probability. Our problem is formulated as learning from scratch — everything is learned from data rather than assuming access to a good reward estimate, and thus differs from the existing inference-time alignment theory. Our framework shares similarity to the recent work of Joshi et al., (arXiv:2510.15464), where for each prompt, there could be multiple good responses. Our definition of the alignment learnability follows the standard PAC learning principle. We introduce a novel combinatorial dimension of the reward class which we call the alignment dimension, and show that it completely characterizes the alignment learnability — a reward class is alignment learnable if and only if its alignment dimension is finite. The core of our learning procedure works by learning a pairwise comparator and then running a tournament over candidate responses. We believe that our results might shed light toward establishing a complete theoretical understanding of alignment.

Topological deep learning (TDL) methods rely on lifting raw data into higher-order discrete domains such as simplicial complexes, cell complexes, and hypergraphs. In practice, this lifting step is often treated as a black box: practitioners select a lifting and then tune architectures, with limited visibility into whether the induced higher-order connectivity is meaningful for the downstream task. To address this missing diagnostic layer, we propose a visualization technique called TopoExplorer that leverages the strictly augmented Hasse graph form of topological datasets for exploratory data analysis. For the first time, practitioners can easily visualize the incidence- and adjacency-based neighborhoods that define the lifted dataset, as well as read off key graph metrics that describe its structural and feature landscape. Via an extensive set of experiments across many datasets and liftings, we show that several of these metrics correlate with downstream model performance, suggesting they can help inform TDL preprocessing design. Our perspective reframes the TDL workflow from lift-train to lift-look-design-train, enabling more principled, interpretable, and efficient model development. TopoExplorer is hosted at https://topoexplorer.pagekite.me, and its source code is available at github.com/geometric-intelligence/topoexplorer.

Sparse mixture-of-experts (MoE) language models reduce arithmetic by activating only a small subset of experts per token, yet deployment still requires storing and moving the full expert bank. We present ExactMoE, an inference design that applies symmetric group-128 four-bit weight quantization only to routed experts, stores those experts in kernel-native MARLIN form in pinned host memory, and executes all selected experts through a configurable GPU-resident slot cache and fused grouped MoE kernels. The router, attention, embeddings, normalization layers, and language-model head remain in BF16. "Exact" refers to complete expert availability and an unchanged top-k routing procedure: no expert is pruned, substituted, or forced to execute on the CPU. It does not imply numerical identity with the BF16 model. On OLMoE-1B-7B-0924-Instruct, evaluated on a single NVIDIA L4, a 16-slot configuration reduces peak reserved GPU memory from 14.168 to 1.836 GiB (87.04%) while retaining 81.85% of BF16 decode throughput. A fully resident 64-slot configuration reaches 31.923 tokens/s versus 21.662 tokens/s for BF16 while reserving 4.061 GiB. Across 12,450 zero-shot multiple-choice questions, ExactMoE obtains 70.3534% normalized accuracy versus 70.8996% for BF16, retaining 99.23% of the baseline accuracy. In a matched 16-token ablation, fused grouped execution is 1.97x as fast as a sequential W4 reference. These results identify a practical memory-transfer-throughput frontier for complete-expert MoE inference.

Low-Rank Adaptation (LoRA) enables efficient federated fine-tuning of large language models, but its factorized parameterization creates a tension between accurate aggregation of local updates and continuity of locally optimized factors. Factor-wise aggregation incurs aggregation mismatch but better preserves factor continuity, whereas product-space reconstruction reduces this mismatch at the cost of greater factor-level initialization mismatch from newly reconstructed factors. We propose FedPA-LoRA, a product-aligned federated LoRA framework that jointly addresses these limitations and provably converges under both homogeneous and heterogeneous client ranks. Each client preserves its local factors across communication rounds and aligns its product toward a rank-specific global reference, maintaining local optimization continuity while promoting global consistency under data heterogeneity. The server aggregates heterogeneous-rank updates in the common product space and efficiently reconstructs a rank-constrained global adapter without forming the dense aggregate. This design supports client-specific computation and communication budgets. Experiments on natural language understanding and generation tasks show that FedPA-LoRA consistently outperforms representative baselines across varying levels of data heterogeneity and homogeneous- and heterogeneous-rank settings, with up to a $6.82$ percentage-point improvement in average GLUE accuracy under heterogeneous client ranks.

