Papers

We present a concise survey of matrix completion methods and associated implementations of several fundamental algorithms. Our study covers both passive and adaptive strategies. We further illustrate the behavior of a simple adaptive sampling scheme through controlled synthetic experiments.

Recent work \cite{arifgroup} introduced Federated Proximal Gradient \textbf{(FedProxGrad)} for solving non-convex composite optimization problems in group fair federated learning. However, the original analysis established convergence only to a noise-dominated neighborhood of stationarity, with explicit dependence on a variance-induced noise floor. In this work, we provide an improved asymptotic convergence analysis for a generalized FedProxGrad-type analytical framework with inexact local proximal solutions and explicit fairness regularization. We call this extended analytical framework \textbf{DS FedProxGrad} (Decay Step Size FedProxGrad). Under a Robbins-Monro step-size schedule \cite{robbins1951stochastic} and a mild decay condition on local inexactness, we prove that $\liminf_{r\to\infty} \mathbb{E}[\|\nabla F(\mathbf{x}^r)\|^2] = 0$, i.e., the algorithm is asymptotically stationary and the convergence rate does not depend on a variance-induced noise floor.

In oncological clinical trials, overall survival (OS) is the gold-standard endpoint, but long follow-up and treatment switching can delay or dilute detectable effects. Progression-free survival (PFS) often provides earlier evidence and is therefore frequently used together with OS as multiple primary endpoints. Since in certain scenarios trial success may be defined if one of the two hypotheses involved can be rejected, a correction for multiple testing may be deemed necessary. Because PFS and OS are generally highly dependent, their test statistics are typically correlated. Ignoring this dependency (e.g. via a simple Bonferroni correction) is not power optimal. We develop a group-sequential testing procedure for the multiple primary endpoints PFS and OS that fully exhausts the family-wise error rate (FWER) by exploiting their dependence. Specifically, we characterize the joint asymptotic distribution of log-rank statistics across endpoints and multiple event-driven analysis cutoffs. Furthermore, we show that we can consistently estimate the covariance structure. Embedding these results in a closed testing procedure, we can recalculate critical values of the test statistics in order to spend the available type I error optimally. An important extension to the current literature is that we allow for both interim and final analysis to be event-driven. Simulations based on illness-death multi-state models empirically confirm FWER control for moderate to large sample sizes. Compared with a simple Bonferroni correction, the proposed methods recover roughly two thirds of the power loss for OS, increase disjunctive and conjunctive power, and enable meaningful early stopping. In planning, these gains translate into about 5% fewer OS events required to reach the targeted power. We also discuss practical issues in the implementation of such designs and possible extensions of the introduced method.

In this article, we introduce a Topological Data Analysis (TDA) pipeline for neural spike train data. Understanding how the brain transforms sensory information into perception and behavior requires analyzing coordinated neural population activity. Modern electrophysiology enables simultaneous recording of spike train ensembles, but extracting meaningful information from these datasets remains a central challenge in neuroscience. A fundamental question is how ensembles of neurons discriminate between different stimuli or behavioral states, particularly when individual neurons exhibit weak or no stimulus selectivity, yet their coordinated activity may still contribute to network-level encoding. We describe a TDA framework that identifies stimulus-discriminative structure in spike train ensembles recorded from the mouse insular cortex during presentation of deionized water stimuli at distinct non-nociceptive temperatures. We show that population-level topological signatures effectively differentiate oral thermal stimuli even when individual neurons provide little or no discrimination. These findings demonstrate that ensemble organization can carry perceptually relevant information that standard single-unit analysis may miss. The framework builds on a mathematical representation of spike train ensembles that enables persistent homology to be applied to collections of point processes. At its core is the widely-used Victor-Purpura (VP) distance. Using this metric, we construct persistence-based descriptors that capture multiscale topological features of ensemble geometry. Two key theoretical results support the method: a stability theorem establishing robustness of persistent homology to perturbations in the VP metric parameter, and a probabilistic stability theorem ensuring robustness of topological signatures.

