The challenge of \textbf{imbalanced regression} arises when standard Empirical Risk Minimization (ERM) biases models toward high-frequency regions of the data distribution, causing severe degradation on rare but high-impact ``tail'' events. Existing strategies uch as loss re-weighting or synthetic over-sampling often introduce noise, distort the underlying distribution, or add substantial algorithmic complexity. We introduce \textbf{PARIS} (Pruning Algorithm via the Representer theorem for Imbalanced Scenarios), a principled framework that mitigates imbalance by \emph{optimizing the training set itself}. PARIS leverages the representer theorem for neural networks to compute a \textbf{closed-form representer deletion residual}, which quantifies the exact change in validation loss caused by removing a single training point \emph{without retraining}. Combined with an efficient Cholesky rank-one downdating scheme, PARIS performs fast, iterative pruning that eliminates uninformative or performance-degrading samples. We use a real-world space weather example, where PARIS reduces the training set by up to 75\% while preserving or improving overall RMSE, outperforming re-weighting, synthetic oversampling, and boosting baselines. Our results demonstrate that representer-guided dataset pruning is a powerful, interpretable, and computationally efficient approach to rare-event regression.

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.

In this paper, we examine the convergence landscape of multi-agent learning under uncertainty. Specifically, we analyze two stochastic models of regularized learning in continuous games -- one in continuous and one in discrete time with the aim of characterizing the long-run behavior of the induced sequence of play. In stark contrast to deterministic, full-information models of learning (or models with a vanishing learning rate), we show that the resulting dynamics do not converge in general. In lieu of this, we ask instead which actions are played more often in the long run, and by how much. We show that, in strongly monotone games, the dynamics of regularized learning may wander away from equilibrium infinitely often, but they always return to its vicinity in finite time (which we estimate), and their long-run distribution is sharply concentrated around a neighborhood thereof. We quantify the degree of this concentration, and we show that these favorable properties may all break down if the underlying game is not strongly monotone -- underscoring in this way the limits of regularized learning in the presence of persistent randomness and uncertainty.

Kernel density estimation is a key component of a wide variety of algorithms in machine learning, Bayesian inference, stochastic dynamics and signal processing. However, the unsupervised density estimation technique requires tuning a crucial hyperparameter: the kernel bandwidth. The choice of bandwidth is critical as it controls the bias-variance trade-off by over- or under-smoothing the topological features. Topological data analysis provides methods to mathematically quantify topological characteristics, such as connected components, loops, voids et cetera, even in high dimensions where visualization of density estimates is impossible. In this paper, we propose an unsupervised learning approach using a topology-based loss function for the automated and unsupervised selection of the optimal bandwidth and benchmark it against classical techniques -- demonstrating its potential across different dimensions.

Stellar and AGN-driven feedback processes affect the distribution of gas on a wide range of scales, from within galaxies well into the intergalactic medium. Yet, it remains unclear how feedback, through its connection to key galaxy properties, shapes the radial gas density profile in the host halo. We tackle this question using suites of the EAGLE, IllustrisTNG, and Simba cosmological hydrodynamical simulations, which span a variety of feedback models. We develop a random forest algorithm that predicts the radial gas density profile within haloes from the total halo mass and five global properties of the central galaxy: gas and stellar mass; star formation rate; mass and accretion rate of the central black hole (BH). The algorithm reproduces the simulated gas density profiles with an average accuracy of $\sim$80-90% over the halo mass range 10^{9.5} \, \mathrm{M}_{\odot} < M_{\rm 200c} < 10^{15} \, \mathrm{M}_{\odot} and redshift interval $0

We revisit the signal denoising problem through the lens of optimal transport: the goal is to recover an unknown scalar signal distribution $X \sim P$ from noisy observations $Y = X + \sigma Z$, with $Z$ being standard Gaussian independent of $X$ and $\sigma>0$ a known noise level. Let $Q$ denote the distribution of $Y$. We introduce a hierarchy of denoisers $T_0, T_1, \ldots, T_\infty : \mathbb{R} \to \mathbb{R}$ that are agnostic to the signal distribution $P$, depending only on higher-order score functions of $Q$. Each denoiser $T_K$ is progressively refined using the $(2K-1)$-th order score function of $Q$ at noise resolution $\sigma^{2K}$, achieving better denoising quality measured by the Wasserstein metric $W(T_K \sharp Q, P)$. The limiting denoiser $T_\infty$ identifies the optimal transport map with $T_\infty \sharp Q = P$. We provide a complete characterization of the combinatorial structure underlying this hierarchy through Bell polynomial recursions, revealing how higher-order score functions encode the optimal transport map for signal denoising. We study two estimation strategies with convergence rates for higher-order scores from i.i.d. samples drawn from $Q$: (i) plug-in estimation via Gaussian kernel smoothing, and (ii) direct estimation via higher-order score matching. This hierarchy of agnostic denoisers opens new perspectives in signal denoising and empirical Bayes.

The optimal transport (OT) map is a geometry-driven transformation between high-dimensional probability distributions which underpins a wide range of tasks in statistics, applied probability, and machine learning. However, existing statistical theory for OT map estimation is quite restricted, hinging on Brenier's theorem (quadratic cost, absolutely continuous source) to guarantee existence and uniqueness of a deterministic OT map, on which various additional regularity assumptions are imposed to obtain quantitative error bounds. In many real-world problems these conditions fail or cannot be certified, in which case optimal transportation is possible only via stochastic maps that can split mass. To broaden the scope of map estimation theory to such settings, this work introduces a novel metric for evaluating the transportation quality of stochastic maps. Under this metric, we develop computationally efficient map estimators with near-optimal finite-sample risk bounds, subject to easy-to-verify minimal assumptions. Our analysis further accommodates common forms of adversarial sample contamination, yielding estimators with robust estimation guarantees. Empirical experiments are provided which validate our theory and demonstrate the utility of the proposed framework in settings where existing theory fails. These contributions constitute the first general-purpose theory for map estimation, compatible with a wide spectrum of real-world applications where optimal transport may be intrinsically stochastic.

