Papers

We study the problem of learning a $n$-variables $k$-CNF formula $\Phi$ from its i.i.d. uniform random solutions, which is equivalent to learning a Boolean Markov random field (MRF) with $k$-wise hard constraints. Revisiting Valiant's algorithm (Commun. ACM'84), we show that it can exactly learn (1) $k$-CNFs with bounded clause intersection size under Lovász local lemma type conditions, from $O(\log n)$ samples; and (2) random $k$-CNFs near the satisfiability threshold, from $\widetilde{O}(n^{\exp(-\sqrt{k})})$ samples. These results significantly improve the previous $O(n^k)$ sample complexity. We further establish new information-theoretic lower bounds on sample complexity for both exact and approximate learning from i.i.d. uniform random solutions.

This paper investigates global and local laws for sample covariance matrices with general growth rates of dimensions. The sample size $N$ and population dimension $M$ can have the same order in logarithm, which implies that their ratio $M/N$ can approach zero, a constant, or infinity. These theories are utilized to determine the convergence rate of spiked eigenvalue estimates.

Concept drift and label scarcity are two critical challenges limiting the robustness of predictive models in dynamic industrial environments. Existing drift detection methods often assume global shifts and rely on dense supervision, making them ill-suited for regression tasks with local drifts and limited labels. This paper proposes an adaptive sampling framework that combines residual-based exploration and exploitation with EWMA monitoring to efficiently detect local concept drift under labeling budget constraints. Empirical results on synthetic benchmarks and a case study on electricity market demonstrate superior performance in label efficiency and drift detection accuracy.

We present a suite of packages in R, Python, Julia, and C++ that efficiently solve the Sorted L-One Penalized Estimation (SLOPE) problem. The packages feature a highly efficient hybrid coordinate descent algorithm that fits generalized linear models (GLMs) and supports a variety of loss functions, including Gaussian, binomial, Poisson, and multinomial logistic regression. Our implementation is designed to be fast, memory-efficient, and flexible. The packages support a variety of data structures (dense, sparse, and out-of-memory matrices) and are designed to efficiently fit the full SLOPE path as well as handle cross-validation of SLOPE models, including the relaxed SLOPE. We present examples of how to use the packages and benchmarks that demonstrate the performance of the packages on both real and simulated data and show that our packages outperform existing implementations of SLOPE in terms of speed.

All Resolutions Inference (ARI) is a post hoc inference method for functional Magnetic Resonance Imaging (fMRI) data analysis that provides valid lower bounds on the proportion of truly active voxels within any, possibly data-driven, cluster. As such, it addresses the paradox of spatial specificity encountered with more classical cluster-extent thresholding methods. It allows the cluster-forming threshold to be increased in order to locate the signal with greater spatial precision without overfitting, also known as the drill-down approach. Notip and pARI are two recent permutation-based extensions of ARI designed to increase statistical power by accounting for the strong dependence structure typical of fMRI data. A recent comparison between these papers based on large voxel clusters concluded that pARI outperforms Notip. We revisit this conclusion by conducting a systematic comparison of the two. Our reanalysis of the same fMRI data sets from the Neurovault database demonstrates the existence of complementary performance regimes: while pARI indeed achieves higher sensitivity for large clusters, Notip provides more informative and robust results for smaller clusters. In particular, while Notip supports informative “drill-down” exploration into subregions of activation, pARI often yields non-informative bounds in such cases, and can even underperform the baseline ARI method.

Score-based Generative Models (SGMs) have achieved impressive performance in data generation across a wide range of applications and benefit from strong theoretical guarantees. Recently, methods inspired by statistical mechanics, in particular, Hamiltonian dynamics, have introduced Critically-damped Langevin Diffusions (CLDs), which define diffusion processes on extended spaces by coupling the data with auxiliary variables. These approaches, along with their associated score-matching and sampling procedures, have been shown to outperform standard diffusion-based samplers numerically. In this paper, we analyze a generalized dynamic that extends classical CLDs by introducing an additional hyperparameter controlling the noise applied to the data coordinate, thereby better exploiting the extended space. We further derive a novel upper bound on the sampling error of CLD-based generative models in the Wasserstein metric. This additional hyperparameter influences the smoothness of sample paths, and our discretization error analysis provides practical guidance for its tuning, leading to improved sampling performance.

