Papers

We consider the problem of quickest changepoint detection under the Average Run Length (ARL) constraint where the pre-change and post-change laws lie in composite families $\mathscr{P}$ and $\mathscr{Q}$ respectively. In such a problem, a massive challenge is characterizing the best possible detection delay when the "hardest" pre-change law in $\mathscr{P}$ depends on the unknown post-change law $Q\in\mathscr{Q}$. And typical simple-hypothesis likelihood-ratio arguments for Page-CUSUM and Shiryaev-Roberts do not at all apply here. To that end, we derive a universal sharp lower bound in full generality for any ARL-calibrated changepoint detector in the low type-I error ($\gamma\to\infty$ regime) of the order $\log(\gamma)/\mathrm{KL}_{\mathrm{inf}}(Q,\mathscr{P})$. We show achievability of this universal lower bound by proving a tight matching upper bound (with the same sharp $\log\gamma$ constant) in the important bounded mean detection setting. In addition, for separated mean shifts, we also we derive a uniform minimax guarantee of this achievability over the alternatives.

The population $\mathrm{KL}_{\inf}$ is a fundamental quantity that appears in lower bounds for (asymptotically) optimal regret of pure-exploration stochastic bandit algorithms, and optimal stopping time of sequential tests. Motivated by this, an empirical $\mathrm{KL}_{\inf}$ statistic is frequently used in the design of (asymptotically) optimal bandit algorithms and sequential tests. While nonasymptotic concentration bounds for the empirical $\mathrm{KL}_{\inf}$ have been developed, their optimality in terms of constants and rates is questionable, and their generality is limited (usually to bounded observations). The fundamental limits of nonasymptotic concentration are often described by the asymptotic fluctuations of the statistics. With that motivation, this paper presents a tight (upper and lower) law of the iterated logarithm for empirical $\mathrm{KL}_{\inf}$ applying to extremely general (unbounded) data.

Credible microscopic traffic simulation requires car-following models that capture both the average response and the substantial variability observed across drivers and situations. However, most data-driven calibrations remain deterministic, producing a single best-fit parameter vector and offering limited guidance for uncertainty-aware prediction, risk-sensitive evaluation, and population-level simulation. Bayesian calibration addresses this gap by inferring a posterior distribution over parameters, but per-trajectory sampling methods such as Markov chain Monte Carlo (MCMC) are computationally infeasible for modern large-scale naturalistic driving datasets. This paper proposes an active simulation-based inference framework for scalable car-following model calibration. The approach combines (i) a residual-augmented car-following simulator with two alternatives for the residual process and (ii) an amortized conditional density estimator that maps an observed leader–follower trajectory directly to a driver-specific posterior over model parameters with a single forward pass at test time. To reduce simulation cost during training, we introduce a joint active design strategy that selects informative parameter proposals together with representative driving contexts, focusing simulations where the current inference model is most uncertain while maintaining realism. Experiments on the HighD dataset show improved predictive accuracy and closer agreement between simulated and observed trajectory distributions relative to Bayesian calibration baselines, with convergence and ablation studies supporting the robustness of the proposed design choices. The framework enables scalable, uncertainty-aware driver population modeling for traffic flow simulation and risk-sensitive transportation analysis.

While regression models capture the relationship between predictors and the response variable, they often lack intuitive accompanying methods to understand the influence of predictors on the outcome. To address this, we introduce an interpretability method called Impact Range Assessment (IRA), which quantifies the maximal influence of each predictor by measuring the total potential change in the response variable, across the predictor range. Validation using synthetic linear and nonlinear datasets demonstrates that relevant predictors produced higher IRA values than irrelevant ones. Moreover, repeated evaluations produced results closely aligned with those from the single-execution analysis, confirming the robustness of the method. A case study using a model that predicts pellet quality demonstrated that the IRA provides a simple and intuitive approach to interpret and rank predictor influence, thereby improving model transparency and reliability.

