Carlsson, Singh and Memoli's TDA mapper takes a point cloud dataset and outputs a graph that depends on several parameter choices. Dey, Memoli, and Wang developed Multiscale Mapper for abstract topological spaces so that parameter choices can be analyzed via persistent homology. However, when applied to actual data, one does not always obtain filtrations of mapper graphs. DBSCAN, one of the most common clustering algorithms used in the TDA mapper software, has two parameters, \textbf{$\epsilon$} and \textbf{MinPts}. If \textbf{MinPts = 1} then DBSCAN is equivalent to single linkage clustering with cutting height \textbf{$\epsilon$}. We show that if DBSCAN clustering is used with \textbf{MinPts $>$ 2}, a filtration of mapper graphs may not exist except in the absence of free-border points; but such filtrations exist if DBSCAN clustering is used with \textbf{MinPts = 1} or \textbf{2} as the cover size increases, \textbf{$\epsilon$} increases, and/or \textbf{MinPts} decreases. However, the 1-dimensional filtration is unstable. If one adds noise to a data set so that each data point has been perturbed by a distance at most \textbf{$\delta$}, the persistent homology of the mapper graph of the perturbed data set can be significantly different from that of the original data set. We show that we can obtain stability by increasing both the cover size and \textbf{$\epsilon$} at the same time. In particular, we show that the bi-filtrations of the homology groups with respect to cover size and $\epsilon$ between these two datasets are \textbf{2$\delta$}-interleaved.

This paper advances the understanding of how the size of a machine learning model affects its vulnerability to poisoning, despite state-of-the-art defenses. Given isotropic random honest feature vectors and the geometric median (or clipped mean) as the robust gradient aggregator rule, we essentially prove that, perhaps surprisingly, linear and logistic regressions with $D \geq 169 H^2/P^2$ parameters are subject to arbitrary model manipulation by poisoners, where $H$ and $P$ are the numbers of honestly labeled and poisoned data points used for training. Our experiments go on exposing a fundamental tradeoff between augmenting model expressivity and increasing the poisoners' attack surface, on both synthetic data, and on MNIST & FashionMNIST data for linear classifiers with random features. We also discuss potential implications for source-based learning and neural nets.

This study introduces a novel approach that integrates agricultural census data with remotely sensed time series to develop precise predictive models for paddy rice yield across various regions of Peru. By utilizing sparse regression and Elastic-Net regularization techniques, the study identifies causal relationships between key remotely sensed variables-such as NDVI, precipitation, and temperature-and agricultural yield. To further enhance prediction accuracy, the first- and second-order dynamic transformations (velocity and acceleration) of these variables are applied, capturing non-linear patterns and delayed effects on yield. The findings highlight the improved predictive performance when combining regularization techniques with climatic and geospatial variables, enabling more precise forecasts of yield variability. The results confirm the existence of causal relationships in the Granger sense, emphasizing the value of this methodology for strategic agricultural management. This contributes to more efficient and sustainable production in paddy rice cultivation.

Whether deterministic or stochastic, models can be viewed as functions designed to approximate a specific quantity of interest. We propose a data-driven framework that aggregates predictions from diverse models into a single, more accurate output. This aggregation approach exploits each model's strengths to enhance overall accuracy. It is non-intrusive - treating models as black-box functions - model-agnostic, requires minimal assumptions, and can combine outputs from a wide range of models, including those from machine learning and numerical solvers. We argue that the aggregation process should be point-wise linear and propose two methods to find an optimal aggregate: Minimal Error Aggregation (MEA), which minimizes the aggregate's prediction error, and Minimal Variance Aggregation (MVA), which minimizes its variance. While MEA is inherently more accurate when correlations between models and the target quantity are perfectly known, Minimal Empirical Variance Aggregation (MEVA), an empirical version of MVA - consistently outperforms Minimal Empirical Error Aggregation (MEEA), the empirical counterpart of MEA, when these correlations must be estimated from data. The key difference is that MEVA constructs an aggregate by estimating model errors, while MEEA treats the models as features for direct interpolation of the quantity of interest. This makes MEEA more susceptible to overfitting and poor generalization, where the aggregate may underperform individual models during testing. We demonstrate the versatility and effectiveness of our framework in various applications, such as data science and partial differential equations, showing how it successfully integrates traditional solvers with machine learning models to improve both robustness and accuracy.

In this paper, we provide a non-asymptotic analysis of the convergence of the stochastic gradient Hamiltonian Monte Carlo (SGHMC) algorithm to a target measure in Wasserstein-1 and Wasserstein-2 distance. Crucially, compared to the existing literature on SGHMC, we allow its stochastic gradient to be discontinuous. This allows us to provide explicit upper bounds, which can be controlled to be arbitrarily small, for the expected excess risk of non-convex stochastic optimization problems with discontinuous stochastic gradients, including, among others, the training of neural networks with ReLU activation function. To illustrate the applicability of our main results, we consider numerical experiments on quantile estimation and on several optimization problems involving ReLU neural networks relevant in finance and artificial intelligence.