We first study the generalization error of models that use a fixed feature representation (frozen intermediate layers) followed by a trainable readout layer. This setting encompasses a range of architectures, from deep random-feature models to echo-state networks (ESNs) with recurrent dynamics. Working in the high-dimensional regime, we apply Random Matrix Theory to derive a closed-form expression for the asymptotic generalization error. We then apply this analysis to recurrent representations and obtain concise formula that characterize their performance. Surprisingly, we show that a linear ESN is equivalent to ridge regression with an exponentially time-weighted (”memory”) input covariance, revealing a clear inductive bias toward recent inputs. Experiments match predictions: ESNs win in low-sample, short-memory regimes, while ridge prevails with more data or long-range dependencies. Our methodology provides a general framework for analyzing overparameterized models and offers insights into the behavior of deep learning networks.

This work addresses the problem of efficient sampling of Markov random fields (MRF). The sampling of Potts or Ising MRF is most often based on Gibbs sampling, and is thus computationally expensive. We consider in this work how to circumvent this bottleneck through a link with Gaussian Markov Random fields. The latter can be sampled in several cost-effective ways, and we introduce a mapping from real-valued GMRF to discrete-valued MRF. The resulting new class of MRF benefits from a few theoretical properties that validate the new model. Numerical results show the drastic performance gain in terms of computational efficiency, as we sample at least 35x faster than Gibbs sampling using at least 37x less energy, all the while exhibiting empirical properties close to classical MRFs.

Preconditioning is a key component of MCMC algorithms that improves sampling efficiency by facilitating exploration of geometrically complex target distributions through an invertible map. While linear preconditioners are often sufficient for moderately complex target distributions, recent work has explored nonlinear preconditioning with invertible neural networks as components of normalizing flows (NFs). However, empirical and theoretical studies show that overparameterized NF preconditioners can degrade sampling efficiency and fit quality. Moreover, existing NF-based approaches do not adapt their architectures to the target distribution. Related work outside of MCMC similarly finds that suitably parameterized NFs can achieve comparable or superior performance with substantially less training time or data. We propose a factorized preconditioning architecture that reduces NF complexity by combining a linear component with a conditional NF, improving adaptability to target geometry. The linear preconditioner is applied to dimensions that are approximately Gaussian, as estimated from warmup samples, while the conditional NF models more complex dimensions. Our method yields significantly better tail samples on two complex synthetic distributions and consistently better performance on a sparse logistic regression posterior across varying likelihood and prior strengths. It also achieves higher effective sample sizes on hierarchical Bayesian model posteriors with weak likelihoods and strong funnel geometries. This approach is particularly relevant for hierarchical Bayesian model analyses with limited data and could inform current theoretical and software strides in neural MCMC design.

We introduce Cycle-Sync, a robust and global framework for estimating camera poses (both rotations and locations). Our core innovation is a location solver that adapts message-passing least squares (MPLS) – originally developed for group synchronization – to camera location estimation. We modify MPLS to emphasize cycle-consistent information, redefine cycle consistencies using estimated distances from previous iterations, and incorporate a Welsch-type robust loss. We establish the strongest known deterministic exact-recovery guarantee for camera location estimation, showing that cycle consistency alone – without access to inter-camera distances – suffices to achieve the lowest sample complexity currently known. To further enhance robustness, we introduce a plug-and-play outlier rejection module inspired by robust subspace recovery, and we fully integrate cycle consistency into MPLS for rotation synchronization. Our global approach avoids the need for bundle adjustment. Experiments on synthetic and real datasets show that Cycle-Sync consistently outperforms leading pose estimators, including full structure-from-motion pipelines with bundle adjustment.