Since the 1990s, considerable empirical work has been carried out to train statistical models, such as neural networks (NNs), as learned heuristics for combinatorial optimization (CO) problems. When successful, such an approach eliminates the need for experts to design heuristics per problem type. Due to their structure, many hard CO problems are amenable to treatment through reinforcement learning (RL). Indeed, we find a wealth of literature training NNs using value-based, policy gradient, or actor-critic approaches, with promising results, both in terms of empirical optimality gaps and inference runtimes. Nevertheless, there has been a paucity of theoretical work undergirding the use of RL for CO problems. To this end, we introduce a unified framework to model CO problems through Markov decision processes (MDPs) and solve them using RL techniques. We provide easy-to-test assumptions under which CO problems can be formulated as equivalent undiscounted MDPs that provide optimal solutions to the original CO problems. Moreover, we establish conditions under which value-based RL techniques converge to approximate solutions of the CO problem with a guarantee on the associated optimality gap. Our convergence analysis provides: (1) a sufficient rate of increase in batch size and projected gradient descent steps at each RL iteration; (2) the resulting optimality gap in terms of problem parameters and targeted RL accuracy; and (3) the importance of a choice of state-space embedding. Together, our analysis illuminates the success (and limitations) of the celebrated deep Q-learning algorithm in this problem context.

This study explores the application of deep learning to improve and automate pollen grain detection and classification in both optical and holographic microscopy images, with a particular focus on veterinary cytology use cases. We used YOLOv8s for object detection and MobileNetV3L for the classification task, evaluating their performance across imaging modalities. The models achieved 91.3% mAP50 for detection and 97% overall accuracy for classification on optical images, whereas the initial performance on greyscale holographic images was substantially lower. We addressed the performance gap issue through dataset expansion using automated labeling and bounding box area enlargement. These techniques, applied to holographic images, improved detection performance from 2.49% to 13.3% mAP50 and classification performance from 42% to 54%. Our work demonstrates that, at least for image classification tasks, it is possible to pair deep learning techniques with cost-effective lensless digital holographic microscopy devices.

To study gap acceptance behaviour one needs the distribution (or probability density function) of gaps in the opposing stream. Further, in these times of widespread availability of large computing powers, traffic simulation has emerged as a popular analysis and design tool. Such simulations rely on randomly generating the arriving vehicles in a way that statistically resembles real-world streams. The generation process for disorderly streams requires information on gap distributions. A study of past literature reveals that very little work has been done to determine the distribution of gaps on multi-lane orderly and disorderly streams. This study aims to develop an analytical framework to specify the distribution of gaps for such streams. This analytical framework is built using the Renewal Process Theory. A maximum likelihood based process for the estimation of the parameters of the analytically derived distribution is also described. Later, real-world gap data from three different sites covering orderly and disorderly streams are used to show how the derived distribution function (using the proposed method) ably describes the observed gap distributions.

We consider an adaptive experiment for treatment choice and design a minimax and Bayes optimal adaptive experiment with respect to regret. Given binary treatments, the experimenter's goal is to choose the treatment with the highest expected outcome through an adaptive experiment, in order to maximize welfare. We consider adaptive experiments that consist of two phases, the treatment allocation phase and the treatment choice phase. The experiment starts with the treatment allocation phase, where the experimenter allocates treatments to experimental subjects to gather observations. During this phase, the experimenter can adaptively update the allocation probabilities using the observations obtained in the experiment. After the allocation phase, the experimenter proceeds to the treatment choice phase, where one of the treatments is selected as the best. For this adaptive experimental procedure, we propose an adaptive experiment that splits the treatment allocation phase into two stages, where we first estimate the standard deviations and then allocate each treatment proportionally to its standard deviation. We show that this experiment, often referred to as Neyman allocation, is minimax and Bayes optimal in the sense that its regret upper bounds exactly match the lower bounds that we derive. To show this optimality, we derive minimax and Bayes lower bounds for the regret using change-of-measure arguments. Then, we evaluate the corresponding upper bounds using the central limit theorem and large deviation bounds.