Inverse physics-informed neural networks (PINNs) can reconstruct a field accurately while returning an incorrect physical parameter. We introduce a two-axis post-training diagnosis that separates finite-sample resolution under a specified observation-and-estimation protocol from the signed parameter preference encoded by the final learned field and residual metric. The first axis repeatedly fits noisy observations with a matched forward estimator. At known synthetic truth, the second freezes the field and residual view and computes a local score displacement toward a nearby residual-profile minimum. Endpoint consistency then tests whether joint training delivers that preference under the same final view. Across three synthetic one-dimensional, scalar-parameter PDEs, matched-forward mean absolute relative error ranges from 2.34 percent to 17.46 percent. The displacement tracks frozen-profile minima across locked seeds, architectures, and fresh-noise retraining (r from .945 to .982), and it tracks delivered signed log-error in 240 fresh-noise RBA runs (r = .994; 237/240 correct directions). A coupled two-parameter Darcy check validates the full matrix calculation. The axes are complementary diagnostic coordinates, not additive error components or a deployable oracle-free estimator. Together, they route follow-up work toward observations, residual evidence, or endpoint delivery.

Regret minimization (RM) and best-arm identification (BAI) are two fundamental objectives in multi-armed bandits. Among regret-minimizing algorithms, $1/2$-Tsallis-INF is a canonical best-of-both-worlds FTRL algorithm: it achieves logarithmic pseudo-regret in stochastic bandits while retaining minimax-optimal regret in adversarial bandits, without knowing the environment in advance. This raises a natural question: can the same algorithm, without additional exploration, also identify the best arm reliably? We study this question in stochastic bandits by analyzing the failure probability $\operatorname{Err}_t$, defined as the probability that the empirical best arm determined by the cumulative importance-weighted loss estimates of 1/2-Tsallis-INF differs from the true optimal arm. The main difficulty is that, at the logarithmic-regret scale, suboptimal arms are sampled with probability heuristically of order $1/t$. Consequently, importance weighting causes the cumulative estimator to fluctuate on the same linear scale as its mean separation. To overcome this obstacle, guided by a diffusion toy model, we construct a Lyapunov function for the gap process between the estimated cumulative loss of the optimal arm and that of the best competing arm. This leads to polynomial upper bounds on $\operatorname{Err}_t$: for learning rate $\eta_t=\alpha/\sqrt t$, $\operatorname{Err}_t$ decays at rate $t^{-2+\alpha^2\mu_{i_*}/4+\rho}$ for any $\rho>0$, where $\mu_{i_*}$ denotes the mean loss of the true optimal arm. We also establish a lower bound $\Omega(t^{-2-\varepsilon})$ for any $\varepsilon>0$, showing that the exponent $2$ is essentially tight.

We propose a methodology based on the standard ReLU Deep Neural Networks (DNN) to make predictions and quantify their uncertainty. Classically, people rely on linear, non-linear, or non-parametric kernel methods to fit and then predict the time series. As the universal approximation ability was revealed for DNN, its application has become more and more popular for prediction tasks in various scientific areas. However, the corresponding uncertainty quantification has not been studied thoroughly. Particularly, the uncertainty in prediction will consist of two parts: (1) the future variability; (2) the estimation variability within training data. To capture both variabilities, we build the so-called pertinent prediction interval (PPI) with the DNN model estimator. We first explore the consistency property of the DNN estimator with beta-mixing dependent data. Subsequently, we show that the implied forward bootstrap series is still beta-mixing and possesses the same stationary distribution as the original time series in probability, which is a key condition to enable the PPI. Lastly, the desired PPI is built after imposing minimal conditions on the limiting distribution of predictive roots. Simulations and real-data analysis are deployed to challenge our approach with standard non-parametric methods.

While Parameter-Efficient Fine-Tuning (PEFT) has substantially reduced the hardware cost of adapting Large Language Models (LLMs) by decreasing the number of trainable parameters, recent studies have sought to further improve PEFT through parameter sharing. However, these approaches either employ uniform parameter sharing across layers, which can delay convergence, or rely on dynamic masking strategies, which add computational overhead. The potential of sharing patterns inspired by the inherent hierarchical structure of Transformer architectures remains unexplored in PEFT. To address this gap, we introduce SAPE (Sandwich Adapters for Parameter Efficiency), a PEFT framework based on a sandwich-style hard weight-sharing topology. SAPE routes intermediate Transformer layers through balanced shared group adapters while strictly isolating the input embedding and final projection boundary transformations to prevent gradient interference. This design significantly reduces memory consumption while eliminating the computational overhead associated with dynamic parameter-sharing methods. Extensive evaluations across encoder-only and causal decoder architectures demonstrate that SAPE achieves state-of-the-art performance in low-parameter regimes. On natural language understanding, SAPE outperforms proPETL on RoBERTa-large while utilizing only 10% of the baseline's parameter budget. On natural language generation and world knowledge reasoning with LLaMA-3.2 (3B) under a strict ~0.6M parameter constraint, SAPE outperforms AdaLoRA, yielding absolute improvements of +4.85% on GSM8K and +3.11% on CommonsenseQA. Furthermore, through comprehensive topological ablations, we formalize an inherent capacity trade-off: while hard parameter sharing strongly regularizes semantic generalization, it slightly smooths the sharp layer-wise transformations required for rigid multi-step arithmetic reasoning.

