Papers

We develop a moduli-theoretic view of psychometric test batteries for AI agents and connect it explicitly to the AAI score developed previously. First, we make precise the notion of an AAI functional on a battery and set out axioms that any reasonable autonomy/general intelligence score should satisfy. Second, we show that the composite index ('AAI-Index') defined previously is a special case of our AAI functional. Third, we introduce the notion of a cognitive core of an agent relative to a battery and define the associated AAI$_{\textrm{core}}$ score as the restriction of an AAI functional to that core. Finally, we use these notions to describe invariants of batteries under evaluation-preserving symmetries and outline how moduli of equivalent batteries are organized.

Searching large and complex design spaces for a global optimum can be infeasible and unnecessary. A practical alternative is to iteratively refine the neighborhood of an initial design using local optimization methods such as gradient descent. We propose local entropy search (LES), a Bayesian optimization paradigm that explicitly targets the solutions reachable by the descent sequences of iterative optimizers. The algorithm propagates the posterior belief over the objective through the optimizer, resulting in a probability distribution over descent sequences. It then selects the next evaluation by maximizing mutual information with that distribution, using a combination of analytic entropy calculations and Monte-Carlo sampling of descent sequences. Empirical results on high-complexity synthetic objectives and benchmark problems show that LES achieves strong sample efficiency compared to existing local and global Bayesian optimization methods.

Transformations of covariates are widely used in applied statistics to improve interpretability and to satisfy assumptions required for valid inference. More broadly, feature engineering encompasses a wider set of practices aimed at enhancing predictive performance, and is typically performed as part of a data pre-processing step. In contrast, this paper integrates a substantial component of the feature engineering process directly into the modelling stage. This is achieved by introducing a novel general framework for embedding interpretable covariate transformations within multi-parameter Generalised Additive Models (GAMs). Our framework accommodates any sufficiently differentiable scalar-valued transformation of potentially high-dimensional and complex covariates. These transformations are treated as integral model components, with their parameters estimated jointly with regression coefficients via maximum a posteriori (MAP) methods, and joint uncertainty quantified via approximate Bayesian techniques. Smoothing parameters are selected in an empirical Bayes framework using a Laplace approximation to the marginal likelihood, supported by efficient computation based on implicit differentiation methods. We demonstrate the flexibility and practical value of the proposed methodology through applications to forecasting electricity net-demand in Great Britain and to modelling house prices in London. Methods for building and fitting GAMs with nested transformations are provided by the gamFactory R package, available at https://github.com/mfasiolo/gamFactory, while the code for reproducing the results in this paper is available at https://doi.org/10.5281/zenodo.19239350.

This paper introduces a novel Kalman filter framework designed to achieve robust state estimation under both process and measurement noise. Inspired by the Weighted Observation Likelihood Filter (WoLF), which provides robustness against measurement outliers, we applied generalized Bayesian approach to build a framework considering both process and measurement noise outliers.

Masked Diffusion Models (MDMs) have emerged as one of the most promising paradigms for generative modeling over discrete domains. It is known that MDMs effectively train to decode tokens in a random order, and that this ordering has significant performance implications in practice. This observation raises a fundamental question: can we design a training framework that optimizes for a favorable decoding order? We answer this in the affirmative, showing that the continuous-time variational objective of MDMs, when equipped with multivariate noise schedules, can identify and optimize for a decoding order during training. We establish a direct correspondence between decoding order and the multivariate noise schedule and show that this setting breaks invariance of the MDM objective to the noise schedule. Furthermore, we prove that the MDM objective decomposes precisely into a weighted auto-regressive losses over these orders, which establishes them as auto-regressive models with learnable orders.