Linear attention methods offer Transformers $O(N)$ complexity but typically underperform standard softmax attention. We identify two fundamental limitations affecting these approaches: the restriction to convex combinations that only permits additive information blending, and uniform accumulated weight bias that dilutes attention in long contexts. We propose Zero-Sum Linear Attention (ZeroS), which addresses these limitations by removing the constant zero-order term $1/t$ and reweighting the remaining zero-sum softmax residuals. This modification creates mathematically stable weights, enabling both positive and negative values and allowing a single attention layer to perform contrastive operations. While maintaining $O(N)$ complexity, ZeroS theoretically expands the set of representable functions compared to convex combinations. Empirically, it matches or exceeds standard softmax attention across various sequence modeling benchmarks.

Gradient flows of the Kullback–Leibler (KL) divergence, such as the Fokker–Planck equation and Stein Variational Gradient Descent, evolve a distribution toward a target density known only up to a normalizing constant. We introduce new gradient flows of the KL divergence with a remarkable combination of properties: they admit accurate interacting-particle approximations in high dimensions, and the per-step cost scales linearly in both the number of particles and the dimension. These gradient flows are based on new transportation-based Riemannian geometries on the space of probability measures: the Radon–Wasserstein geometry and the related Regularized Radon–Wasserstein (RRW) geometry. We define these geometries using the Radon transform so that the gradient-flow velocities depend only on one-dimensional projections. This yields interacting-particle-based algorithms whose per-step cost follows from efficient Fast Fourier Transform-based evaluation of the required 1D convolutions. We additionally provide numerical experiments that study the performance of the proposed algorithms and compare convergence behavior and quantization. Finally, we prove some theoretical results including well-posedness of the flows and long-time convergence guarantees for the RRW flow.

Central banks rely on density forecasts from professional surveys to assess inflation risks and communicate uncertainty. A central challenge in using these surveys is irregular participation: forecasters enter and exit, skip rounds, and reappear after long gaps. In the European Central Bank's Survey of Professional Forecasters, turnover and missingness vary substantially over time, causing the set of submitted predictions to change from quarter to quarter. Standard aggregation rules – such as equal-weight pooling, renormalization after dropping missing forecasters, or ad hoc imputation – can generate artificial jumps in combined predictions driven by panel composition rather than economic information, complicating real-time interpretation and obscuring forecaster performance. We develop coherent Bayesian updating rules for forecast combination under sporadic participation that maintain a well-defined latent predictive state for each forecaster even when their forecast is unobserved. Rather than relying on renormalization or imputation, the combined predictive distribution is updated through the implied conditional structure of the panel. This approach isolates genuine performance differences from mechanical participation effects and yields interpretable dynamics in forecaster influence. In the ECB survey, it improves predictive accuracy relative to equal-weight benchmarks and delivers smoother and better-calibrated inflation density forecasts, particularly during periods of high turnover.

We consider the problem of estimating the Fréchet and conditional Fréchet mean from data taking values in separable metric spaces. Unlike Euclidean spaces, where well-established methods are available, there is no practical estimator that works universally for all metric spaces. Therefore, we introduce a computable estimator for the Fréchet mean based on random quantization techniques and establish its universal consistency across any separable metric spaces. Additionally, we propose another estimator for the conditional Fréchet mean, leveraging data-driven partitioning and quantization, and demonstrate its universal consistency when the output space is any Banach space.