Nominal LoRA rank is a design parameter; calibrated spectral evidence is a separate inferential quantity. This article develops a finite-sample framework for inferring effective rank structure in public foundation-model adapters. The theoretical core is an exact chi-square divergence for the fixed-dimensional Gaussian rank-one reference experiment, with an unknown signal direction integrated under a rotation-invariant reference prior. The resulting series yields a computable finite-sample Le Cam bound at concrete layer sizes, an explicit remainder bound for numerical truncation, and the rectangular Baik-Ben Arous-Peche (BBP) limit. A compact-manifold Laplace expansion shows that finite-sample likelihood evidence also depends on leading spectral gaps through the factor $s_1^{|m-n|}\prod_{i\ge2}(s_1^2-s_i^2)$, motivating joint calibration of clustered singular values. Building on these results, we introduce an empirical-null workflow for PEFT LoRA adapters: factor reconstruction, Monte Carlo $p$-values, stagewise and block testing, and module-wise and corpus-level BH reporting. In an audit of 26 public adapters, 684 modules, six architecture families, and 31,770 public-checkpoint spectra rows, calibrated effective rank is typically much smaller than nominal rank and differs systematically from 95% energy retention. A measured RoBERTa-RTE slice on $n=24$ examples illustrates the measurement path from calibrated ranks to task evaluation, without treating the slice as a utility study. The main empirical finding is that calibrated effective rank is usually far below nominal rank, and that energy retention and statistical surprise answer different questions.

We disprove the expectation stated by Xin, Yu, and Ronney that the physical positive part strain $G$-equation should possess an effective burning velocity in cellular flows. For the standard cellular flow in dimension two $V_A(x_1,x_2)=A(-\sin x_1\cos x_2,\cos x_1\sin x_2)$, if $0<d<20/399$ and $\sqrt{1+4d^2}<Ad\le1+d/10$, then for every unit planar slope the periodic correction develops oscillations at least linearly in time. The solution remains bounded below on an explicit horizontal channel through $(\pi,0)$, while at $(\pi/2,0)$ it decreases at rate at least $CA/\log A$, with $C>0$ universal. The same conclusions hold for arbitrary continuous periodic perturbations of planar initial data. Under the physical scaling $V_A(x/\varepsilon)$ and $d_\varepsilon=\varepsilon d$, an order one value gap persists between points at distance $O(\varepsilon)$ at every positive macroscopic time, so the rescaled solutions have no locally uniformly convergent subsequence. The proof uses the Hamiltonian sandwich $H_{\mathrm{unc}}\le H_+\le\widehat H$. The upper comparator $\widehat H$ is a rectangular support function, equivalently an upper expectation over a state-dependent credal set, whose reversed control dynamics possess an invariant comparison channel. We also prove that for any $C^2$ incompressible periodic flow, every $\varepsilon$-outward barrier certificate has covering radius at most $2d\varepsilon$ for all sufficiently small $\varepsilon$. We further discuss implications for statistics and machine learning: rectangular, time-consistent local uncertainty need not imply forgetting of the initial state in the long run, so additional global stability or ergodicity conditions are needed in robust sequential decision making. Two Lean 4 appendices record conditional formalizations of a sufficient $p=e_1$ subregime and of the logical assembly of the rigidity theorem for barrier certificates.

In this paper, we propose SHOPCA (Shape Operator-based Principal Component Analysis), a novel method for unsupervised metric learning and dimensionality reduction that incorporates differential geometric information into the covariance structure of classical PCA. SHOPCA regularizes the global covariance matrix using the mean shape operator, defined as the average of the absolute local shape operators estimated from the data manifold, steering principal components toward directions of both maximum variance and informative curvature. A single trace-normalized mixing coefficient $\alpha$ controls the regularization, recovering standard PCA at $\alpha = 0$ and a curvature-driven embedding as $\alpha \to \infty$. We further introduce a fully unsupervised criterion for selecting $\alpha$ based on the spectral eigengap of the regularized covariance matrix, maximizing the relative separation between the top-$d$ and remaining eigenvalues without using class labels. We evaluate SHOPCA on more than 50 real-world benchmark datasets, comparing it with PCA, ISOMAP, and UMAP using Adjusted Rand Index (ARI), Normalized Mutual Information (NMI), Fowlkes-Mallows index (FM), and V-measure. Results show that SHOPCA consistently improves clustering quality over PCA across a broad range of datasets and surpasses UMAP on small-sample settings, where iterative neighborhood-based manifold estimation can degrade. SHOPCA is computationally tractable, parameter-efficient, and applicable to domains requiring fully unsupervised, geometry-aware dimensionality reduction.