Small-area mortality estimation is inherently difficult, as random fluctuations from low death counts can obscure real geographic differences. We introduce a flexible model that borrows strength across age, space, and time to estimate mortality schedules and trends in very small populations. The approach ensures smooth patterns across these dimensions while allowing localized breaks from the spatial structure, capturing broad trajectories as well as sharp local contrasts. We implement our model within a Penalized Spline framework and estimate it using Generalized Linear Array Model techniques, resulting in a computationally fast, interpretable, and parsimonious method. Crucially, it can readily incorporate sudden mortality shocks, such as the Covid-19 pandemic, making it highly versatile for real-world demographic and epidemiological challenges. We demonstrate its application by estimating life expectancy and age-specific mortality inequalities in over 4,800 small areas across the Greater London Authority from 2002 to 2024.

We propose a new formulation of the maximum score estimator that uses compositions of rectified linear unit (ReLU) functions, instead of indicator functions as in Manski (1975,1985), to encode the sign alignment restrictions. Since the ReLU function is Lipschitz, our new ReLU-based maximum score criterion function is substantially easier to optimize using standard gradient-based optimization pacakges. We also show that our ReLU-based maximum score (RMS) estimator can be generalized to an umbrella framework defined by multi-index single-crossing (MISC) conditions, while the original maximum score estimator cannot be applied. We establish the $n^{-s/(2s+1)}$ convergence rate and asymptotic normality for the RMS estimator under order-$s$ Holder smoothness. In addition, we propose an alternative estimator using a further reformulation of RMS as a special layer in a deep neural network (DNN) architecture, which allows the estimation procedure to be implemented via state-of-the-art software and hardware for DNN.

We study core stability in non-centroid clustering under the max-loss objective, where each agent's loss is the maximum distance to other members of their cluster. We prove that for all $k\geq 3$ there exist metric instances with $n\ge 9$ agents, with $n$ divisible by $k$, for which no clustering lies in the $\alpha$-core for any $\alpha<2^{\frac{1}{5}}\sim 1.148$. The bound is tight for our construction. Using a computer-aided proof, we also identify a two-dimensional Euclidean point set whose associated lower bound is slightly smaller than that of our general construction. This is, to our knowledge, the first impossibility result showing that the core can be empty in non-centroid clustering under the max-loss objective.

Understanding how the causal effect of a treatment evolves over time, including the potential for waning, is important for informed decisions on treatment discontinuation or repetition. For example, waning vaccine protection influences booster dose recommendations, while cost-effectiveness analyses require accounting for long-term efficacy of treatments. However, there is no consensus on the methodology to assess and account for treatment effect waning. Even in randomized controlled trials, the common naïve comparison of hazard functions can lead to misleading causal conclusions due to inherent selection bias. Although comparing survival curves is sometimes recommended as a safer measure of causal effect, it only represents a cumulative effect over time and does not address treatment effect waning. We also explore recent formulations of causal hazard ratios, based on the principal stratification approach or the controlled direct effect. These causal hazard ratios cannot be identified without strong modeling assumptions, but bounds can be derived accounting for unobserved heterogeneity and one could try to use them to detect treatment effect waning. However, we illustrate that an increase in causal hazard ratios towards one does not necessarily mean that the protective effect of the treatment is fading. Furthermore, the same survival functions may correspond to both scenarios with and without waning, which shows that treatment effect waning cannot be identified from standard time-to-event data without strong untestable modeling assumptions.

Unbalanced optimal transport (UOT) provides a flexible way to match or compare nonnegative finite Radon measures. However, UOT requires a predefined ground transport cost, which may misrepresent the data's underlying geometry. Choosing such a cost is particularly challenging when datasets live in heterogeneous spaces, often motivating practitioners to adopt Gromov-Wasserstein formulations. To address this challenge, we introduce cost-regularized unbalanced optimal transport (CR-UOT), a framework that allows the ground cost to vary while allowing mass creation and removal. We show that CR-UOT incorporates unbalanced Gromov-Wasserstein type problems through families of inner-product costs parameterized by linear transformations, enabling the matching of measures or point clouds across Euclidean spaces. We develop algorithms for such CR-UOT problems using entropic regularization and demonstrate that this approach improves the alignment of heterogeneous single-cell omics profiles, especially when many cells lack direct matches.