Riemannian flow matching (RFM) extends flow-based generative modeling to data supported on manifolds by learning a time-dependent tangent vector field whose flow-ODE transports a simple base distribution to the data law. We develop a nonasymptotic Total Variation (TV) convergence analysis for RFM samplers that use a learned vector field together with Euler discretization on manifolds. Our key technical ingredient is a differential inequality governing the evolution of TV between two manifold ODE flows, which expresses the time-derivative of TV through the divergence of the vector-field mismatch and the score of the reference flow; controlling these terms requires establishing new bounds that explicitly account for parallel transport and curvature. Under smoothness assumptions on the population flow-matching field and either uniform (compact manifolds) or mean-square (Hadamard manifolds) approximation guarantees for the learned field, we obtain explicit bounds of the form $\mathrm{TV}\le C_{\mathrm{Lip}}\,h + C_{\varepsilon}\,\varepsilon$ (with an additional higher-order $\varepsilon^2$ term on compact manifolds), cleanly separating numerical discretization and learning errors. Here, $h$ is the step-size and $\varepsilon$ is the target accuracy. Instantiations yield explicit polynomial iteration complexities on the hypersphere $S^d$, and on the SPD$(n)$ manifolds under mild moment conditions.

We derive finite-particle rates for the regularized Stein variational gradient descent (R-SVGD) algorithm introduced by He et al. (2024) that corrects the constant-order bias of the SVGD by applying a resolvent-type preconditioner to the kernelized Wasserstein gradient. For the resulting interacting $N$-particle system, we establish explicit non-asymptotic bounds for time-averaged (annealed) empirical measures, illustrating convergence in the true (non-kernelized) Fisher information and, under a $\mathrm{W}_1\mathrm{I}$ condition on the target, corresponding $\mathrm{W}_1$ convergence for a large class of smooth kernels. Our analysis covers both continuous- and discrete-time dynamics and yields principled tuning rules for the regularization parameter, step size, and averaging horizon that quantify the trade-off between approximating the Wasserstein gradient flow and controlling finite-particle estimation error.

Short-term forecasting of vegetation dynamics is a key enabler for data-driven decision support in precision agriculture. Normalized Difference Vegetation Index (NDVI) forecasting from satellite observations, however, remains challenging due to sparse and irregular sampling caused by cloud masking, as well as the heterogeneous climatic conditions under which crops evolve. In this work, we propose a probabilistic forecasting framework for field-level NDVI prediction under sparse, irregular clear-sky acquisitions. The architecture separates the encoding of historical NDVI and meteorological observations from future exogenous covariates, fusing both representations for multi-step quantile prediction. To address irregular revisit patterns and horizon-dependent uncertainty, we introduce a temporal-distance weighted quantile loss that aligns the training objective with the effective forecasting horizon. In addition, we incorporate cumulative and extreme-weather feature engineering to capture delayed meteorological effects relevant to vegetation response. Experiments on European satellite data show that the proposed approach outperforms statistical, deep learning, and time-series baselines on both pointwise and probabilistic evaluation metrics. Ablation studies confirm that target history is the primary driver of performance, with meteorological covariates providing additional gains in the full multimodal setting. The code is available at https://github.com/arco-group/ndvi-forecasting.

We extend the recently developed entropic and conservative variance reduction framework [M. Sadr, N. G. Hadjiconstantinou, A variance-reduced direct Monte Carlo simulation method for solving the Boltzmann equation over a wide range of rarefaction, Journal of Computational Physics 472 (2023) 111677.] to the particle-in-cell (PIC) method of solving Vlasov-Poisson equation. We show that a zeroth-order approximation that freezes the importance weights during the velocity-space kick is stable at the expense of introducing bias. Then, we propose a correction for the weight distribution using maximum cross-entropy formulation to ensure conservation laws while minimizing the introduced bias. In several test cases including Sod's shock tube and Landau damping we show that the proposed method maintains the substantial speed-up of variance reduction method compared to the PIC simulations in the low signal regime with minimal changes to the simulation code.