We examine the non-asymptotic properties of robust density ratio estimation (DRE) in contaminated settings. Weighted DRE is the most promising among existing methods, exhibiting doubly strong robustness from an asymptotic perspective. This study demonstrates that Weighted DRE achieves sparse consistency even under heavy contamination within a non-asymptotic framework. This method addresses two significant challenges in density ratio estimation and robust estimation. For density ratio estimation, we provide the non-asymptotic properties of estimating unbounded density ratios under the assumption that the weighted density ratio function is bounded. For robust estimation, we introduce a non-asymptotic framework for doubly strong robustness under heavy contamination, assuming that at least one of the following conditions holds: (i) contamination ratios are small, and (ii) outliers have small weighted values. This work provides the first non-asymptotic analysis of strong robustness under heavy contamination.

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.

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.

In this paper, we examine the robustness of Nash equilibria in continuous games, under both strategic and dynamic uncertainty. Starting with the former, we introduce the notion of a robust equilibrium as those equilibria that remain invariant to small -- but otherwise arbitrary -- perturbations to the game's payoff structure, and we provide a crisp geometric characterization thereof. Subsequently, we turn to the question of dynamic robustness, and we examine which equilibria may arise as stable limit points of the dynamics of "follow the regularized leader" (FTRL) in the presence of randomness and uncertainty. Despite their very distinct origins, we establish a structural correspondence between these two notions of robustness: strategic robustness implies dynamic robustness, and, conversely, the requirement of strategic robustness cannot be relaxed if dynamic robustness is to be maintained. Finally, we examine the rate of convergence to robust equilibria as a function of the underlying regularizer, and we show that entropically regularized learning converges at a geometric rate in games with affinely constrained action spaces.

Axis-aligned decision trees are fast and stable but struggle on datasets with rotated or interaction-dependent decision boundaries, where informative splits require linear combinations of features rather than single-feature thresholds. Oblique forests address this with per-node hyperplane splits, but at added computational cost and implementation complexity. We propose a simple alternative: JARF, Jacobian-Aligned Random Forests. Concretely, we first fit an axis-aligned forest to estimate class probabilities or regression outputs, compute finite-difference gradients of these predictions with respect to each feature, aggregate them into an expected Jacobian outer product that generalizes the expected gradient outer product (EGOP), and use it as a single global linear preconditioner for all inputs. This supervised preconditioner applies a single global rotation of the feature space, then hands the transformed data back to a standard axis-aligned forest, preserving off-the-shelf training pipelines while capturing oblique boundaries and feature interactions that would otherwise require many axis-aligned splits to approximate. The same construction applies to any model that provides gradients, though we focus on random forests and gradient-boosted trees in this work. On tabular classification and regression benchmarks, this preconditioning consistently improves axis-aligned forests and often matches or surpasses oblique baselines while improving training time. Our experimental results and theoretical analysis together indicate that supervised preconditioning can recover much of the accuracy of oblique forests while retaining the simplicity and robustness of axis-aligned trees.

Learned image reconstruction has become a pillar in computational imaging and inverse problems. Among the most successful approaches are learned iterative networks, which are formulated by unrolling classical iterative optimisation algorithms for solving variational problems. While the underlying algorithm is usually formulated in the functional analytic setting, learned approaches are often viewed as purely discrete. In this chapter we present a unified operator view for learned iterative networks. Specifically, we formulate a learned reconstruction operator, defining how to compute, and separately the learning problem, which defines what to compute. In this setting we present common approaches and show that many approaches are closely related in their core. We review linear as well as nonlinear inverse problems in this framework and present a short numerical study to conclude.

Decision making often occurs in the presence of incomplete information, leading to the under- or overestimation of risk. Leveraging the observable information to learn the complete information is called nowcasting. In practice, incomplete information is often a consequence of reporting or observation delays. In this paper, we propose an expectation-maximisation (EM) framework for nowcasting that uses machine learning techniques to model both the occurrence as well as the reporting process of events. We allow for the inclusion of covariate information specific to the occurrence and reporting periods as well as characteristics related to the entity for which events occurred. We demonstrate how the maximisation step and the information flow between EM iterations can be tailored to leverage the predictive power of neural networks and (extreme) gradient boosting machines (XGBoost). With simulation experiments, we show that we can effectively model both the occurrence and reporting of events when dealing with high-dimensional covariate information. In the presence of non-linear effects, we show that our methodology outperforms existing EM-based nowcasting frameworks that use generalised linear models in the maximisation step. Finally, we apply the framework to the reporting of Argentinian Covid-19 cases, where the XGBoost-based approach again is most performant.

Access to multiple predictive models trained for the same task, whether in regression or classification, is increasingly common in many applications. Aggregating their predictive uncertainties to produce reliable and efficient uncertainty quantification is therefore a critical but still underexplored challenge, especially within the framework of conformal prediction (CP). While CP methods can generate individual prediction sets from each model, combining them into a single, more informative set remains a challenging problem. To address this, we propose SACP (Symmetric Aggregated Conformal Prediction), a novel method that aggregates nonconformity scores from multiple predictors. SACP transforms these scores into e-values and combines them using any symmetric aggregation function. This flexible design enables a robust, data-driven framework for selecting aggregation strategies that yield sharper prediction sets. We also provide theoretical insights that help justify the validity and performance of the SACP approach. Extensive experiments on diverse datasets show that SACP consistently improves efficiency and often outperforms state-of-the-art model aggregation baselines.