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arXiv2026

DUA-D2C: Dynamic Uncertainty Aware Method for Overfitting Remediation in Deep Learning

Overfitting remains a significant challenge in deep learning, often arising from data outliers, noise, and limited training data. To address this, we previously proposed the Divide2Conquer (D2C) method, which partitions training data into multiple subsets and trains identical models independently on each. This strategy enables learning more consistent patterns while minimizing the influence of individual outliers and noise. D2C's standard aggregation typically treats all subset models equally or based on fixed heuristics (like data size), potentially underutilizing information about their varying generalization capabilities. Building upon this foundation, we introduce Dynamic Uncertainty-Aware Divide2Conquer (DUA-D2C), an advanced technique that refines the aggregation process. DUA-D2C dynamically weights the contributions of subset models based on their performance on a shared validation set, employing a novel composite score of accuracy and normalized prediction entropy. This intelligent aggregation allows the central model to preferentially learn from subsets yielding more generalizable and confident edge models, thereby more effectively combating overfitting. In this work, we provide a rigorous theoretical justification for this approach, analytically demonstrating how dynamic parameter fusion reduces model variance. Empirical evaluations on benchmark datasets spanning image, audio, and text domains demonstrate that DUA-D2C significantly improves generalization. Our analysis includes evaluations of decision boundaries, loss curves, and ablation studies, highlighting that DUA-D2C provides additive performance gains even when applied on top of standard regularizers like Dropout. This study establishes it as a theoretically grounded and effective approach to combating overfitting in deep learning. Our code is publicly available at: https://github.com/Saiful185/DUAD2C.

cs.LG↗
arXiv2026

DDPM Score Matching and Distribution Learning

Score estimation is the backbone of score-based generative models (SGMs), especially denoising diffusion probabilistic models (DDPMs). A key result in this area shows that with accurate score estimates, SGMs can efficiently generate samples from any realistic data distribution (Chen et al., ICLR'23; Lee et al., ALT'23). This distribution learning result, where the learned distribution is implicitly that of the sampler's output, does not explain how score estimation relates to classical tasks of parameter and density estimation. This paper introduces a framework that reduces score estimation to these two tasks, with various implications for statistical and computational learning theory: Parameter Estimation: Koehler et al. (ICLR'23) demonstrate that a score-matching variant is statistically inefficient for the parametric estimation of multimodal densities common in practice. In contrast, we show that under mild conditions, denoising score-matching in DDPMs is asymptotically efficient. Density Estimation: By linking generation to score estimation, we lift existing score estimation guarantees to $(ε,δ)$-PAC density estimation, i.e., a function approximating the target log-density within $ε$ on all but a $δ$-fraction of the space. We provide (i) minimax rates for density estimation over Hölder classes and (ii) a quasi-polynomial PAC density estimation algorithm for the classical Gaussian location mixture model, building on and addressing an open problem from Gatmiry et al. (COLT'26). Lower Bounds for Score Estimation: Our framework offers the first principled method to prove computational lower bounds for score estimation across general distributions. As an application, we establish cryptographic lower bounds for score estimation in general Gaussian mixture models, conceptually recovering Song's (NeurIPS'24) result and advancing his key open problem.

stat.ML↗
arXiv2026

GyroSwin: 5D Surrogates for Gyrokinetic Plasma Turbulence Simulations

Nuclear fusion plays a pivotal role in the quest for reliable and sustainable energy production. A major roadblock to viable fusion power is understanding plasma turbulence, which significantly impairs plasma confinement, and is vital for next-generation reactor design. Plasma turbulence is governed by the nonlinear gyrokinetic equation, which evolves a 5D distribution function over time. Due to its high computational cost, reduced-order models are often employed in practice to approximate turbulent transport of energy. However, they omit nonlinear effects unique to the full 5D dynamics. To tackle this, we introduce GyroSwin, the first scalable 5D neural surrogate that can model 5D nonlinear gyrokinetic simulations, thereby capturing the physical phenomena neglected by reduced models, while providing accurate estimates of turbulent heat transport. GyroSwin (i) extends hierarchical Vision Transformers to 5D, (ii) introduces cross-attention and integration modules for latent 3D$\leftrightarrow$5D interactions between electrostatic potential fields and the distribution function, and (iii) performs channelwise mode separation inspired by nonlinear physics. We demonstrate that GyroSwin outperforms widely used reduced numerics on heat flux prediction, captures the turbulent energy cascade, and reduces the cost of fully resolved nonlinear gyrokinetics by three orders of magnitude while remaining physically verifiable. GyroSwin shows promising scaling laws, tested up to one billion parameters, paving the way for scalable neural surrogates for gyrokinetic simulations of plasma turbulence.