Do cryptocurrency markets respond differently to infrastructure and regulatory shocks? We study returns and conditional variance on a shared sample of 50 events and six assets (January 2019–August 2025), using GJR-GARCH-X models and dependence-aware inference. Treating event inclusion as a design parameter, we trace the variance differential across inclusion screens. Curated high-salience events yield a $3.49\times$ point-estimate multiplier, whereas a mechanical impact filter on a broad reconstructed candidate pool yields approximately $0.5$–$1.6\times$. This pattern is descriptive and selection-conditional, not an inferential comparison between screens. The curated variance differential is not significant against its fitted sharp per-asset-equality null under the conditional fixed-path Student-$t$-copula bootstrap ($p\approx0.39$). Floored recursive sensitivity gives one-sided $p=0.025$ at baseline and $0.041$ with asset-specific high-variance-regime controls, so the verdict depends on inference scheme and implementation. Across reported dependence inputs, the six-contrast effective sample size is $1.33$–$2.35$; one-sided effective-df sensitivity gives $p=0.044$–$0.116$. Earlier significance from treating correlated per-asset coefficients as independent samples is not robust to dependence and heavy-tail corrections. The cumulative-abnormal-return difference is $+8.69$ percentage points (event-level block-bootstrap $p=0.202$). The asymmetry remains directional, selection-conditional and unresolved. The contribution is an inference ladder and an internal Monte-Carlo calibration study, demonstrated through correction of the author's earlier significance claim.

Scarcity of labeled training data remains the long pole in the tent for building performant language technology and generative AI models. Transformer models – particularly LLMs – are increasingly being used to mitigate the data scarcity problem via synthetic data generation. However, because the models are black boxes, the properties of the synthetic data are difficult to predict. In practice it is common for language technology engineers to 'fiddle' with the LLM temperature setting and hope that what comes out the other end improves the downstream model. Faced with this uncertainty, here we propose Data Kernel Perspective Space (DKPS) to provide the foundation for mathematical analysis yielding concrete statistical guarantees for the quality of the outputs of transformer models. We first show the mathematical derivation of DKPS and how it provides performance guarantees. Next we show how DKPS performance guarantees can elucidate performance of a downstream task, such as neural machine translation models or LLMs trained using Contrastive Preference Optimization (CPO). Limitations of the current work and future research are also discussed.

In recent years, explaining decisions made by complex machine learning models has become essential in high-stakes domains such as energy systems, healthcare, finance, and autonomous systems. However, the reliability of these explanations, namely, whether they remain stable and consistent under realistic, non-adversarial changes, remains largely unmeasured. Widely used methods such as SHAP and Integrated Gradients (IG) are well-motivated by axiomatic notions of attribution, yet their explanations can vary substantially even under system-level conditions, including small input perturbations, correlated representations, and minor model updates. Such variability undermines explanation reliability, as reliable explanations should remain consistent across equivalent input representations and small, performance-preserving model changes. We introduce the Explanation Reliability Index (ERI), a family of metrics that quantifies explanation stability under four reliability axioms: robustness to small input perturbations, consistency under feature redundancy, smoothness across model evolution, and resilience to mild distributional shifts. For each axiom, we derive formal guarantees, including Lipschitz-type bounds and temporal stability results. We further propose ERI-T, a dedicated measure of temporal reliability for sequential models, and introduce ERI-Bench, a benchmark designed to systematically stress-test explanation reliability across synthetic and real-world datasets. Experimental results reveal widespread reliability failures in popular explanation methods, showing that explanations can be unstable under realistic deployment conditions. By exposing and quantifying these instabilities, ERI enables principled assessment of explanation reliability and supports more trustworthy explainable AI (XAI) systems.

A large body of theory and empirical work hypothesizes a connection between the flatness of a neural network's loss landscape during training and its performance. However, there have been conceptually opposite pieces of evidence regarding when SGD prefers flatter or sharper solutions during training. In this work, we partially but causally clarify the flatness-seeking behavior of SGD by identifying and exactly solving an analytically solvable model that exhibits both flattening and sharpening behavior during training. In this model, the SGD training has no a priori preference for flatness, but only a preference for minimal gradient fluctuations. This leads to the insight that, at least within this model, it is data distribution that uniquely determines the sharpness at convergence, and that a flat minimum is preferred if and only if the noise in the labels is isotropic across all output dimensions. When the noise in the labels is anisotropic, the model instead prefers sharpness and can converge to an arbitrarily sharp solution, depending on the imbalance in the noise in the labels spectrum. We reproduce this key insight in controlled settings with different model architectures such as MLP, RNN, and transformers.