The Lasso has been widely used as a method for variable selection, valued for its simplicity and empirical performance. However, Lasso's selection stability deteriorates in the presence of correlated predictors. Several approaches have been developed to mitigate this limitation. In this paper, we provide a brief review of existing approaches, highlighting their limitations. We then propose a technique to improve the selection stability of Lasso by integrating a weighting scheme into the Lasso penalty function, where the weights are defined as an increasing function of a correlation-adjusted ranking that reflects the predictive power of predictors. Empirical evaluations on both simulated and real-world datasets demonstrate the efficacy of the proposed method. Additional numerical results demonstrate the effectiveness of the proposed approach in stabilizing other regularization-based selection methods, indicating its potential as a general-purpose solution.

To achieve the United Nations Sustainable Development Goals, coordinated action across their interlinked indicators is required. Although most of the research on the interlinkages of the SDGs is done at the goal level, policies are usually made and implemented at the level of indicators (or targets). Our study examines the existing literature on SDG interlinkages and indicator (or target) prioritization, highlighting important drawbacks of current methodologies. To address these limitations, we propose a generic network-based model that can quantify the importance of the SDG indicators and help policymakers in identifying indicators for maximum synergistic impact. Our model applies to any country, offering a tool for national policymakers. We illustrate the application of this model using data from India, identifying important indicators that are crucial for accelerating progress in the SDGs. While our main contribution lies in developing this network-theoretic methodology, we also provide supporting empirical evidence from existing literature for selected key observations.

Probabilistic relaxations of graph cuts offer a differentiable alternative to spectral clustering, enabling end-to-end and online learning without eigendecompositions, yet prior work centered on RatioCut and lacked general guarantees and principled gradients. We present a unified probabilistic framework that covers a wide class of cuts, including Normalized Cut. Our framework provides tight analytic upper bounds on expected discrete cuts via integral representations and Gauss hypergeometric functions with closed-form forward and backward. Together, these results deliver a rigorous, numerically stable foundation for scalable, differentiable graph partitioning covering a wide range of clustering and contrastive learning objectives.

We introduce a unified operator-theoretic framework for analyzing mixing times of finite-state ergodic Markov chains that applies to both reversible and non-reversible dynamics. The central object in our analysis is the projected transition operator $PU_{\perp 1}$, where $P$ is the transition kernel and $U_{\perp 1}$ is orthogonal projection onto mean-zero subspace in $\ell^{2}(\pi)$, where $\pi$ is the stationary distribution. We show that explicitly computable matrix norms of $(PU_{\perp 1})^k$ gives non-asymptotic mixing times/distance to stationarity, and bound autocorrelations at lag $k$. We establish, for the first time, submultiplicativity of pointwise chi-squared divergence in the general non-reversible case. We provide for all times $\chi^{2}(k)$ bounds based on the spectrum of $PU_{\perp 1}$, i.e., magnitude of its distinct non-zero eigenvalues, discrepancy between their algebraic and geometric multiplicities, condition number of a similarity transform, and constant coming from smallest atom of stationary distribution(all scientifically computable). Furthermore, for diagonalizable $PU_{\perp 1}$, we provide explict constants satisfying hypocoercivity phenomenon for discrete time Markov Chains. Our framework enables direct computation of convergence bounds for challenging non-reversible chains, including momentum-based samplers for V-shaped distributions. We provide the sharpest known bounds for non-reversible walk on triangle. Our results combined with simple regression reveals a fundamental insight into momentum samplers: although for uniform distributions, $n\log{n}$ iterations suffice for $\chi^{2}$ mixing, for V-shaped distributions they remain diffusive as $n^{1.969}\log{n^{1.956}}$ iterations are sufficient. The framework shows that for ergodic chains relaxation times $\tau_{rel}=\|\sum_{k=0}^{\infty}P^{k}U_{\perp 1}\|_{\ell^{2}(\pi)}$.