We study the following distribution clustering problem: Given a hidden partition of $k$ distributions into two groups, such that the distributions within each group are the same, and the two distributions associated with the two clusters are $\varepsilon$-far in total variation, the goal is to recover the partition. We establish upper and lower bounds on the sample complexity for two fundamental cases: (1) when one of the cluster's distributions is known, and (2) when both are unknown. Our upper and lower bounds characterize the sample complexity's dependence on the domain size $n$, number of distributions $k$, size $r$ of one of the clusters, and distance $\varepsilon$. In particular, we achieve tightness with respect to $(n,k,r,\varepsilon)$ (up to an $O(\log k)$ factor) for all regimes.

Datasets may contain observations with multiple labels. If the labels are not mutually exclusive, and if the labels vary greatly in frequency, obtaining a sample that includes sufficient observations with scarcer labels to make inferences about those labels, and which deviates from the population frequencies in a known manner, creates challenges. In this paper, we consider a multivariate Bernoulli distribution as our underlying distribution of a multi-label problem. We present a novel sampling algorithm that takes label dependencies into account. It uses observed label frequencies to estimate multivariate Bernoulli distribution parameters and calculates weights for each label combination. This approach ensures the weighted sampling acquires target distribution characteristics while accounting for label dependencies. We applied this approach to a variety of datasets, including a sample of research articles from Web of Science labeled with 64 biomedical topic categories. We aimed to preserve category frequency order, reduce frequency differences between most and least common categories, and account for category dependencies. This approach produced a more balanced sub-sample, enhancing the representation of minority categories.

Input features are conventionally represented as vectors, matrices, or third order tensors in the real field, for color image classification. Inspired by the success of quaternion data modeling for color images in image recovery and denoising tasks, we propose a novel classification method for color image classification, named as the Low-rank Support Quaternion Matrix Machine (LSQMM), in which the RGB channels are treated as pure quaternions to effectively preserve the intrinsic coupling relationships among channels via the quaternion algebra. For the purpose of promoting low-rank structures resulting from strongly correlated color channels, a quaternion nuclear norm regularization term, serving as a natural extension of the conventional matrix nuclear norm to the quaternion domain, is added to the hinge loss in our LSQMM model. An Alternating Direction Method of Multipliers (ADMM)-based iterative algorithm is designed to effectively resolve the proposed quaternion optimization model. Experimental results on multiple color image classification datasets demonstrate that our proposed classification approach exhibits advantages in classification accuracy, robustness and computational efficiency, compared to several state-of-the-art methods using support vector machines, support matrix machines, and support tensor machines.

The extreme value index (EVI) characterizes the tail behavior of a distribution and is crucial for extreme value theory. Inference on the EVI is challenging due to data scarcity in the tail region. We propose a novel method for constructing confidence intervals for the EVI using synthetic exceedances generated via perturbation. Rather than perturbing the entire sample, we add noise to exceedances above a high threshold and apply the generalized Pareto distribution (GPD) approximation. Confidence intervals are derived by simulating the distribution of pivotal statistics from the perturbed data. We show that the pivotal statistic is consistent, ensuring the proposed method provides consistent intervals for the EVI. Additionally, we demonstrate that the perturbed data is differentially private. When the GPD approximation is inadequate, we introduce a refined perturbation method. Simulation results show that our approach outperforms existing methods, providing robust and reliable inference.