When units in observational studies are clustered in groups, such as students in schools or patients in hospitals, researchers often address confounding by adjusting for cluster-level covariates or cluster membership. In this paper, we develop a unified weighting framework that clarifies how different estimation methods control two distinct sources of imbalance: global balance (differences between treated and control units across clusters) and local balance (differences within clusters). We show that inverse propensity score weighting (IPW) with a random effects propensity score model – the current standard in the literature – targets only global balance and constant level shifts across clusters, but imposes no constraints on local balance. We then present two approaches that target both forms of balance. First, hierarchical balancing weights directly control global and local balance through a constrained optimization problem. Second, building on the recently proposed Generalized Mundlak approach, we develop a novel Mundlak balancing weights estimator that adjusts for cluster-level sufficient statistics rather than cluster indicators; this approach can accommodate small clusters where all units are treated or untreated. Critically, these approaches rest on different assumptions: hierarchical balancing weights require only that treatment is ignorable given covariates and cluster membership, while Mundlak methods additionally require an exponential family structure. We then compare these methods in a simulation study and in two applications in education and health services research that exhibit very different cluster structures.

Modern training and inference pipelines in statistical learning and deep learning repeatedly invoke linear-system solves as inner loops, yet high-accuracy deterministic solvers can be prohibitively expensive when solves must be repeated many times or when only partial information (selected components or linear functionals) is required. We position Monte Carlo boosting as a practical alternative in this regime, surveying random-walk estimators and sequential residual correction in a unified notation (Neumann-series representation, forward/adjoint estimators, and Halton-style sequential correction), with extensions to overdetermined/least-squares problems and connections to IRLS-style updates in data augmentation and EM/ECM algorithms. Empirically, we compare Jacobi and Gauss–Seidel iterations with plain Monte Carlo, exact sequential Monte Carlo, and a subsampled sequential variant, illustrating scaling regimes that motivate when Monte Carlo boosting can be an enabling compute primitive for modern statistical learning workflows.

The problem of combining multiple forecasts of related quantities that obey expected equality and additivity constraints, often referred to a hierarchical forecast reconciliation, is naturally stated as a simple optimization problem. In this paper we explore optimization-based point forecast reconciliation at scales faced by large retailers. We implement and benchmark several algorithms to solve the forecast reconciliation problem, showing efficacy when the dimension of the problem exceeds four billion forecasted values. To the best of our knowledge, this is the largest forecast reconciliation problem, and perhaps on-par with the largest constrained least-squares-problem ever solved. We also make several theoretical contributions. We show that for a restricted class of problems and when the loss function is weighted appropriately, least-squares forecast reconciliation is equivalent to share-based forecast reconciliation. This formalizes how the optimization based approach can be thought of as a generalization of share-based reconciliation, applicable to multiple, overlapping data hierarchies.

Synthetic vehicle speed trajectory generation is essential for evaluating vehicle control algorithms and connected vehicle technologies. Traditional Markov chain approaches suffer from discretization artifacts and limited expressiveness. This paper proposes a physics-informed diffusion framework for conditional micro-trip synthesis, combining a dual-channel speed-acceleration representation with soft physics constraints that resolve optimization conflicts inherent to hard-constraint formulations. We compare a 1D U-Net architecture against a transformer-based Conditional Score-based Diffusion Imputation (CSDI) model using 6,367 GPS-derived micro-trips. CSDI achieves superior distribution matching (Wasserstein distance 0.30 for speed, 0.026 for acceleration), strong indistinguishability from real data (discriminative score 0.49), and validated utility for downstream energy assessment tasks. The methodology enables scalable generation of realistic driving profiles for intelligent transportation systems (ITS) applications without costly field data collection.