Multimodal data collected by heterogeneous devices are used for collaborative training, where federated learning (FL) serves as a key paradigm for effective distributed modeling with data privacy preservation. However, local training suffers from the forgetting of previously learned global knowledge under cross-client data heterogeneity, which leads to significant declines in both performance and convergence speed. Most previous studies rely on global alignment strategies to retain global knowledge, which hinder local optimization and lead to inadequate supervision of missing classes. Some studies introduce proxy datasets to supplement supervision for missing classes. However, it remains a challenge to balance class-wise global consistency and local optimization objectives without proxy datasets. In this work, we propose FedADB, a Class Anchor-Driven Dual-Branch FL framework. Specifically, the server generates class anchors optimized in a differentiable input space, which are shared across clients. These class anchors serve as global references that provide supervision for missing classes during local training. A dual-branch collaborative training mechanism is designed for clients. In this mechanism, the anchor-based global branch focuses on learning with global consistency, achieving global knowledge alignment by class-anchor balanced sampling. The local calibration branch focuses on learning discriminative local features, mitigating the degradation of local representations caused by excessive global alignment. Extensive experiments across multiple medical and natural datasets demonstrate that FedADB achieves significant improvements in both accuracy and convergence speed.

High-throughput single-cell and spatial transcriptomic technologies provide high-resolution snapshots of heterogeneous cellular states, but their destructive nature prevents repeated measurements of the same cells over time. Consequently, temporal and spatial dynamics must be inferred from independently sampled, unaligned cell populations, making it challenging to reconstruct developmental trajectories. Optimal transport (OT) offers a geometric framework for aligning cell populations and inferring developmental trajectories, but many existing approaches focus on modeling the evolution of distributions of cells in gene expression space rather than the relational structure encoded by gene expression networks. To address this limitation, we introduce a geometric framework for analyzing the spatiotemporal evolution of gene expression networks through embeddings in Gromov–Wasserstein (GW) space. By representing each developmental stage as a graph combining gene expression and spatial proximity, our approach enables comparisons of network structure across time, continuous interpolation between developmental stages via GW geodesics, and quantification of network-level changes using Ollivier-Ricci curvature. We evaluate our framework on a spatiotemporal transcriptomic Drosophila dataset and show that GW geodesic interpolations reproduce main trends in curvature dynamics observed in empirical gene expression networks. Agreement with higher-order Co-Optimal Transport (COOT) distances, which jointly represent spatial and temporal information, further validates the framework and suggests that hypernetwork representations successfully record salient biological changes across time. In general, our approach provides a unified geometric approach to study dynamically evolving biological networks.

Sparsely-activated Mixture-of-Experts (MoE) Transformers universally fix the same number of routed experts across all layers, a convention that ignores the well-documented heterogeneity in layer-wise redundancy. We demonstrate that this uniformity is systematically suboptimal and propose MAPLE, a plug-and-play framework that reallocates the routed-expert budget heterogeneously across layers of any pretrained MoE LLM, without modifying weights or requiring retraining. Our core contribution is a closed-form sensitivity-guided allocation: we probe each layer's response to variation in expert count, quantify sensitivity using three measures, and derive an analytically optimal budget assignment that directs capacity towards sensitive layers and absorbs reductions in redundant layers. This closed-form solution is further refined by a sensitivity-constrained genetic search that uses layer-wise sensitivity as a prior to guide exploration, yielding faster convergence and superior allocation quality. On four MoE models spanning different scales and architectures, MAPLE outperforms uniform and pruning-based baselines under a 75% routed-expert budget. Notably, on DeepSeek-MoE-16B, MAPLE uses only 75% of the experts yet surpasses the original 100% expert-uniform baseline on ARC-E, ARC-C, and BoolQ, improving accuracy from 65.09 to 71.40, 48.49 to 51.50, and 80.03 to 82.38, respectively. These accuracy gains translate into measured deployment efficiency: implementing MAPLE in SGLang reduces single-GPU end-to-end serving latency by 32.2% and improves throughput by 47.4%. These results show that well-designed heterogeneous allocation can be more effective than simply activating more experts, establishing it as a principled and practical axis for improving MoE efficiency.