We study nonconvex optimization for phase retrieval and the more general problem of semidefinite low-rank matrix sensing; in particular, we focus on the global nonconvex landscape of overparametrized versions of the nonsmooth amplitude least-squares loss as well as a smooth reformulation of this loss based on the PhaseCut approach. We first give a general, deterministic result on properties of second-order critical points for a general class of loss functions; we then specialize this result to the nonsmooth amplitude loss and, additionally, prove nearly identical results for a smooth reformulation (similar to PhaseCut) as a synchronization problem over spheres. Finally, we show the usefulness of these tools by proving high-probability landscape guarantees in two settings: (1) phase retrieval with isotropic sub-Gaussian measurements, and (2) phase retrieval in a general (possibly infinite-dimensional) Hilbert space with Gaussian measurements. In both cases, our results give state-of-the-art and statistically optimal guarantees with only a constant amount of overparametrization (in the well-studied case of isotropic sub-Gaussian measurements, such statistical guarantees had previously required greater degrees of overparametrization/relaxation); this demonstrates the potential of overparametrized nonconvex optimization as a principled and scalable algorithmic approach to phase retrieval.

Extreme event attribution (EEA), which assesses the extent to which disasters are caused by climate change, is crucial for informing climate policy and legal proceedings. Machine learning is increasingly used for EEA by modeling rare weather events too complex or computationally intensive for traditional methods. However, its validity remains unclear, as machine learning applications are criticized for bias and lack of robustness. Here we evaluate machine learning for EEA using California wildfire data from 2003-2020. We identify three major threats to validity: (1) individual event attribution estimates are highly sensitive to algorithmic design choices; (2) common performance metrics like Brier score are not strongly correlated with attribution error, facilitating suboptimal model selection; and (3) distribution shift across climate scenarios substantially degrades predictive performance. We propose a more robust attribution analysis using aggregate estimates and additional evaluation metrics for predictive performance and distribution shift.

Opportunistic Networks (OppNets) employ the Store-Carry-Forward (SCF) paradigm to maintain communication during intermittent connectivity. However, routing performance suffers due to dynamic topology changes, unpredictable contact patterns, and resource constraints including limited energy and buffer capacity. These challenges compromise delivery reliability, increase latency, and reduce node longevity in highly dynamic environments. This paper proposes Cluster-based Routing using Deep Reinforcement Learning (CR-DRL), an adaptive routing approach that integrates an Actor-Critic learning framework with a heuristic function. CR-DRL enables real-time optimal relay selection and dynamic cluster overlap adjustment to maintain connectivity while minimizing redundant transmissions and enhancing routing efficiency. Simulation results demonstrate significant improvements over state-of-the-art baselines. CR-DRL extends node lifetimes by up to 21%, overall energy use is reduced by 17%, and nodes remain active for 15% longer. Communication performance also improves, with up to 10% higher delivery ratio, 28.5% lower delay, 7% higher throughput, and data requiring 30% fewer transmission steps across the network.

The mixture model is undoubtedly one of the greatest contributions to clustering. For continuous data, Gaussian models are often used and the Expectation-Maximization (EM) algorithm is particularly suitable for estimating parameters from which clustering is inferred. If these models are particularly popular in various domains including image clustering, they however suffer from the dimensionality and also from the slowness of convergence of the EM algorithm. However, the Classification EM (CEM) algorithm, a classifying version, offers a fast convergence solution while dimensionality reduction still remains a challenge. Thus we propose in this paper an algorithm combining simultaneously and non-sequentially the two tasks –Data embedding and Clustering– relying on Principal Component Analysis (PCA) and CEM. We demonstrate the interest of such approach in terms of clustering and data embedding. We also establish different connections with other clustering approaches.