physics.plasm-ph↗
arXiv2026

Generalized infinite dimensional Alpha-Procrustes based geometries

This work extends the recently introduced Alpha-Procrustes family of Riemannian metrics for symmetric positive definite (SPD) matrices by incorporating generalized versions of the Bures-Wasserstein (GBW), Log-Euclidean, and Wasserstein distances. While the Alpha-Procrustes framework has unified many classical metrics in both finite- and infinite- dimensional settings, it previously lacked the structural components necessary to realize these generalized forms. We introduce a formalism based on unitized Hilbert-Schmidt operators and an extended Mahalanobis norm that allows the construction of robust, infinite-dimensional generalizations of GBW and Log-Hilbert-Schmidt distances. Our approach also incorporates a learnable regularization parameter that enhances geometric stability in high-dimensional comparisons. Preliminary experiments reproducing benchmarks from the literature demonstrate the improved performance of our generalized metrics, particularly in scenarios involving comparisons between datasets of varying dimension and scale. This work lays a theoretical and computational foundation for advancing robust geometric methods in machine learning, statistical inference, and functional data analysis.

stat.ML↗
arXiv2026

Advancing Subseasonal Forecasting with Machine Learning

Decision-makers rely on weather forecasts to plant crops, manage wildfires, allocate water and energy, and prepare for weather extremes. Today, such forecasts enjoy unprecedented accuracy out to two weeks thanks to steady advances in physics-based dynamical models and data-driven artificial intelligence (AI) models. However, model skill drops precipitously at subseasonal timescales (2 - 6 weeks ahead), due to compounding errors, systemic model biases, and the chaotic nature of the atmosphere. To counter this degradation, we introduce probabilistic bias correction (PBC), a machine learning framework that substantially reduces systematic error by learning to correct historical probabilistic forecasts. When applied to the leading dynamical and AI models from the European Centre for Medium-Range Weather Forecasts (ECMWF), PBC doubles the modest subseasonal skill of the AI Forecasting System and improves the skill of the operationally-debiased dynamical model for 91% of pressure, 92% of temperature, and 98% of precipitation targets. We designed PBC for operational deployment, and, in ECMWF's 2025 real-time forecasting competition, its global forecasts placed first for all weather variables and lead times, outperforming the dynamical models from six operational forecasting centers, an international dynamical multi-model ensemble, ECMWF's AI Forecasting System, and the forecasting systems of 34 teams worldwide. These probabilistic skill gains translate into more accurate prediction of extreme events and have the potential to improve agricultural planning, energy management, and disaster preparedness in vulnerable communities.

cs.LG↗
arXiv2026

Deep Learning as Neural Low-Degree Filtering: A Spectral Theory of Hierarchical Feature Learning

Understanding how deep neural networks learn useful internal representations from data remains a central open problem in the theory of deep learning. We introduce Neural Low-Degree Filtering (Neural LoFi), a stylized limit of gradient-based training in which hierarchical feature learning becomes an explicit iterative spectral procedure. In this limit, the dynamics at each layer decouple: given the current representation, the next layer selects directions with maximal accessible low-degree correlation to the label. This yields a tractable surrogate mechanism for deep learning, together with a natural kernel-space interpretation. Neural LoFi provides a mathematically explicit framework for studying multi-layer feature learning beyond the lazy regime. It predicts how representations are selected layer by layer, explains how emergence of concepts arises with given sample complexity, and gives a concrete mechanism by which depth progressively constructs new features from old ones through low-degree compositionality. We complement the theory with mechanistic experiments on fully connected and convolutional architectures, showing that Neural LoFi improves over lazy random-feature baselines, recovers meaningful structured filters, and predicts representations aligned with early gradient-descent feature discovery with real datasets.