The increasing availability of large and complex datasets across many scientific disciplines has led to widespread adoption of machine learning (ML) for prediction. However, most ML algorithms focus on point estimation and provide limited information about predictive uncertainty or the conditional distribution of the response, restricting their ability to characterize rare or extreme outcomes. We develop QXGB, a quantile-based gradient boosting framework, and introduce a convolution smoothed loss within it that estimates conditional quantiles for constructing dense cumulative distribution functions (CDFs), exceedance probabilities, and tail behaviour relevant to extreme outcomes. This approach preserves the computational efficiency of extreme gradient boosting while restoring the Hessian information XGBoost relies on for tree splitting, in turn providing interpretable measures of extreme value and exceedance probability predictions. We derive the gradients and Hessians needed to integrate convolution smoothed quantile loss with different kernel specifications into XGBoost, and with simulated data, benchmark this approach against alternative smoothed quantile regression losses, the native quantile objective in the XGBoost Python package, and independent versus multi-output tree estimation. The practical relevance is illustrated in an application predicting fine particulate matter (PM$_{2.5}$) in northern California, including periods where levels were elevated due to wildfire smoke. Our results show that convolution smoothed QXGB, particularly when paired with multi-output trees, delivers accurate predictions with near-zero quantile crossing, well-calibrated CDF and exceedance probability estimates, and useful tail characterization for extreme values. Interval estimation is also evaluated as a measure of data spread.

We introduce AgentRelBench, an environment-agnostic reliability instrument that computes ground-truth, severity-priced damage from database state diffs across repeated runs, with no LLM in the measurement path, demonstrated on EnterpriseOps-Gym. Across 2,128 evaluation runs spanning nine models in six families (four development, three pre-registered held-out, plus a frontier pass on two frontier-tier models that the pre-registration designates exploratory), we find: (1) damage on irreversible actions is universal across the families we measured and stochastic within them on pinned, single-provider stacks. (2) No task damaged on every run: zero always-fail cells across 42 confirmatory held-out damage events. A single clean run misses a damage-producing (model, task) pair 0.80 of the time on the development pool (13 pairs); the held-out pool is descriptively consistent (0.575 over 5 pairs, pair-weighted) but sits below our pre-registered power floor and is reported as underpowered, not as confirmation. (3) Damage-producing task count falls with model capability, from 7 of 20 tasks for an 8B model to 1 of 20 for the most capable; capability is confounded with family and training, so this is an observed gradient, not a causal claim. The residual damage does not change in character: in the exploratory frontier pass, the most capable model's one damaging task damages at $\hat{p} = 0.16$ per run, inside the same demonstrably-stochastic band, and a single audit misses it 84% of the time. (4) One model family committed the gated irreversible change while declaring it had refused: transcript- and judge-based grading scores those runs as safe refusals, only state diffs as damage. All confirmatory findings were pre-registered with per-claim demote criteria; one demoted our own initially favored finding, which we report.

Earth observation foundation models (EOFMs) are emerging as reusable representation frameworks for data-driven retrieval, prediction and process modelling within ecohydrology, which integrate EO, meteorological forcing and process models to characterise coupled water, energy and carbon dynamics in vegetation and soil across scales. However, there is yet to be an ecohydrology-specific synthesis assessing the EOFM relevance, application evidence or evaluation requirements under uncertain reference data, scale mismatch and temporal dependence. Here, we develop a framework for determining when EOFMs support interpretable inference and identify a mismatch between EOFMs and ecohydrological requirements. Firstly, an observation-to-inference hierarchy shows that relevance depends on target-specific sensing pathways, spatial-temporal support and traceable uncertainty. Secondly, a meta-analysis shows that pretraining is dominated by reflected optical and active-microwave data, with sparse thermal coverage and no passive-microwave-emission sources. Thirdly, our synthesis of ecohydrological applications finds strongest support for spatial context, label-efficient adaptation and hybrid workflows. Evidence declines with inference depth; independent validation of fluxes, coupled dynamics, event trajectories, calibrated uncertainty and decision benefits remains sparse. Fourthly, our benchmark audit finds stronger coverage of fair adaptation and reproducibility in general EOFM suites, and of process targets, direct reference evidence and distribution shifts in ecohydrological evaluations; physical consistency and uncertainty remain weakly assessed. These findings motivate a process-aware framework aligning EOFM design and evaluation with the target variable, observation pathway and process timescale, supporting trustworthy monitoring and interpretation of coupled water, energy and carbon dynamics.