This paper studies the high-dimensional scaling limits of online stochastic gradient descent (SGD). Building on the work of Ben Arous, Gheissari, and Jagannath on the effective dynamics of SGD, we study the critical scaling regime of the step size for single-layer networks. Below this regime the effective dynamics are governed by deterministic (ballistic) limits, whereas at the critical scale a correction term emerges that changes the phase diagram. Near the fixed points of these dynamics, we show that the diffusive (SDE) limit of the rescaled correlation is an Ornstein-Uhlenbeck process. More precisely, it is mean-reverting whenever the information exponent is at least three. At information exponent two the drift has no universal sign, and the fixed point may become repelling; we show this explicitly for phase retrieval, where the sign is determined by the step size and the noise level. These results illustrate the limitations of deterministic scaling limits in capturing stochastic fluctuations in high-dimensional learning dynamics.

Accurately analyzing NMR and MRI diffusion experimental data relies on the theoretical expression used for signal attenuation or phase evolution. In a complex system, the encountered magnetic field is often inhomogeneous, which may be represented by a linear combination of z^n gradient fields, where n is the order. Additionally, the higher the order of the nonlinear gradient field, the more sensitive the phase variances are to differences in diffusion coefficients and delay times. Hence, studying higher-order fields has both theoretical and experimental importance, but this is a challenge for traditional methods. The recently proposed phase diffusion method proposed a general way to overcome the challenge. This method is used and demonstrated in detail in this paper to determine the phase evolution in a quadric field (n = 4). Three different types of phase evolution in the quadric gradient field are obtained. Moreover, a general signal attenuation expression is proposed to describe the signal attenuation for spin diffusion from the origin of the nonlinear gradient field. This approximation is based on the short gradient pulse (SGP) approximation but is extended to include the finite gradient pulse width (FGPW) effect by using the mean square phase. Compared to other forms of signal attenuation, such as Gaussian and Lorentzian, this method covers a broader range of attenuation, from small to relatively large. Additionally, this attenuation is easier to understand than the Mittag-Leffler function-based attenuation. The results, particularly the phase and signal attenuation expressions obtained in this paper, potentially advance PFG diffusion research in nonlinear gradient fields in NMR and MRI.

In this paper, we consider diffusion index forecasting with both tensor and non-tensor predictors, where the tensor structure is preserved with a Canonical Polyadic (CP) tensor factor model. When the number of non-tensor predictors is small, we study the asymptotic properties of the least squares estimator in this tensor factor-augmented regression, allowing for factors with different strengths. We derive an analytical formula for prediction intervals that accounts for the estimation uncertainty of the latent factors. In addition, we propose a novel thresholding estimator for the high-dimensional covariance matrix that is robust to cross-sectional dependence. When the number of non-tensor predictors exceeds or diverges with the sample size, we introduce a multi-source factor-augmented sparse regression model and establish the consistency of the corresponding penalized estimator. Simulation studies validate our theoretical results and an empirical application to U.S. trade flows demonstrates the advantages of our approach over other popular methods in the literature.

When releasing binary proportions computed using sensitive data, several government agencies and other data stewards protect confidentiality of the underlying values by ensuring the released statistics satisfy differential privacy. Typically, this is done by adding carefully chosen noise to the sample proportion computed using the confidential data. In this article, we describe and compare methods for turning this differentially private proportion into an interval estimate for an underlying population probability. Specifically, we consider differentially private versions of the Wald and Wilson intervals, Bayesian credible intervals based on denoising the differentially private proportion, and an exact interval motivated by the Clopper-Pearson confidence interval. We examine the repeated sampling performances of the intervals using simulation studies under both the Laplace mechanism and the discrete Gaussian mechanism across a range of privacy guarantees. We find that while several methods can offer reasonable performances, the Bayesian credible intervals are the most attractive.