In many clinical contexts, estimating effects of treatment in time-to-event data is complicated not only by confounding, censoring, and heterogeneity, but also by the presence of a cured subpopulation in which the event of interest never occurs. In such settings, treatment may have distinct effects on (1) the probability of being cured and (2) the event timing among non-cured individuals. Standard survival analysis and causal inference methods typically do not separate cured from non-cured individuals, obscuring distinct treatment mechanisms on cure probability and event timing. To address these challenges, we propose a matching-based framework that constructs distinct match groups to estimate heterogeneous treatment effects (HTE) on cure probability and event timing, respectively. We use mixture cure models to identify feature importance for both estimands, which in turn informs weighted distance metrics for matching in high-dimensional spaces. Within matched groups, Kaplan-Meier estimators provide estimates of cure probability and expected time to event, from which individual-level treatment effects are derived. We provide theoretical guarantees for estimator consistency and distance metric optimality under an equal-scale constraint. We further decompose estimation error into contributions from censoring, model fitting, and irreducible noise. Simulations and real-world data analyses demonstrate that our approach delivers interpretable and robust HTE estimates in time-to-event settings.

Recent progress has rapidly advanced our understanding of the mechanisms underlying in-context learning in modern attention-based neural networks. However, existing results focus exclusively on unimodal data; in contrast, the theoretical underpinnings of in-context learning for multi-modal data remain poorly understood. We introduce a mathematically tractable framework for studying multi-modal learning and explore when transformer-like architectures can recover Bayes-optimal performance in-context. To model multi-modal problems, we assume the observed data arises from a latent factor model. Our first result comprises a negative take on expressibility: we prove that single-layer, linear self-attention fails to recover the Bayes-optimal predictor uniformly over the task distribution. To address this limitation, we introduce a novel, linearized cross-attention mechanism, which we study in the regime where both the number of cross-attention layers and the context length are large. We show that this cross-attention mechanism is provably Bayes optimal when optimized using gradient flow. Our results underscore the benefits of depth for in-context learning and establish the provable utility of cross-attention for multi-modal distributions.

Training modern large language models (LLMs) has become a veritable smorgasbord of algorithms and datasets designed to elicit particular behaviors, making it critical to develop techniques to understand the effects of datasets on the model's properties. This is exacerbated by recent experiments that show datasets can transmit signals that are not directly observable from individual datapoints, posing a conceptual challenge for dataset-centric understandings of LLM training and suggesting a missing fundamental account of such phenomena. Towards understanding such effects, inspired by recent work on the linear structure of LLMs, we uncover a general mechanism through which hidden subtexts can arise in generic datasets. We introduce Logit-Linear-Selection (LLS), a method that prescribes how to select subsets of a generic preference dataset to elicit a wide range of hidden effects. We apply LLS to discover subsets of real-world datasets so that models trained on them exhibit behaviors ranging from having specific preferences, to responding to prompts in a different language not present in the dataset, to taking on a different persona. Crucially, the effect persists for the selected subset, across models with varying architectures, supporting its generality and universality.

Stochastic compartmental models are prevalent tools for describing disease spread, but inference under these models is challenging for many types of surveillance data when the marginal likelihood function becomes intractable due to missing information. To address this, we develop a closed-form likelihood for discretely observed incidence count data under the dynamical survival analysis (DSA) paradigm. The method approximates the stochastic population-level hazard by a large population limit while retaining a count-valued stochastic model, and leads to survival analytic inferential strategies that are both computationally efficient and flexible to model generalizations. Through simulation, we show that parameter estimation is competitive with recent exact but computationally expensive likelihood-based methods in partially observed settings. Previous work has shown that the DSA approximation is generalizable, and we show that the inferential developments here also carry over to models featuring individual heterogeneity, such as frailty models. We consider case studies of both Ebola and COVID-19 data on variants of the model, including a network-based epidemic model and a model with distributions over susceptibility, demonstrating its flexibility and practical utility on real, partially observed datasets.