This study introduces a novel nonparametric approach for detecting Differential Item Functioning (DIF) in binary items through direct comparison of Item Response Curves (IRCs). Building on prior work on nonparametric comparison of regression curves, we extend the methodology to accommodate binary response data, which is typical in psychometric applications. The proposed approach includes a new estimator of the asymptotic variance of the test statistic and derives optimal weight functions that maximise local power. Because the asymptotic distribution of the resulting test statistic is unknown, a wild bootstrap procedure is applied for inference. A Monte Carlo simulation study demonstrates that the nonparametric approach effectively controls Type I error and achieves power comparable to the traditional logistic regression method, outperforming it in cases with multiple intersections of the underlying IRCs. The impact of bandwidth and weight specification is explored. Application to a verbal aggression dataset further illustrates the method's ability to detect subtle DIF patterns missed by parametric models. Overall, the proposed nonparametric framework provides a flexible and powerful alternative for detecting DIF, particularly in complex scenarios where traditional model-based assumptions may not be applicable.

This paper establishes a comprehensive theory of geometric rough paths for mixed fractional Brownian motion (MFBM) and its generalized multi-component extensions. We prove that for a generalized MFBM of the form $M_t^H(a) = \sum_{k=1}^N a_k B_t^{H_k}$ with $\min\{H_k\} > \frac{1}{4}$, there exists a canonical geometric rough path obtained as the limit of smooth rough paths associated with dyadic approximations. This extends the classical result of Coutin and Qian \cite{coutin2002} for single fractional Brownian motion to the mixed case. We provide explicit bounds on the $p$-variation norms and establish a Skorohod integral representation connecting our pathwise construction to the Malliavin calculus framework. Furthermore, we demonstrate applications to rough differential equations driven by MFBM, enabling the use of Lyons' universal limit theorem for this class of processes. Finally, we study the signature of MFBM paths, providing a complete algebraic characterization of their geometric properties. Our approach unifies the treatment of multiple fractional components and reveals the fundamental interactions between different regularity scales, completing the rough path foundation for mixed fractional processes with applications in stochastic analysis and beyond.

X-chromosome association study has specific model uncertainty challenges, such as unknown X-chromosome inactivation status and baseline allele, and considering nonadditive and gene-sex interaction effects in the analysis or not. Although these challenges have been answered for single-locus X-chromosome variants, it remains unclear how to properly perform multilocus association studies when above uncertainties are present. We first carefully investigate the inferential consequences of these uncertainties on existing multilocus association analysis methods, and then propose a theoretically justified framework to analyze multilocus X-chromosome variants while all the uncertainty issues are addressed. We provide separate solutions for common and rare variants, and simulation results show that our solutions are overall more powerful than existing multilocus methods which were proposed to analyze autosomal variants. We finally provide supporting evidences of our approach by revisiting some published X-chromosome association studies.

Cross-subject motor imagery decoding remains a fundamental challenge in EEG-based brain-computer interfaces due to substantial inter-subject variability. Recent approaches have leveraged Riemannian geometry by representing EEG signals as covariance matrices on the symmetric positive definite (SPD) manifold. However, existing methods primarily focus on manifold-based representations while largely overlooking subject-specific variations in covariance dispersion and orientation. In this work, we address these challenges through geometry-aware congruence transformations and propose three complementary models: (i) Discriminative Congruence Transform (DCT), (ii) Deep Linear DCT (DLDCT), and (iii) Deep DCT-UNet (DDCT-UNet). The proposed models are evaluated both as manifold alignment modules for downstream classifiers and as end-to-end discriminative architectures optimized via cross-entropy with a custom logistic regression head. Experiments on challenging cross-subject motor imagery benchmarks demonstrate consistent improvements in transductive decoding performance, achieving 2-3% higher accuracy than strong baselines. These results highlight the effectiveness of geometry-aware congruence learning for mitigating inter-subject variability in EEG decoding.