Reliable outlier detection in high-dimensional data is crucial in modern science, yet it remains a challenging task. Traditional methods often break down in these settings due to their reliance on asymptotic behaviors with respect to sample size under fixed dimension. Furthermore, many modern alternatives introduce sophisticated statistical treatments and computational complexities. To overcome these issues, our approach leverages intuitive geometric properties of high-dimensional space, effectively turning the curse of dimensionality into an advantage. We propose two new outlyingness statistics based on observation's relational patterns with all other points, measured via pairwise distances or inner products. We establish a theoretical foundation for our statistics demonstrating that as the dimension grows, our statistics create a non-vanishing margin that asymptotically separates outliers from non-outliers. Based on this foundation, we develop practical outlier detection procedures, including a simple clustering-based algorithm and a distribution-free test using random rotations. Through simulation experiments and real data applications, we demonstrate that our proposed methods achieve a superior balance between detection power and false positive control, outperforming existing methods and establishing their practical utility in high-dimensional settings.

Early intervention in neurodegenerative diseases requires identifying periods before diagnosis when decline is rapid enough to detect whether a therapy is slowing progression. Since rapid decline typically occurs close to diagnosis, identifying these periods requires knowing each patient's time of diagnosis. Yet many patients exit studies before diagnosis, making time of diagnosis right-censored by time of study exit – creating a right-censored covariate problem when estimating decline. Existing estimators either assume noninformative covariate censoring, where time of study exit is independent of time of diagnosis, or allow informative covariate censoring, but require correctly specifying how these times are related. We developed SPIRE (Semi-Parametric Informative Right-censored covariate Estimator), a super doubly robust estimator that remains consistent without correctly specifying densities governing time of diagnosis or time of study exit. Typical double robustness requires at least one density to be correct; SPIRE requires neither. When both densities are correctly specified, SPIRE achieves semiparametric efficiency. We also developed a test for detecting informative covariate censoring. Simulations with 85% right-censoring demonstrated SPIRE's robustness, efficiency and reliable detection of informative covariate censoring. Applied to Huntington disease data, SPIRE handled informative covariate censoring appropriately and remained consistent regardless of density specification, providing a reliable tool for early intervention.

Wavelet Transforms are a widely used technique for decomposing a signal into coefficient vectors that correspond to distinct frequency/scale bands while retaining time localization. This property enables an adaptive analysis of signals at different scales, capturing both temporal and spectral patterns. By examining how correlations between two signals vary across these scales, we obtain a more nuanced understanding of their relationship than what is possible from a single global correlation measure. In this work, we expand on the theory of wavelet-based correlations already used in the literature and elaborate on wavelet correlograms, partial wavelet correlations, and additive wavelet correlations using the Pearson and Kendall definitions. We use both Orthogonal and Non-decimated discrete Wavelet Transforms, and assess the robustness of these correlations under different wavelet bases. Simulation studies are conducted to illustrate these methods, and we conclude with applications to real-world datasets.

This paper outlines a grammar of data analysis, as distinct from grammars of data manipulation, in which the primitives are metrics and dimensions. We describe a Python implementation of this grammar called Meterstick, which is agnostic to the underlying data source, which may be a DataFrame or a SQL database.

Estimating environmental exposures from multi-source data is central to public health research and policy. Integrating data from satellite products and ground monitors are increasingly used to produce exposure surfaces. However, spatio-temporal misalignment often induced from missing data introduces substantial uncertainty and reduces predictive accuracy. We propose a Bayesian weighted predictor regression framework that models spatio-temporal relationships when predictors are observed on irregular supports or have substantial missing data, and are not concurrent with the outcome. The key feature of our model is a spatio-temporal kernel that aggregates the predictor over local space-time neighborhoods, built directly into the likelihood, eliminating any separate gap-filling or forced data alignment stage. We introduce a numerical approximation using a Voronoi-based spatial quadrature combined with irregular temporal increments for estimation under data missingness and misalignment. We showed that misspecification of the spatial and temporal lags induced bias in the mean and parameter estimates, indicating the need for principled parameter selection. Simulation studies confirmed these findings, where careful tuning was critical to control bias and achieve accurate prediction, while the proposed quadrature performed well under severe missingness. As an illustrative application, we estimated fine particulate matter (PM$_{2.5}$) in northern California using satellite-derived aerosol optical depth (AOD) and wildfire smoke plume indicators. Relative to a traditional collocated linear model, our approach improved out-of-sample predictive performance, reduced uncertainty, and yielded robust temporal predictions and spatial surface estimation. Our framework is extensible to additional spatio-temporally varying covariates and other kernel families.