We study causal effect estimation under interference from network data. We work under the chain-graph formulation pioneered in Tchetgen Tchetgen et. al (2021). Our first result shows that polynomial time evaluation of treatment effects is computationally hard in this framework without additional assumptions on the underlying chain graph. Subsequently, we assume that the interactions among the study units are governed either by (i) a dense graph or (ii) an i.i.d. Gaussian matrix. In each case, we show that the treatment effects have well-defined limits as the population size diverges to infinity. Additionally, we develop polynomial time algorithms to consistently evaluate the treatment effects in each case. Finally, we estimate the unknown parameters from the observed data using maximum pseudo-likelihood estimates, and establish the stability of our causal effect estimators under this perturbation. Our algorithms provably approximate the causal effects in polynomial time even in low-temperature regimes where the canonical MCMC samplers are slow mixing. For dense graphs, our results use the notion of regularity partitions; for Gaussian interactions, our approach uses ideas from spin glass theory and Approximate Message Passing.

A Wishart kernel density estimator (KDE) is introduced for density estimation in the cone of positive definite matrices. The estimator is boundary-aware and mitigates the boundary bias suffered by conventional KDEs, while remaining simple to implement. Its mean squared error, uniform strong consistency on expanding compact sets, and asymptotic normality are established under the Lebesgue measure and suitable mixing conditions. This work represents the first study of density estimation for dependent data on this space under any metric. For independent observations, an asymptotic upper bound on the mean absolute error is also derived. A simulation study compares the performance of the Wishart KDE with that of the log-Gaussian KDE, another boundary-aware estimator based on the matrix-variate lognormal distribution proposed by Schwartzman [Int. Stat. Rev., 2016, 84(3), 456–486], and with the naive Gaussian KDE on the ambient Euclidean space. When estimating the stationary marginal density of a Wishart autoregressive process for several autoregressive coefficient matrices and innovation covariance matrices, the Wishart KDE exhibits the best overall accuracy and stability. The practical utility of the Wishart KDE is illustrated by estimating the marginal density of a one-year time series of realized covariance matrices computed from 5-minute intra-day returns on Amazon Corp. shares and on the Standard & Poor's 500 exchange-traded fund. All code is publicly available via the R package ksm to facilitate implementation of the method and reproducibility of the findings.

This paper examines the relationship between the genderedness of authors' first names and citation distributions in US-affiliated scholarly production from 2010 to 2019. Merging a first name genderedness table derived from Wikidata with Web of Science publication data, we develop a relative distributional framework that compares name, article, and citation distributions along a continuous genderedness spectrum. The lexical structure of the corpus proves stable across author roles, while productivity diverges substantially by disciplinary group: physical sciences show a consistent masculine lean, social sciences a feminine one, with amplitude remaining stable across groups despite these directional differences. Citation analyses net of publication volume reveal a pervasive deficit across disciplinary groups and author roles: only unambiguously masculine names accumulate citations in excess of their publication volume, while feminine, neutral, and moderately masculine names all fall short of it. This asymmetry is strongest in the life sciences and weakest in the physical sciences except among middle authors, where the pattern reflects the gender composition of large collaborative teams rather than evaluative bias in citing behavior. Taken together, these results are consistent with the hypothesis that first name genderedness shapes citation recognition through implicit bias operating in low-deliberation evaluative contexts.

Theorem 5.1 in the monograph by Hall (1992) provides rigorous in-probability justification of Edgeworth expansions of bootstrap distributions. Proving this result was rather challenging because bootstrap distributions do not satisfy the classical Cramér condition and therefore classical methods for justifying Edgeworth expansions, e.g. Bhattacharya and Rao (1976) and Bhattacharya and Ghosh (1978), are not available. Hall's (1992) theorem is for a univariate statistic which can be expressed as a smooth function of means, though the underlying population can be multivariate. However, there are a number of applications where a multivariate version of Hall's theorem is needed, and generalizing the proof from the univariate case to the multivariate case is not immediate. Our primary purpose in this article is to fill this gap by stating a multivariate version of the theorem and sketching the modifications to the proof of Hall's (1992) Theorem 5.1 that are needed.

