Causal inference is central to statistics and scientific discovery, enabling researchers to identify cause-and-effect relationships beyond associations. While traditionally studied within Euclidean spaces, contemporary applications increasingly involve complex, non-Euclidean data structures that reside in abstract metric spaces, known as random objects, such as images, shapes, networks, and distributions. This paper introduces a novel framework for causal inference with continuous treatments applied to non-Euclidean data. To address the challenges posed by the lack of linear structures, we leverage Hilbert space embeddings of the metric spaces to facilitate Fréchet mean estimation and causal effect mapping. Motivated by a study on the impact of exposure to fine particulate matter on age-at-death distributions across U.S. counties, we propose a nonparametric, doubly-debiased causal inference approach for outcomes as random objects with continuous treatments. Our framework can accommodate moderately high-dimensional vector-valued confounders and derive efficient influence functions for estimation to ensure both robustness and interpretability. We establish rigorous asymptotic properties of the cross-fitted estimators and employ conformal inference techniques for counterfactual outcome prediction. Validated through numerical experiments and applied to real-world environmental data, our framework extends causal inference methodologies to complex data structures, broadening its applicability across scientific disciplines.

In many modern applications, positive definite matrices are often used to summarize the marginal covariance structure among sets of variables. Examples include medical imaging (Dryden et al., 2009; Fillard et al., 2007), neuroscience (Friston, 2011; Kong et al., 2020; Hu et al., 2021) and gene coexpression analysis in single cell genomics. A central challenge in these fields is how to perform regression analysis where the covariance matrix serves as the outcome variable in relation to a set of Euclidean covariates and how to test for the association between these matrix and covariates. Several regression approaches for covariance matrix outcomes have been proposed. Chiu et al. (1996) developed a method that models the elements of the logarithm of the covariance matrix as a linear function of the covariates, but this approach requires estimating a large number of parameters. Hoff & Niu (2012) proposed a regression model where the covariance matrix is expressed as a quadratic function of the explanatory variables. Zou et al. (2017) linked the matrix outcome to a linear combination of similarity matrices derived from the covariates and examined the asymptotic properties of different estimators under this framework. Xu & Li (2025) introduced Fr echet regression with covariate matrix as the outcome.

Approximate Message Passing (AMP) algorithms enable precise characterization of certain classes of random objects in the high-dimensional limit, and have found widespread applications in fields such as signal processing, statistics, and communications. In this work, we introduce Multi-Environment Generalized Long AMP, a novel AMP framework that applies to transfer learning problems with multiple data sources and distribution shifts. We rigorously establish state evolution for multi-environment GLAMP. We demonstrate the utility of this framework by precisely characterizing the risk of three Lasso-based transfer learning estimators for the first time: the Stacked Lasso, the Model Averaging Estimator, and the Second Step Estimator. We also demonstrate the remarkable finite sample accuracy of our theory via extensive simulations.

Quantum Neural Networks (QNNs), a prominent approach in Quantum Machine Learning (QML), are emerging as a powerful alternative to classical machine learning methods. Recent studies have focused on the applicability of QNNs to various tasks, such as time-series forecasting, prediction, and classification, across a wide range of applications, including cybersecurity and medical imaging. With the increased use of smart grids driven by the integration of renewable energy systems, machine learning plays an important role in predicting power demand and detecting system disturbances. This study provides an in-depth investigation of QNNs for predicting the power output of a wind turbine. We assess the predictive performance and simulation time of six QNN configurations that are based on the Z Feature Map for data encoding and varying ansatz structures. Through detailed cross-validation experiments and tests on an unseen hold-out dataset, we experimentally demonstrate that QNNs can achieve predictive performance that is competitive with, and in some cases marginally better than, the benchmarked classical approaches. Our results also reveal the effects of dataset size and circuit complexity on predictive performance and simulation time. We believe our findings will offer valuable insights for researchers in the energy domain who wish to incorporate quantum machine learning into their work.

This tutorial provides a concise introduction to modern causal modeling by integrating potential outcomes and graphical methods. We motivate causal questions such as counterfactual reasoning under interventions and define binary treatments and potential outcomes. We discuss causal effect measures-including average treatment effects on the treated and on the untreated-and choices of effect scales for binary outcomes. We derive identification in randomized experiments under exchangeability and consistency, and extend to stratification and blocking designs. We present inverse probability weighting with propensity score estimation and robust inference via sandwich estimators. Finally, we introduce causal graphs, d-separation, the backdoor criterion, single-world intervention graphs, and structural equation models, showing how graphical and potential-outcome approaches complement each other. Emphasis is placed on clear notation, intuitive explanations, and practical examples for applied researchers.