Greedy sampling produces a compact yet representative summary of normal data, which is essential for reliable anomaly detection that relies on measuring distance from normality. For continual anomaly detection where tasks arrive sequentially, extending greedy sampling is straightforward with unbounded memory through coreset accumulation. However, practical deployment requires fixed memory where the coreset size remains constant regardless of task count. We observe that continued greedy sampling, which iteratively applies greedy selection over previously greedy-sampled sets, effectively preserves representativeness under strict memory limits. Despite discarding data at each step to satisfy the memory constraint, coreset quality degrades gracefully rather than catastrophically, enabling reliable anomaly detection across the tasks. We provide theoretical justification by showing that resulting greedy-continued coreset approximates the oracle coreset within a bounded gap. We instantiate this principle in ContCore, which constructs a greedy-continued coreset through greedy expansion on new task features followed by greedy consolidation to enforce the memory budget. Unlike neural methods susceptible to catastrophic forgetting or naive coreset accumulation requiring unbounded memory, ContCore maintains fixed memory with theoretical guarantees. Empirically, ContCore achieves state-of-the-art performance across 11 task schedules on MVTecAD and VisA, and extends effectively to online continual AD settings where prior methods degrade significantly. Code: https://github.com/jungyg/ContCore

Functional connectome analysis examines brain-region interactions to understand and identify disorders such as autism spectrum disorder and Alzheimer's disease. Existing methods typically use GNNs and Transformers to model the full functional connectivity matrix. However, processing tens of thousands of connections introduces redundancy and noise, increases computational cost, and limits connection-level interpretability. This raises a central question: do we really need complex interaction modeling, or is identifying a small set of disease-relevant connectivity patterns sufficient? To answer this question, we propose BrainLinear, a lightweight geometry-aware framework for mining disease-discriminative connectome patterns. BrainLinear first maps each functional connectivity matrix to a shared tangent space centered at the Fréchet mean of the training set, capturing subject-specific deviations while respecting matrix geometry. It then scores each ROI-pair tangent direction by its classification contribution and disease–control difference, retaining Top-$K$ directions as a compact representation. Finally, a shallow multilayer perceptron performs classification on the selected representation. Experiments on ABIDE and ADNI show that BrainLinear matches or exceeds strong GNN and Transformer baselines at a fraction of their cost: it improves AUC and ACC over the best baseline for each metric by up to $3.54$ and $1.39$ percentage points, while reducing runtime and peak GPU memory by $84.0\%$ and $68.4\%$ relative to the closest baseline in AUC. The selected directions are directionally consistent with between-group displacements and organized across major functional systems, supporting connection-level interpretation.

Collaborative training in distributed semantic communication (DSC) networks typically relies on decentralized federated learning (DFL). However, pushing topology-agnostic aggregation into heterogeneous, multi-task environments creates a fundamental bottleneck: it drives negative transfer and overconsensus bias (OCB). This paper introduces a personalized DSC framework that cuts off this cross-task interference. At the node level, a policy-driven multi-path routing mechanism separates task-specific features from shared representations to preserve local fidelity. Across the network, we deploy a "communicationwhile- aggregation" protocol. It calibrates a column-stochastic consensus matrix using task affinities. This limits the system to absorbing complementary knowledge while actively blocking mismatched parameter updates. To bound the convergence, we derive a unified Lyapunov drift analysis. We reveal a strict Ushaped trade-off: deeper topological mixing reduces variance but amplifies structural OCB. Resolving this tension yields a closed-form expression for the optimal aggregation depth. We evaluate the proposed framework on NYU-v2, where the results reveal a clear trade-off between insufficient aggregation and excessive topological mixing. At the analytically derived optimal aggregation depth, our method achieves a 4.77% global relative improvement over the no-aggregation baseline and outperforms decentralized FedAvg, FedAMP, and heuristic max aggregation. We further evaluate the framework on Taskonomy and imperfect wireless links to examine the effects of network-size variation and wireless-link reliability.

Power-law anisotropy in internal representations has been observed across a wide range of biological and artificial neural systems, from state-of-the-art language models to the mouse cerebral cortex. This anisotropy is a key geometric property of high-dimensional information processing and underlies a variety of theoretical analyses. However, the mechanism by which it emerges from input structure and task-driven learning has remained unclear. Here, we characterize this formation process by exactly solving the learning dynamics of a wide two-layer linear neural network in a teacher–student setting with power-law input and teacher structures. We show that, in the feature-learning regime, the local power-law exponent of the internal-representation spectrum evolves nonmonotonically over the course of training and exhibits up to four distinct asymptotic regimes across modes and training times. By contrast, in the lazy regime, the exponent remains essentially unchanged. We further demonstrate numerically that similar exponent dynamics arise in more realistic nonlinear networks. Together, these results suggest a general mechanism by which the dynamic interaction between input statistics and task structure gives rise to power-law internal representations.