We study the estimation of quadratic Sobolev-type integral functionals of an unknown density on the unit sphere. The functional is defined through fractional powers of the Laplace–Beltrami operator and provides a global measure of smoothness and spectral energy. Our approach relies on spherical needlet frames, which yield a localized multiscale decomposition while preserving tight frame properties in the natural square-integrable function space on the sphere. We construct unbiased estimators of suitably truncated versions of the functional and derive sharp oracle risk bounds through an explicit bias–variance analysis. When the smoothness of the density is unknown, we propose a Lepski-type data-driven selection of the resolution level. The resulting adaptive estimator achieves minimax-optimal rates over Sobolev classes, without resorting to nonlinear or sparsity-based methods.

We study sequential change-point detection for spatio-temporal point processes, where actionable detection requires not only identifying when a distributional change occurs but also localizing where it manifests in space. While classical quickest change detection methods provide strong guarantees on detection delay and false-alarm rates, existing approaches for point-process data predominantly focus on temporal changes and do not explicitly infer affected spatial regions. We propose a likelihood-free, score-based detection framework that jointly estimates the change time and the change region in continuous space-time without assuming parametric knowledge of the pre- or post-change dynamics. The method leverages a localized and conditionally weighted Hyvärinen score to quantify event-level deviations from nominal behavior and aggregates these scores using a spatio-temporal CUSUM-type statistic over a prescribed class of spatial regions. Operating sequentially, the procedure outputs both a stopping time and an estimated change region, enabling real-time detection with spatial interpretability. We establish theoretical guarantees on false-alarm control, detection delay, and spatial localization accuracy, and demonstrate the effectiveness of the proposed approach through simulations and real-world spatio-temporal event data.

Nonnegative matrix factorization (NMF) is a popular data embedding technique. Given a nonnegative data matrix $X$, it aims at finding two lower dimensional matrices, $W$ and $H$, such that $X\approx WH$, where the factors $W$ and $H$ are constrained to be element-wise nonnegative. The factor $W$ serves as a basis for the columns of $X$. In order to obtain more interpretable and unique solutions, minimum-volume NMF (MinVol NMF) minimizes the volume of $W$. In this paper, we consider the dual approach, where the volume of $H$ is maximized instead; this is referred to as maximum-volume NMF (MaxVol NMF). MaxVol NMF is identifiable under the same conditions as MinVol NMF in the noiseless case, but it behaves rather differently in the presence of noise. In practice, MaxVol NMF is much more effective to extract a sparse decomposition and does not generate rank-deficient solutions. In fact, we prove that the solutions of MaxVol NMF with the largest volume correspond to clustering the columns of $X$ in disjoint clusters, while the solutions of MinVol NMF with smallest volume are rank deficient. We propose two algorithms to solve MaxVol NMF. We also present a normalized variant of MaxVol NMF that exhibits better performance than MinVol NMF and MaxVol NMF, and can be interpreted as a continuum between standard NMF and orthogonal NMF. We illustrate our results in the context of hyperspectral unmixing. The code is available from https://gitlab.com/vuthanho/maxvolmf.jl.