The increasing use of machine learning in sensitive applications demands algorithms that simultaneously preserve data privacy and ensure fairness across potentially sensitive sub-populations. While privacy and fairness have each been extensively studied, their joint treatment remains poorly understood. Existing research often frames them as conflicting objectives, with multiple studies suggesting that strong privacy notions such as differential privacy inevitably compromise fairness. In this work, we challenge that perspective by showing that differential privacy can be integrated into a fairness-enhancing pipeline with minimal impact on fairness guarantees. We design a postprocessing algorithm, called DP2DP, that enforces both demographic parity and differential privacy. Our analysis reveals that our algorithm converges towards its demographic parity objective at essentially the same rate (up logarithmic factor) as the best non-private methods from the literature. Experiments on both synthetic and real datasets confirm our theoretical results, showing that the proposed algorithm achieves state-of-the-art accuracy/fairness/privacy trade-offs.

Fractional cumulative residual inaccuracy (FCRI) measure allows to determine regions of discrepancy between systems, depending on their respective fractional and chaotic map parameters. Most of the theoretical results and applications related to the FCRI of the lifetime random variable are based on the distribution function approach. However, there are situations in which the distribution function may not be available in explicit form but has a closed-form quantile function (QF), an alternative method of representing a probability distribution. Motivated by these, the present study is devoted to introduce a quantile-based FCRI and study its various properties. We also deal with non-parametric estimation of quantile-based FCRI and examine its validity using simulation studies and illustrate its usefulness in measuring the discrepancy between chaotic systems and in measuring the discrepancy in two different time regimes using Nifty 50 dataset.

Over the last few months, AI models including large language models have improved greatly. There are now several documented examples where they have helped professional mathematical scientists prove new results, sometimes even helping resolve known open problems. In this short note, we add another example to the list, by documenting how we were able to solve a previously unsolved research problem in robust mathematical statistics with crucial help from GPT-5. Our problem concerns robust density estimation, where the observations are perturbed by Wasserstein-bounded contaminations. In a previous preprint (Chao and Dobriban, 2023, arxiv:2308.01853v2), we have obtained upper and lower bounds on the minimax optimal estimation error; which were, however, not sharp. Starting in October 2025, making significant use of GPT-5 Pro, we were able to derive the minimax optimal error rate (reported in version 3 of the above arxiv preprint). GPT-5 provided crucial help along the way, including by suggesting calculations that we did not think of, and techniques that were not familiar to us, such as the dynamic Benamou-Brenier formulation, for key steps in the analysis. Working with GPT-5 took a few weeks of effort, and we estimate that it could have taken several months to get the same results otherwise. At the same time, there are still areas where working with GPT-5 was challenging: it sometimes provided incorrect references, and glossed over details that sometimes took days of work to fill in. We outline our workflow and steps taken to mitigate issues. Overall, our work can serve as additional documentation for a new age of human-AI collaborative work in mathematical science.

Persistent homology (PH) is a crucial concept in computational topology, providing a multiscale topological description of a space. It is particularly significant in topological data analysis, which aims to make statistical inference from a topological perspective. In this work, we introduce a new topological summary for Bayesian neural networks, termed the predictive topological uncertainty (pTU). The proposed pTU measures the uncertainty in the interaction between the model and the inputs. It provides insights from the model perspective: if two samples interact with a model in a similar way, then they are considered identically distributed. We also show that the pTU is insensitive to the model architecture. As an application, pTU is used to solve the out-of-distribution (OOD) detection problem, which is critical to ensure model reliability. Failure to detect OOD input can lead to incorrect and unreliable predictions. To address this issue, we propose a significance test for OOD based on the pTU, providing a statistical framework for this issue. The effectiveness of the framework is validated through various experiments, in terms of its statistical power, sensitivity, and robustness.