Reliable post-hoc evaluation asks whether already generated text satisfies a target criterion after generation. In this paper we study a focused frozen-embedding setting using principle-evaluation proxy tasks: toxicity detection, fine-grained emotion categorization, and ordinal review rating. General-purpose text embeddings are widely deployed for such tasks, but broad semantic similarity can place semantically similar yet task-distinct examples in overlapping regions of the representation space. We introduce Prototype-Guided Contrastive Learning (PGCL), a prototype-guided geometric regularization module built on top of frozen text embeddings. The module combines a semantic stream, a prototype-anchor attention stream, supervised contrastive learning, offset-based prototype-margin regularization, and stream regularization to produce a compact task-adapted representation without updating the base encoder. Controlled experiments show that PGCL improves over raw frozen embeddings on all three datasets and gives the clearest direct-baseline margin on AmazonReviews, while remaining competitive with strong direct frozen metric-learning baselines on GoEmotions and ToxicComment. We also add supervised residual-adapter, encoder-LoRA, full fine-tuning, objective ablation, sensitivity, and fully logged few-shot LLM protocol diagnostics to define the boundary of the claim. The theoretical analysis is revised as a sufficient-condition account for prototype-margin behavior under explicit assumptions in the prototype-mapping space, rather than as an unconditional training or final-embedding separation guarantee.

Token-level knowledge distillation (KD) matches two conditional distributions per position, yet the standard objectives compare them pointwise: a Kullback-Leibler gradient is blind to which wrong token receives probability mass. We develop a distributional view in which the teacher is represented not by a single softened output but by a family of multi-temperature views - marginals of the annealing path of its logits - and the student is trained against a geometry-aware aggregate of these views under an embedding-based ground cost. We formalize the resulting design space (mixtures, log-linear pooling, entropic Wasserstein barycenters, and a debiased Sinkhorn-divergence flagship in hub and path forms), prove an exact collapse result showing log-linear pooling of tempered views is equivalent to a single temperature, and give a multi-marginal Schrodinger-bridge reading that yields falsifiable predictions. On instruction-tuned Pythia pairs, experiments yield three empirical laws: (i) dispersion law - the benefit of multi-temperature aggregation grows monotonically with the effective temperature dispersion of the views, not with their number; (ii) dispersed views unlock the aggregation operator - the barycenter separates from the arithmetic mixture exactly when transport-based aggregation starts to beat averaging; and (iii) two-regime picture governed by the ceiling gap $\Gamma=\mathrm{PPL}_{\mathrm{SFT}}-\mathrm{PPL}_{T}$: when the fine-tuned teacher barely beats a supervised student the gentle transport objective is the best KD loss but no KD beats supervised fine-tuning, whereas at a real ceiling the ranking inverts - and the sign of the fidelity-generalization correlation flips. We argue that "which distillation loss is the best" is not a fixed property of the loss but a function of $\Gamma$.

Parameter identification of stochastic dynamical systems driven by mixed noises is challenging due to intractable likelihood functions. We propose PENN-GMD, a parameter estimation neural network that maps partially observed trajectories to a Gaussian mixture distribution (GMD) over the system parameters. Unlike conventional uncertainty estimates, the GMD employs full covariance matrices to explicitly reveal parameter couplings and multi-modal likelihood structures. The network is trained by minimizing the negative log-likelihood via a surjective parameterization that hard-encodes all GMD constraints, thereby approximating the true likelihood. We validate the method on five numerical examples with increasing complexity, including systems driven by fractional Gaussian and Lévy noises, oscillators with colored noise, coupled neurons under different observability, and an aeroelastic airfoil with unidentifiable stochastic disturbances. Results demonstrate that PENN-GMD accurately recovers likelihood distributions, captures parameter couplings, and naturally diagnoses non-identifiability through variance broadening or mode splitting. These capabilities establish PENN-GMD as a practical tool for uncertainty-aware parameter identification in complex stochastic systems where conventional likelihood-based methods are infeasible.

Scientific machine learning for partial differential equations commonly targets solution fields, as in physics-informed neural networks, or solution maps, as in neural operators. We study a third target: the propagator itself, a phase and amplitude in phase space. The motivation is a gap in regularity. A transported discontinuity is nonsmooth in space and time, yet the rule that moves it can be a polynomial phase carrying unit amplitude, so the object that generates an evolution can be far smoother than the field it generates. The microlocal neural operator (MiNO) learns that object, using the eikonal equation for the phase and the transport equation for the amplitude, and recovers the solution by an oscillatory integral. Sharp fronts and caustics then belong to propagation geometry rather than to a field fitted pointwise. Small residuals certify more than the reconstructed field. They place the learned canonical relation, the geometry that carries singularities, close to the exact one, and they separate trainable error from the frequency-truncation tail. On a matched-budget discontinuous-advection benchmark, MiNO stops improving within 10,000 steps at the accuracy limit of its finite reconstruction window, a limit predicted in closed form, whereas a physics-informed neural network with neural-tangent-kernel loss balancing stays near its initial error. On smooth advection, the mean error is $3.84\times10^{-3}$ for MiNO and $3.12\times10^{-2}$ for a supervised Fourier neural operator. Single-branch MiNO is the smallest model compared, and one trained generator serves five unseen initial conditions without retraining.