To address the computational issue in empirical likelihood methods with massive data, this paper proposes a grouped empirical likelihood (GEL) method. It divides $N$ observations into $n$ groups, and assigns the same probability weight to all observations within the same group. GEL estimates the $n\ (\ll N)$ weights by maximizing the empirical likelihood ratio. The dimensionality of the optimization problem is thus reduced from $N$ to $n$, thereby lowering the computational complexity. We prove that GEL possesses the same first order asymptotic properties as the conventional empirical likelihood method under the estimating equation settings and the classical two-sample mean problem. A distributed GEL method is also proposed with several servers. Numerical simulations and real data analysis demonstrate that GEL can keep the same inferential accuracy as the conventional empirical likelihood method, and achieves substantial computational acceleration compared to the divide-and-conquer empirical likelihood method. We can analyze a billion data with GEL in tens of seconds on only one PC.

We study estimation of the conditional law $P(Y|X = x)$ and continuous measurable maps of it when $Y \in \mathcal{Y}$ takes values in a locally compact Polish space (e.g., $\mathbb{R}^d$), $X \in \mathbb{R}^p$, and the observations arise from a complex survey design: a single- or multi-stage sampling scheme that may involve unequal selection, stratification, and clustering. We propose a survey-calibrated distributional random forest (SDRF) that incorporates complex-design features via the pseudo-population bootstrap, PSU-level honesty, and a Maximum Mean Discrepancy (MMD) split criterion computed from kernel mean embeddings of design-weighted node distributions. We provide a framework for analyzing forest-based estimators under various survey designs; establish consistency for both finite- and super-population conditional laws under explicit conditions on the design, kernel, resampling multipliers, and tree partitions. As far as we are aware, these are the first results on model-free estimation of conditional distributions under survey designs. Simulations under a stratified two-stage cluster design expose the systematic bias incurred by ignoring survey structure. We illustrate the broad applicability of SDRF on NHANES, estimating the conditional joint tolerance regions for two diabetes biomarkers, revealing subgroup-level distributional heterogeneity relevant to diabetes risk profiling in the U.S. population.

Worst-case generation plays a critical role in evaluating robustness and stress-testing systems under distribution shifts, in applications ranging from machine learning models to power grids and medical prediction systems. We develop a generative modeling framework for worst-case generation for a pre-specified risk, based on min-max optimization over continuous probability distributions, namely the Wasserstein space. Unlike traditional discrete distributionally robust optimization approaches, which often suffer from scalability issues, limited generalization, and costly worst-case inference, our framework exploits the Brenier theorem to characterize the least favorable (worst-case) distribution as the pushforward of a transport map from a continuous reference measure, enabling a continuous and expressive notion of risk-induced generation beyond classical discrete DRO formulations. Based on the min-max formulation, we propose a Gradient Descent Ascent (GDA)-type scheme that updates the decision model and the transport map in a single loop, establishing global convergence guarantees under mild regularity assumptions and possibly without convexity-concavity. We also propose to parameterize the transport map using a neural network that can be trained simultaneously with the GDA iterations by matching the transported training samples, thereby achieving a simulation-free approach. The efficiency of the proposed method as a risk-induced worst-case generator is validated by numerical experiments on synthetic and image data.

In recent years, mixture cure models have gained increasing popularity in survival analysis as an alternative to the Cox proportional hazards model, particularly in settings where a subset of patients is considered cured. The proportional hazards mixture cure model is especially advantageous when the presence of a cured fraction can be reasonably assumed, providing a more accurate representation of long-term survival dynamics. In this study, we propose a novel hierarchical Bayesian framework for the semiparametric mixture cure model, which accommodates both the inclusion and exclusion of a frailty component, allowing for greater flexibility in capturing unobserved heterogeneity among patients. Samples from the posterior distribution are obtained using a Markov chain Monte Carlo method, leveraging a hierarchical structure inspired by Bayesian Lasso. Comprehensive simulation studies are conducted across diverse scenarios to evaluate the performance and robustness of the proposed models. Bayesian model comparison and assessment are performed using various criteria. Finally, the proposed approaches are applied to two well-known datasets in the cure model literature: the E1690 melanoma trial and a colon cancer clinical trial.