This paper presents a comprehensive analysis of the well-known extragradient (EG) method for solving both equations and inclusions. First, we unify and generalize EG for [non]linear equations to a wider class of algorithms, encompassing various existing schemes and potentially new variants. Next, we analyze both sublinear ``best-iterate'' and ``last-iterate'' convergence rates for the entire class of algorithms, and derive new convergence results for two well-known instances. Second, we extend our EG framework above to ``monotone'' inclusions, introducing a new class of algorithms and its corresponding convergence results. Third, we also unify and generalize Tseng's forward-backward-forward splitting (FBFS) method to a broader class of algorithms to solve [non]linear inclusions when a weak-Minty solution exists, and establish its ``best-iterate'' convergence rate. Fourth, to complete our picture, we also investigate sublinear rates of two other common variants of EG using our EG analysis framework developed here: the reflected forward-backward splitting and the golden ratio methods. Finally, we conduct an extensive numerical experiment to validate our theoretical findings. Our results demonstrate that several new variants of our proposed algorithms outperform existing schemes in the majority of examples.

In this paper, we address conditional testing problems through the conformal inference framework. We define the localized conformal p-values by inverting prediction intervals and prove their theoretical properties. These defined p-values are then applied to several conditional testing problems to illustrate their practicality. Firstly, we propose a conditional outlier detection procedure to test for outliers in the conditional distribution with finite-sample false discovery rate (FDR) control. We also introduce a novel conditional label screening problem with the goal of screening multivariate response variables and propose a screening procedure to control the family-wise error rate (FWER). Finally, we consider the two-sample conditional distribution test and define a weighted U-statistic through the aggregation of localized p-values. Numerical simulations and real-data examples validate the superior performance of our proposed strategies.

Structural transformation, the shift from agrarian economies to more diversified industrial and service-based systems, is a key driver of economic development. However, in low- and middle-income countries (LMICs), data scarcity and unreliability hinder accurate assessments of this process. This paper presents a novel statistical framework designed to address these challenges by integrating Bayesian hierarchical modeling, machine learning-based data imputation, and factor analysis. The framework is specifically tailored for conditions of data sparsity and is capable of providing robust insights into sectoral productivity and employment shifts across diverse economies. By utilizing Bayesian models, uncertainties in data are effectively managed, while machine learning techniques impute missing data points, ensuring the integrity of the analysis. Factor analysis reduces the dimensionality of complex datasets, distilling them into core economic structures. The proposed framework has been validated through extensive simulations, demonstrating its ability to predict structural changes even when up to 60\% of data is missing. This approach offers policymakers and researchers a valuable tool for making informed decisions in environments where data quality is limited, contributing to the broader understanding of economic development in LMICs.

Multi-task learning is a popular machine learning approach that enables simultaneous learning of multiple related tasks, improving algorithmic efficiency and effectiveness. In the hard parameter sharing approach, an encoder shared through multiple tasks generates data representations passed to task-specific predictors. Therefore, it is crucial to have a shared encoder that provides decent representations for every and each task. However, despite recent advances in multi-task learning, the question of how to improve the quality of representations generated by the shared encoder remains open. To address this gap, we propose a novel approach called Dummy Gradient norm Regularization that aims to improve the universality of the representations generated by the shared encoder. Specifically, the method decreases the norm of the gradient of the loss function with repect to dummy task-specific predictors to improve the universality of the shared encoder's representations. Through experiments on multiple multi-task learning benchmark datasets, we demonstrate that DGR effectively improves the quality of the shared representations, leading to better multi-task prediction performances. Applied to various classifiers, the shared representations generated by DGR also show superior performance compared to existing multi-task learning methods. Moreover, our approach takes advantage of computational efficiency due to its simplicity. The simplicity also allows us to seamlessly integrate DGR with the existing multi-task learning algorithms.

In the context of global energy strategy, accurate natural gas demand forecasting is crucial for ensuring efficient resource allocation and operational planning. Traditional forecasting methods struggle to cope with the growing complexity and variability of gas consumption patterns across diverse industries and commercial sectors. To address these challenges, we propose the first foundation model specifically tailored for natural gas demand forecasting. Foundation models, known for their ability to generalize across tasks and datasets, offer a robust solution to the limitations of traditional methods, such as the need for separate models for different customer segments and their limited generalization capabilities. Our approach leverages contrastive learning to improve prediction accuracy in real-world scenarios, particularly by tackling issues such as noise in historical consumption data and the potential misclassification of similar data samples, which can lead to degradation in the quaility of the representation and thus the accuracy of downstream forecasting tasks. By integrating advanced noise filtering techniques within the contrastive learning framework, our model enhances the quality of learned representations, leading to more accurate predictions. Furthermore, the model undergoes industry-specific fine-tuning during pretraining, enabling it to better capture the unique characteristics of gas consumption across various sectors. We conducted extensive experiments using a large-scale dataset from ENN Group, which includes data from over 10,000 industrial, commercial, and welfare-related customers across multiple regions. Our model outperformed existing state-of-the-art methods, demonstrating a relative improvement in MSE by 3.68\% and in MASE by 6.15\% compared to the best available model.