Many quantities characterizing infectious disease outbreaks - like the effective reproduction number ($R_t$), defined as the average number of secondary infections a newly infected individual will cause over the course of their infection - need to be modeled as time-varying parameters. It is common practice to use Gaussian random walks as priors for estimating such functions in Bayesian analyses of pathogen surveillance data. In this setting, however, the random walk prior may be too permissive, as it fails to capture prior scientific knowledge about the estimand and results in high posterior variance. We propose several Gaussian Markov process priors for $R_t$ inference, including the Integrated Brownian Motion (IBM), which can be represented as a Markov process when augmented with its corresponding Brownian Motion component, and is therefore computationally efficient and simple to implement and tune. We use simulated outbreak data to compare the performance of these proposed priors with the Gaussian random walk prior and another state-of-the-art Gaussian process prior based on an approximation to a Matérn covariance function. We find that IBM can match or exceed the performance of other priors, and we show that it produces epidemiologically reasonable and precise results when applied to county-level SARS-CoV-2 data.

We study the problem of excess risk evaluation for empirical risk minimization (ERM) under convex losses. We show that by leveraging the idea of wild refitting, one can upper bound the excess risk through the so-called "wild optimism," without relying on the global structure of the underlying function class but only assuming black box access to the training algorithm and a single dataset. We begin by generating two sets of artificially modified pseudo-outcomes created by stochastically perturbing the derivatives with carefully chosen scaling. Using these pseudo-labeled datasets, we refit the black-box procedure twice to obtain two wild predictors and derive an efficient excess risk upper bound under the fixed design setting. Requiring no prior knowledge of the complexity of the underlying function class, our method is essentially model-free and holds significant promise for theoretically evaluating modern opaque deep neural networks and generative models, where traditional learning theory could be infeasible due to the extreme complexity of the hypothesis class.

Class imbalance remains a critical challenge in semi-supervised learning (SSL), especially when distributional mismatches between labeled and unlabeled data lead to biased classification. Although existing methods address this issue by adjusting logits based on the estimated class distribution of unlabeled data, they often handle model imbalance in a coarse-grained manner, conflating data imbalance with bias arising from varying class-specific learning difficulties. To address this issue, we propose a unified framework, SC-SSL, which suppresses model bias through decoupled sampling control. During training, we identify the key variables for sampling control under ideal conditions. By introducing a classifier with explicit expansion capability and adaptively adjusting sampling probabilities across different data distributions, SC-SSL mitigates feature-level imbalance for minority classes. In the inference phase, we further analyze the weight imbalance of the linear classifier and apply post-hoc sampling control with an optimization bias vector to directly calibrate the logits. Extensive experiments across various benchmark datasets and distribution settings validate the consistency and state-of-the-art performance of SC-SSL.

Statistical inference from data generated by multi-armed bandit (MAB) algorithms is challenging due to their adaptive, non-i.i.d. nature. A classical manifestation is that sample averages of arm rewards under bandit sampling may fail to satisfy a central limit theorem. Lai and Wei's stability condition provides a sufficient, and essentially necessary criterion, for asymptotic normality in bandit problems. While the celebrated Upper Confidence Bound (UCB) algorithm satisfies this stability condition, it is not minimax optimal, raising the question of whether minimax optimality and statistical stability can be achieved simultaneously. In this paper, we analyze the stability properties of a broad class of bandit algorithms that are based on the optimism principle. We establish general structural conditions under which such algorithms violate the Lai-Wei stability criterion. As a consequence, we show that widely used minimax-optimal UCB-style algorithms, including MOSS, Anytime-MOSS, Vanilla-MOSS, ADA-UCB, OC-UCB, KL-MOSS, KL-UCB++, KL-UCB-SWITCH, and Anytime KL-UCB-SWITCH, are unstable. We further complement our theoretical results with numerical simulations demonstrating that, in all these cases, the sample means fail to exhibit asymptotic normality. Overall, our findings suggest a fundamental tension between stability and minimax optimal regret, raising the question of whether it is possible to design bandit algorithms that achieve both. Understanding whether such simultaneously stable and minimax optimal strategies exist remains an important open direction.