cs.LG↗
arXiv2026

Expressivity of Contradiction Graphs

We study the contradiction graphs associated with a binary concept class. For a class $H\subseteq\{0,1\}^X$, the order-$m$ contradiction graph $G_m(H)$ has as vertices the $H$-realizable labeled sequences of length $m$, with two vertices adjacent when the two sequences assign opposite labels to some common domain point. First, we identify a graph-theoretic property that determines the threshold predicate $\operatorname{VCdim}(H)\ge m$. Consequently, the sequence $(G_m(H))_{m\ge1}$ determines the exact VC dimension and, in particular, distinguishes finite from infinite VC dimension, answering a question posed by Alon et al. (2024). We then generalize this result by proving that the sequence of contradiction graphs determines, up to signed relabeling, the realizable datasets of any fixed length. Thus, any learning-theoretic property determined by the realizable datasets of a fixed length and invariant under signed relabeling can be recovered from the contradiction graph sequence. As an application, we explicitly provide a characterization of Littlestone dimension.

stat.ML↗
arXiv2026

Automatic knot selection in smooth additive models

B-spline regression constitutes a widely used framework for nonparametric modeling. The performance of this methodology depends on specifying the number and placement of changepoints, known as knots, prior to the estimation process. Such knot sequence determines the dimension of the B-spline basis used to represent the regression function and the number of coefficients to be estimated. Therefore, the knots' choice affects the model's flexibility, influencing its smoothness and goodness-of-fit. Traditionally, this problem has been addressed either by explicitly selecting knots, via knot-selection algorithms, or by regularization methods, such as P-splines, which automatically tune the regressor's smoothness. The latter have become the standard in generalized additive models (GAMs). In contrast, knot-selection techniques, frequently neglected because of computational or modeling limitations, provide certain advantages which can be valuable in some contexts. In this work, we introduce a novel explicit knot-selection technique for GAMs based on an extension of the adaptive splines (A-splines) knot selection methodology, combined with a customized Fellner-Schall scheme for tuning the associated parameters. Our approach is evaluated on various synthetic and real datasets and compared with P-splines and state-of-the-art knot-selection techniques. The results indicate comparable performance, while producing models built on a substantially smaller number of basis elements.

stat.ML↗
arXiv2026

Interpretability for Turing Machines

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.

cs.LG↗
arXiv2026

Minimax Lower Bound for Estimating Diffusion-based Local Intrinsic Dimension

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 $σ^d$, 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 $O(σ^2)$. We then establish a minimax lower bound of order $(nσ^d)^{-1}$ for estimating this finite-scale field from $n$ observations, for $n^{-1/(2α+d)}\lesssimσ\leσ_0$. At the smallest scale covered by our lower-bound construction, the bound becomes the nonparametric rate $n^{-2α/(2α+d)}$.

stat.ML↗
arXiv2026

An Analysis of Self-supervised Pre-training with Dependent Samples

Self-supervised learning relies on so-called data augmentations $ϕ(x)$ 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 $\{ ϕ_l(x_i) \}$ are pooled together to learn $\cal V$, despite obvious inter-dependencies between different augmentations $ϕ_l(x), ϕ_k(x)$ 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 $ϕ_l(x), ϕ_k(x)$ 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.

stat.ML↗
arXiv2026

Confounding-Valid Conformal Inference for Counterfactual KPIs in Wireless Networks

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.

cs.LG↗
arXiv2026

FluxDisco: Symbolic Regression for Stoichiometric Dynamical Systems via Monte Carlo Graph Search

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.

stat.ML↗
arXiv2026

PAC-Bayesian Reconstruction Guarantees for Time Series Variational Autoencoders

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.

stat.ML↗
arXiv2026

Learning from VAE Errors to support ECG-based Differential Diagnosis of Myocardial Scar

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 $β$-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 $β$-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 $β$-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.

cs.LG↗
arXiv2026

Bias-Corrected Subspace Intersection: Minimax-Optimal Shared Subspace Estimation in Multi-View Data

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.

stat.ME↗
arXiv2026

Connecting Score Matching, Maximum Likelihood, and Expectation-Maximization in Mixed Linear Regression

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.

cs.LG↗
arXiv2026

Gaussian Linear Functional Manifold Method for Massive Point Cloud Data

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.

cs.CV↗