Accurate quantification of pavement crack width plays a pivotal role in assessing structural integrity and guiding maintenance interventions. However, achieving precise crack width measurements presents significant challenges due to: (1) the complex, non-uniform morphology of crack boundaries, which limits the efficacy of conventional approaches, and (2) the demand for rapid measurement capabilities from arbitrary pixel locations to facilitate comprehensive pavement condition evaluation. To overcome these limitations, this study introduces a cascaded framework integrating Principal Component Analysis (PCA) and Robust PCA (RPCA) for efficient crack width extraction from digital images. The proposed methodology comprises three sequential stages: (1) initial crack segmentation using established detection algorithms to generate a binary representation, (2) determination of the primary orientation axis for quasi-parallel cracks through PCA, and (3) extraction of the Main Propagation Axis (MPA) for irregular crack geometries using RPCA. Comprehensive evaluations were conducted across three publicly available datasets, demonstrating that the proposed approach achieves superior performance in both computational efficiency and measurement accuracy compared to existing state-of-the-art techniques.

Time-series forecasting increasingly demands not only accurate observational predictions but also causal forecasting under interventional and counterfactual queries in multivariate systems. We present DoFlow, a flow-based generative model defined over a causal Directed Acyclic Graph (DAG) that delivers coherent observational and interventional predictions, as well as counterfactuals through the natural encoding-decoding mechanism of continuous normalizing flows (CNFs). We also provide a supporting counterfactual recovery theory under certain assumptions. Beyond forecasting, DoFlow provides explicit likelihoods of future trajectories, enabling principled anomaly detection. Experiments on synthetic datasets with various causal DAG structures and real-world hydropower and cancer-treatment time series show that DoFlow achieves accurate system-wide observational forecasting, enables causal forecasting over interventional and counterfactual queries, and effectively detects anomalies. This work contributes to the broader goal of unifying causal reasoning and generative modeling for complex dynamical systems.

Variance-dependent regret bounds have received increasing attention in recent studies on contextual bandits. However, most of these studies are focused on upper confidence bound (UCB)-based bandit algorithms, while sampling based bandit algorithms such as Thompson sampling are still understudied. The only exception is the LinVDTS algorithm (Xu et al., 2023), which is limited to linear reward function and its regret bound is not optimal with respect to the model dimension. In this paper, we present FGTSVA, a variance-aware Thompson Sampling algorithm for contextual bandits with general reward function with optimal regret bound. At the core of our analysis is an extension of the decoupling coefficient, a technique commonly used in the analysis of Feel-good Thompson sampling (FGTS) that reflects the complexity of the model space. With the new decoupling coefficient denoted by $\mathrm{dc}$, FGTS-VA achieves the regret of $\tilde{O}(\sqrt{\mathrm{dc}\cdot\log|\mathcal{F}|\sum_{t=1}^T\sigma_t^2}+\mathrm{dc})$, where $|\mathcal{F}|$ is the size of the model space, $T$ is the total number of rounds, and $\sigma_t^2$ is the subgaussian norm of the noise (e.g., variance when the noise is Gaussian) at round $t$. In the setting of contextual linear bandits, the regret bound of FGTSVA matches that of UCB-based algorithms using weighted linear regression (Zhou and Gu, 2022).