Cell migration often exhibits long-range temporal correlations and anomalous diffusion, even in the absence of external guidance cues such as chemical gradients or topographical constraints. This raises a fundamental question: do such correlations enhance a cell's ability to navigate complex environments? In this work, we explore how temporally correlated noise (modeled using fractional Brownian motion) influences chemotactic search dynamics. Through computational experiments, we show that superdiffusive motion, when combined with gradient-driven migration, enables robust exploration of the chemoattractant landscape. Cells reliably reach the global maximum of the concentration field, even in the presence of spatial noise, secondary cues, or irregular signal geometry. We quantify this behavior by analyzing the distribution of first hitting times under varying degrees of temporal correlation. Notably, our results are consistent across diverse conditions, including flat and curved substrates, and scenarios involving both primary and self-generated chemotactic signals. Beyond biological implications, these findings also offer insight into the design of optimization and sampling algorithms that benefit from structured stochasticity.

Time series anomaly detection is widely used in IoT and cyber-physical systems, yet its evaluation remains challenging due to diverse application objectives and heterogeneous metric assumptions. This study introduces a problem-oriented framework that reinterprets existing metrics based on the specific evaluation challenges they are designed to address, rather than their mathematical forms or output structures. We categorize over twenty commonly used metrics into six dimensions: 1) basic accuracy-driven evaluation; 2) timeliness-aware reward mechanisms; 3) tolerance to labeling imprecision; 4) penalties reflecting human-audit cost; 5) robustness against random or inflated scores; and 6) parameter-free comparability for cross-dataset benchmarking. Comprehensive experiments are conducted to examine metric behavior under genuine, random, and oracle detection scenarios. By comparing their resulting score distributions, we quantify each metric's discriminative ability – its capability to distinguish meaningful detections from random noise. The results show that while most event-level metrics exhibit strong separability, several widely used metrics (e.g., NAB, Point-Adjust) demonstrate limited resistance to random-score inflation. These findings reveal that metric suitability must be inherently task-dependent and aligned with the operational objectives of IoT applications. The proposed framework offers a unified analytical perspective for understanding existing metrics and provides practical guidance for selecting or developing more context-aware, robust, and fair evaluation methodologies for time series anomaly detection.

We consider the problem of joint estimation of the parameters of $m$ linear dynamical systems, given access to single realizations of their respective trajectories, each of length $T$. The linear systems are assumed to reside on the nodes of an undirected and connected graph $G = ([m], \mathcal{E})$, and the system matrices are assumed to either vary smoothly or exhibit small number of “jumps” across the edges. We consider a total variation penalized least-squares estimator and derive non-asymptotic bounds on the mean squared error (MSE) which hold with high probability. In particular, the bounds imply for certain choices of well connected $G$ that the MSE goes to zero as $m$ increases, even when $T$ is constant. The theoretical results are supported by extensive experiments on synthetic and real data.

Global ocean forecasting aims to predict key ocean variables such as temperature, salinity, and currents, which is essential for understanding and describing oceanic phenomena. In recent years, data-driven deep learning-based ocean forecast models, such as XiHe, WenHai, LangYa and AI-GOMS, have demonstrated significant potential in capturing complex ocean dynamics and improving forecasting efficiency. Despite these advancements, the absence of open-source, standardized benchmarks has led to inconsistent data usage and evaluation methods. This gap hinders efficient model development, impedes fair performance comparison, and constrains interdisciplinary collaboration. To address this challenge, we propose OceanForecastBench, a benchmark offering three core contributions: (1) A high-quality global ocean reanalysis data over 28 years for model training, including 4 ocean variables across 23 depth levels and 4 sea surface variables. (2) A high-reliability satellite and in-situ observations for model evaluation, covering approximately 100 million locations in the global ocean. (3) An evaluation pipeline and a comprehensive benchmark with 6 typical baseline models, leveraging observations to evaluate model performance from multiple perspectives. OceanForecastBench represents the most comprehensive benchmarking framework currently available for data-driven ocean forecasting, offering an open-source platform for model development, evaluation, and comparison. The dataset and code are publicly available at: https://github.com/Ocean-Intelligent-Forecasting/OceanForecastBench.