Analyzing multilayer networks is central to understanding complex relational measurements collected across multiple conditions or over time. A pivotal task in this setting is to quantify uncertainty in community structure while appropriately pooling information across layers and accommodating layer-specific heterogeneity. Building on the multilayer degree-corrected mixed-membership (ML-DCMM) model, which captures both stable community membership profiles and layer-specific vertex activity levels, we propose a Bayesian inference framework based on a spectral-assisted likelihood. We then develop a computationally efficient Gaussian variational inference algorithm implemented via stochastic gradient descent. Our theoretical analysis establishes a variational Bernstein–von Mises theorem, which provides a frequentist guarantee for using the variational posterior to construct confidence sets for mixed memberships. We demonstrate the utility of the method on a U.S. airport longitudinal network, where the procedure yields robust estimates, natural uncertainty quantification, and competitive performance relative to state-of-the-art methods.

U.S. state education agencies mark schools displaying achievement gaps between demographic subgroups as needing improvement. Some schools may have few students in these subgroups, such that average end-of-year test scores only noisily measure the average "true" score-the score one would expect if students took the test many times. This, in addition to the masking of small subgroup averages in publicly available assessment data, poses challenges for evaluating interventions aimed at closing achievement gaps. We introduce propensity score estimates designed to achieve balance on subgroup average true scores. These estimates are available even when noisy measurements are not and improve overlap compared to those that ignore measurement error, leading to greater bias reduction of matching estimators. We demonstrate our methods through simulation and an application to a statewide initiative in Texas for curbing summer learning loss.

An important aspect of the performance of algorithms that predict individualized treatment effects (ITE) is moderate calibration, i.e., the average treatment effect among individuals with predicted treatment effect of z being equal to z. The assessment of moderate calibration is challenging on two fronts: counterfactual responses are unobserved, and quantifying the conditional response function for models that generate continuous predicted values requires regularization. Perhaps because of these challenges, there is currently no inferential method for the null hypothesis that an ITE model is moderately calibrated in a population. In this work, we propose non-parametric methods for the assessment of moderate calibration of ITE models for binary outcomes using data from a randomized trial. These methods simultaneously resolve both challenges, resulting in novel graphical, numerical, and inferential methods for the assessment of moderate calibration. The key idea is to formulate a stochastic process for the cumulative prediction errors that obeys a functional central limit theorem, enabling the use of the properties of Brownian motion for asymptotic inference. We propose two approaches to construct this process from a sample: a conditional approach that relies on predicted risks (often an output of ITE models), and a marginal approach based on replacing the cumulative conditional moments with their marginal counterparts. Numerical simulations confirm the desirable properties of both approaches and their ability to detect miscalibration of different forms. We use a case study to provide suggestions on graphical presentation and the interpretation of results. Moderate calibration of predicted ITEs can be assessed without requiring regularization techniques or making assumptions about the functional form of treatment response. The accompanying cumulcalib R package implements this method.

Nonstationarity in spatial and spatio-temporal processes is ubiquitous in environmental datasets, but is not often addressed in practice, due to a scarcity of statistical software packages that implement nonstationary models. In this article, we introduce the R software package deepspat, which allows for modeling, fitting and prediction with nonstationary spatial and spatio-temporal models applied to Gaussian and extremes data. The nonstationary models in our package are constructed using a deep multi-layered deformation of the original spatial or spatio-temporal domain, and are straightforward to implement. Model parameters are estimated using gradient-based optimization of customized loss functions with tensorflow, which implements automatic differentiation. The functionalities of the package are illustrated through simulation studies and an application to Nepal temperature data.

This paper documents a data set of UK rain radar image sequences for use in statistical modeling and machine learning methods for nowcasting. The main dataset contains 1,000 randomly sampled sequences of length 20 steps (15-minute increments) of 2D radar intensity fields of dimension 40x40 (at 5km spatial resolution). Spatially stratified sampling ensures spatial homogeneity despite removal of clear-sky cases by threshold-based truncation. For each radar sequence, additional atmospheric and geographic features are made available, including date, location, mean elevation, mean wind direction and speed and prevailing storm type. New R functions to extract data from the binary "Nimrod" radar data format are provided. A case study is presented to train and evaluate a simple convolutional neural network for radar nowcasting, including self-contained R code.