Objectives: Unsupervised learning with electronic health record (EHR) data has shown promise for phenotype discovery, but approaches typically disregard existing clinical information, limiting interpretability. We operationalize a Bayesian latent class framework for phenotyping that incorporates domain-specific knowledge to improve clinical meaningfulness of EHR-derived phenotypes and illustrate its utility by identifying an asthma sub-phenotype informed by features of Type 2 (T2) inflammation. Materials and methods: We illustrate a framework for incorporating clinical knowledge into a Bayesian latent class model via informative priors to guide unsupervised clustering toward clinically relevant subgroups. This approach models missingness, accounting for potential missing-not-at-random patterns, and provides patient-level probabilities for phenotype assignment with uncertainty. Using reusable and flexible code, we applied the model to a large asthma EHR cohort, specifying informative priors for T2 inflammation-related features and weakly informative priors for other clinical variables, allowing the data to inform posterior distributions. Results and Conclusion: Using encounter data from January 2017 to February 2024 for 44,642 adult asthma patients, we found a bimodal posterior distribution of phenotype assignment, indicating clear class separation. The T2 inflammation-informed class (38.7%) was characterized by elevated eosinophil levels and allergy markers, plus high healthcare utilization and medication use, despite weakly informative priors on the latter variables. These patterns suggest an "uncontrolled T2-high" sub-phenotype. This demonstrates how our Bayesian latent class modeling approach supports hypothesis generation and cohort identification in EHR-based studies of heterogeneous diseases without well-established phenotype definitions.

We develop a Gaussian process framework for learning interaction kernels in multi-species interacting particle systems from trajectory data. Such systems provide a canonical setting for multiscale modeling, where simple microscopic interaction rules generate complex macroscopic behaviors. While our earlier work established a Gaussian process approach and convergence theory for single-species systems, and later extended to second-order models with alignment and energy-type interactions, the multi-species setting introduces new challenges: heterogeneous populations interact both within and across species, the number of unknown kernels grows, and asymmetric interactions such as predator-prey dynamics must be accommodated. We formulate the learning problem in a nonparametric Bayesian setting and establish rigorous statistical guarantees. Our analysis shows recoverability of the interaction kernels, provides quantitative error bounds, and proves statistical optimality of posterior estimators, thereby unifying and generalizing previous single-species theory. Numerical experiments confirm the theoretical predictions and demonstrate the effectiveness of the proposed approach, highlighting its advantages over existing kernel-based methods. This work contributes a complete statistical framework for data-driven inference of interaction laws in multi-species systems, advancing the broader multiscale modeling program of connecting microscopic particle dynamics with emergent macroscopic behavior.

Within the biological, physical, and social sciences, there are two broad quantitative traditions: statistical and mathematical modeling. Both traditions have the common pursuit of advancing our scientific knowledge, but these traditions have developed largely with distinct languages and inferential frameworks. This paper uses the notion of identification from causal inference, a field originating from the statistical modeling tradition, to develop a shared language. I first review foundational identification results for statistical models and then extend these ideas to mathematical models. Central to this framework is the use of bounds, ranges of plausible numerical values, to analyze both statistical and mathematical models. I discuss the implications of this perspective for the interpretation, comparison, and integration of different modeling approaches, and illustrate the framework with a simple pharmacodynamic model for hypertension. To conclude, I describe areas where the approach taken here should be extended in the future. By formalizing connections between statistical and mathematical modeling, this work contributes to a shared framework for quantitative science. My hope is that this work will advance interactions between these two traditions.

In lifelong learning, a learner faces a sequence of tasks with shared structure and aims to identify and leverage it to accelerate learning. We study the setting where such structure is captured by a common representation of data. Unlike multi-task learning or learning-to-learn, where tasks are available upfront to learn the representation, lifelong learning requires the learner to make use of its existing knowledge while continually gathering partial information in an online fashion. In this paper, we consider a generalized framework of lifelong representation learning. We propose a simple algorithm that uses multi-task empirical risk minimization as a subroutine and establish a sample complexity bound based on a new notion we introduce–the task-eluder dimension. Our result applies to a wide range of learning problems involving general function classes. As concrete examples, we instantiate our result on classification and regression tasks under noise.