We present a novel approach to ecological risk assessment by recasting the Species Sensitivity Distribution (SSD) method within a Bayesian nonparametric (BNP) framework. Widely mandated by environmental regulatory bodies globally, SSD has faced criticism due to its historical reliance on parametric assumptions when modeling species variability. By adopting nonparametric mixture models, we address this limitation, establishing a statistically robust foundation for SSD. Our BNP approach offers several advantages, including its efficacy in handling small datasets or censored data, which are common in ecological risk assessment, and its ability to provide principled uncertainty quantification alongside simultaneous density estimation and clustering. We utilize a specific nonparametric prior as the mixing measure, chosen for its robust clustering properties, a crucial consideration given the lack of strong prior beliefs about the number of components. Through simulation studies and analysis of real datasets, we demonstrate the superiority of our BNP-SSD over classical SSD methods. We also provide a BNP-SSD Shiny application, making our methodology available to the Ecotoxicology community. Moreover, we exploit the inherent clustering structure of the mixture model to explore patterns in species sensitivity. Our findings underscore the effectiveness of the proposed approach in improving ecological risk assessment methodologies.

Setting the learning rate (LR) for a deep learning model is a critical part of successful training. Choosing LRs is often done empirically with trial and error. In this work, we explore a solvable model of optimal LR schedules for a powerlaw random feature model trained with stochastic gradient descent (SGD). We consider the optimal schedule $\eta_T^\star(t)$ where $t$ is the current iterate and $T$ is the training horizon. This schedule is computed both as a numerical optimization problem and also analytically using optimal control theory. Our analysis reveals two regimes which we term the easy phase and hard phase. In the easy phase the optimal schedule is a polynomial decay $\eta_T^\star(t) \simeq T^{-\xi} (1-t/T)^{\delta}$ where $\xi$ and $\delta$ depend on the properties of the features and task. In the hard phase, the optimal schedule resembles warmup-stable-decay with constant initial LR and annealing performed over a vanishing fraction of training steps. We investigate joint optimization of LR and batch size and find batch ramps can improve the wall-clock time in the easy phase. Beyond SGD, we derive optimal schedules for momentum parameter $\beta(t)$ and show that it improves the loss-scaling exponent in the hard phase. We compare our optimal schedule to various benchmarks including (1) optimal constant learning rates $\eta_T(t) \sim T^{-\xi}$ (2) optimal power laws $\eta_T(t) \sim T^{-\xi} t^{-\chi}$, finding that our schedule achieves better rates than either of these. Our theory suggests that LR transfer across training horizon depends on the structure of the model and task. For ResNet image classification on CIFAR-5M, the learning curves exhibit hard-phase behavior where optimal base LRs are constant under sufficient annealing. GPT-2 style transformers trained in language modeling exhibit easy-phase behavior where optimal LRs shift even under annealing.

Small and medium-sized islands are acutely exposed to climate change and ecosystem degradation, yet the extent to which uncertainty is systematically addressed in scientific assessments of their ecosystem services remains poorly understood. This study revisits 226 peer-reviewed articles drawn from two global systematic reviews on island ecosystem services and climate change, applying a structured post hoc analysis to evaluate how uncertainty is treated across methods, service categories, ecosystem realms, and decision contexts. Studies were classified according to whether uncertainty was explicitly analysed, just mentioned, or ignored. Only 30 percent of studies incorporated uncertainty explicitly, while more than half did not address it at all. Scenario-based approaches dominated uncertainty assessment, whereas probabilistic and ensemble-based frameworks remained limited. Cultural ecosystem services and extreme climate impacts exhibited the lowest levels of uncertainty integration, and few studies connected uncertainty treatment to policy relevant decision frameworks. Weak or absent treatment of uncertainty emerges as a structural challenge in island systems, where narrow ecological thresholds, strong land-sea coupling, limited spatial buffers, and reduced institutional redundancy amplify the consequences of decision-making under incomplete knowledge. Systematic mapping of how uncertainty is framed, operationalised, or neglected reveals persistent methodological and conceptual gaps and informs concrete directions for strengthening uncertainty integration in future island-focused ecosystem service and climate assessments. Embedding uncertainty more robustly into modelling practices, participatory processes, and policy tools is essential for enhancing scientific credibility, governance relevance, and adaptive capacity in insular socio-ecological systems.