Many travel decisions involve a degree of experience formation, where individuals learn their preferences over time. At the same time, there is extensive scope for heterogeneity across individual travellers, both in their underlying preferences and in how these evolve. The present paper puts forward a Latent Class Reinforcement Learning (LCRL) model that allows analysts to capture both of these phenomena. We apply the model to a driving simulator dataset and estimate the parameters through Variational Bayes. We identify three distinct classes of individuals that differ markedly in how they adapt their preferences: the first displays context-dependent preferences with context-specific exploitative tendencies; the second follows a persistent exploitative strategy regardless of context; and the third engages in an exploratory strategy combined with context-specific preferences.

Regression analysis for responses taking values in general metric spaces has received increasing attention, particularly for settings with Euclidean predictors $X \in \mathbb{R}^p$ and non-Euclidean responses $Y$ in metric spaces. While additive regression is a powerful tool for enhancing interpretability and mitigating the curse of dimensionality in the presence of multivariate predictors, its direct extension is hindered by the absence of vector space operations in general metric spaces. We propose a novel framework for additive optimal transport regression, which incorporates additive structure through optimal geodesic transports. A key idea is to extend the notion of optimal transports in Wasserstein spaces to general geodesic metric spaces. This unified approach accommodates a wide range of responses, including probability distributions, symmetric positive definite (SPD) matrices with various metrics and spherical data. The practical utility of the method is illustrated with correlation matrices derived from resting state fMRI brain imaging data.

Economists often rely on estimates of linear fixed effects models produced by other teams of researchers. Assessing the uncertainty in these estimates can be challenging. I propose a form of sample splitting for networks that partitions the data into statistically independent branches, each of which can be used to compute an unbiased estimate of the parameters of interest in two-way fixed effects models. These branches facilitate uncertainty quantification, moment estimation, and shrinkage. Drawing on results from the graph theory literature on tree packing, I develop algorithms to efficiently extract branches from large networks. I illustrate these techniques using a benchmark dataset from Veneto, Italy that has been widely used to study firm wage effects.

We study the expressivity of one-dimensional (1D) ReLU deep neural networks through the lens of their linear regions. For randomly initialized, fully connected 1D ReLU networks (He scaling with nonzero bias) in the infinite-width limit, we prove that the expected number of linear regions grows as $\sum_{i = 1}^L n_i + \mathop{{o}}\left(\sum_{i = 1}^L{n_i}\right) + 1$, where $n_\ell$ denotes the number of neurons in the $\ell$-th hidden layer. We also propose a function-adaptive notion of sparsity that compares the expected regions used by the network to the minimal number needed to approximate a target within a fixed tolerance.

Intracellular compartmentalization of proteins underpins their function and the metabolic processes they sustain. Various mass spectrometry-based proteomics methods (subcellular spatial proteomics) now allow high throughput subcellular protein localization. Yet, the curation, analysis and interpretation of these data remain challenging, particularly in non-model organisms where establishing reliable marker proteins is difficult, and in contexts where experimental replication and subcellular fractionation are constrained. Here, we develop FSPmix, a semi-supervised functional clustering method implemented as an open-source R package, which leverages partial annotations from a subset of marker proteins to predict protein subcellular localization from subcellular spatial proteomics data. This method explicitly assumes that protein signatures vary smoothly across subcellular fractions, enabling more robust inference under low signal-to-noise data regimes. We applied FSPmix to a subcellular proteomics dataset from a marine diatom, allowing us to assign probabilistic localizations to proteins and uncover potentially new protein functions. Altogether, this work lays the foundation for more robust statistical analysis and interpretation of subcellular proteomics datasets, particularly in understudied organisms.