The increasing complexity of digital financial systems has reshaped financial fraud detection from isolated transaction classification into relational risk reasoning over interconnected financial entities. This shift has motivated graph-based fraud detection, where models identify fraudulent nodes by exploiting dependencies among customers, cards, merchants, categories, and locations. However, despite rapid progress in graph-based methods, existing public benchmarks remain misaligned with real-world financial systems in two important aspects. First, they often simplify financial ecosystems into homogeneous or single-node-type multi-relational graphs, failing to preserve the multi-entity and multi-relational nature of financial data. Second, they rarely provide large-scale heterogeneous financial graph datasets with realistic operating conditions such as extreme class imbalance and limited label availability, making it difficult to assess the practical effectiveness of current methods. To address these gaps, we present FinFraudBench, a heterogeneous graph benchmark for financial fraud detection. FinFraudBench contains two heterogeneous graph datasets (CreditCard-Fraud and BankTrans-Fraud) with up to 8.99M nodes and 89.23M directed typed edges. Each dataset preserves six financial entity types, fourteen directed edge types, and natural fraud rates that mirror deployment constraints. With these datasets, we establish a standardized evaluation protocol covering both ranking and imbalance-sensitive classification metrics, and evaluate representative baselines. Extensive experiments yield empirical insights into current methods' limitations and suggest promising avenues for future research. FinFraudBench is available at https://anonymous.4open.science/r/FinFraudBench-B002.

Population-based adaptive importance sampling (AIS) methods use a set of proposal densities to approximate complex target distributions. Their performance is commonly assessed through effective sample size (ESS) and related weight-based diagnostics, which measure the concentration of normalized importance weights. However, a large ESS only indicates that the normalized sample weights are not strongly concentrated; it does not describe how the proposal components are arranged in the sampling space. In population-based AIS, several proposal components may generate samples in the same region of the target, so the sample weights can appear well balanced even though the effective number of distinct proposal components is small. This letter introduces the effective number of proposals (ENP), a similarity-aware proposal-level diagnostic for population-based AIS. ENP combines the total normalized weight assigned to each proposal with a redundancy measure computed from similarities among target-weighted samples, estimating the number of non-redundant empirical proposal contributions to the approximation. We establish basic effective-number properties and show that ENP detects proposal collapse and duplication missed by standard ESS. We also illustrate its use as a targeted feedback signal for proposal rejuvenation.

Differentially private federated learning must balance privacy protection against model accuracy and training efficiency. Static gradient clipping applies a fixed threshold throughout training and across model layers, which can cause excessive clipping when the threshold is too small or unnecessarily large noise when it is too large. This paper presents DDP-SA-adaptive, an adaptive gradient clipping and noise adding mechanism for differentially private federated learning with secure aggregation. At each communication round, every client determines a separate clipping threshold for each model layer from the median of its per-sample gradient norms. The resulting layer-wise thresholds adapt to the evolving gradient distributions and calibrate the Laplace noise added before the updates are encoded and secret-shared among intermediate aggregation servers. We evaluate the proposed mechanism on a federated regression task in terms of efficiency, accuracy, privacy, convergence, clipping norm, and noise magnitude. Compared with the static DDP-SA baseline, DDP-SA-adaptive reduces the number of communication rounds by 6.81%, total training time by 19.21%, and average per-round training time by 13.33%, leading to improved training efficiency. It also reduces test loss by 98.74% and increases test R2 by 3.41%, leading to improved model accuracy. To attain R2 = 0.99, the adaptive mechanism operates with a privacy budget of approximately epsilon = 0.1, compared with epsilon = 0.4 for static DDP-SA, thus providing stronger privacy protection and achieving stronger privacy guarantees. These results demonstrate that round-wise, layer-wise adaptation can improve the privacy-accuracy-efficiency trade-off of differentially private federated learning.

We revisit Tesauro's TD-Gammon for backgammon money games in the setting of no evaluation-time search. Both checker play and cube action (use of the doubling cube) are learned from scratch via self-play reinforcement learning (RL), with minimal hand-coded logic and no expert features. In this setting, we demonstrate that pure self-play RL suffices to train models that reach near-state-of-the-art playing strength. Specifically, for cubeful money games, our search-free model evaluates faster and is substantially stronger than the open-source engines GNU Backgammon and Open Sage running a one-move (1-ply) look-ahead search. Our method of training a cubeful model also outperforms the standard approach of training a cubeless model and deriving cubeful equity with Janowski-style formulas, when the two models are equally sized.