This study investigates the application of quantum neural networks (QNNs) for propensity score estimation to address selection bias in comparing survival outcomes between laparoscopic and open surgical techniques in a cohort of 1177 colorectal carcinoma patients treated at University Hospital Ostrava (2001-2009). Using a dataset with 77 variables, including patient demographics and tumor characteristics, we developed QNN-based propensity score models focusing on four key covariates (Age, Sex, Stage, BMI). The QNN architecture employed a linear ZFeatureMap for data encoding, a SummedPaulis operator for predictions, and the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) for robust, gradient-free optimization in noisy quantum environments. Variance regularization was integrated to mitigate quantum measurement noise, with simulations conducted under exact, sampling (1024 shots), and noisy hardware (FakeManhattanV2) conditions. QNNs, particularly with simulated hardware noise, outperformed classical logistic regression and gradient boosted machines in small samples (AUC up to 0.750 for n=100), with noise modeling enhancing predictive stability. Propensity score matching and weighting, optimized via genetic matching and matching weights, achieved covariate balance with standardized mean differences of 0.0849 and 0.0869, respectively. Survival analyses using Kaplan-Meier estimation, Cox proportional hazards, and Aalen additive regression revealed no significant survival differences post-adjustment (p-values 0.287-0.851), indicating confounding bias in unadjusted outcomes. These results highlight QNNs' potential, enhanced by CMA-ES and noise-aware strategies, to improve causal inference in biomedical research, particularly for small-sample, high-dimensional datasets.

We investigate the generalization capabilities of small language models under two popular adaptation paradigms: few-shot prompting and supervised fine-tuning. While prompting is often favored for its parameter efficiency and flexibility, it remains unclear how robust this approach is in low-resource settings and under distributional shifts. This paper presents a comparative study of prompting and fine-tuning across task formats, prompt styles, and model scales, with a focus on their behavior in both in-distribution and out-of-distribution (OOD) settings. Beyond accuracy, we analyze the internal representations learned by each approach to assess the stability and abstraction of task-specific features. Our findings highlight critical differences in how small models internalize and generalize knowledge under different adaptation strategies. This work offers practical guidance for model selection in low-data regimes and contributes empirical insight into the ongoing debate over prompting versus fine-tuning. Code for the experiments is available at the following

In recent years, machine learning (ML) techniques have become a powerful tool for improving the accuracy of predictions and decision-making. Machine learning technologies have begun to penetrate all areas, including the real estate sector. Correct forecasting of real estate value plays an important role in the buyer-seller chain, because it ensures reasonableness of price expectations based on the offers available in the market and helps to avoid financial risks for both parties of the transaction. Accurate forecasting is also important for real estate investors to make an informed decision on a specific property. This study helps to gain a deeper understanding of how effective and accurate ensemble machine learning methods are in predicting real estate values. The results obtained in the work are quite accurate, as can be seen from the coefficient of determination (R^2), root mean square error (RMSE) and mean absolute error (MAE) calculated for each model. The Gradient Boosting Regressor model provides the highest accuracy, the Extra Trees Regressor, Hist Gradient Boosting Regressor and Random Forest Regressor models give good results. In general, ensemble machine learning techniques can be effectively used to solve real estate valuation. This work forms ideas for future research, which consist in the preliminary processing of the data set by searching and extracting anomalous values, as well as the practical implementation of the obtained results.

In a regression task, a function is learned from labeled data to predict the labels at new data points. The goal is to achieve small prediction errors. In symbolic regression, the goal is more ambitious, namely, to learn an interpretable function that makes small prediction errors. This additional goal largely rules out the standard approach used in regression, that is, reducing the learning problem to learning parameters of an expansion of basis functions by optimization. Instead, symbolic regression methods search for a good solution in a space of symbolic expressions. To cope with the typically vast search space, most symbolic regression methods make implicit, or sometimes even explicit, assumptions about its structure. Here, we argue that the only obvious structure of the search space is that it contains small expressions, that is, expressions that can be decomposed into a few subexpressions. We show that systematically searching spaces of small expressions finds solutions that are more accurate and more robust against noise than those obtained by state-of-the-art symbolic regression methods. In particular, systematic search outperforms state-of-the-art symbolic regressors in terms of its ability to recover the true underlying symbolic expressions on established benchmark data sets.

Solutions of symbolic regression problems are expressions that are composed of input variables and operators from a finite set of function symbols. One measure for evaluating symbolic regression algorithms is their ability to recover formulae, up to symbolic equivalence, from finite samples. Not unexpectedly, the recovery problem becomes harder when the formula gets more complex, that is, when the number of variables and operators gets larger. Variables in naturally occurring symbolic formulas often appear only in fixed combinations. This can be exploited in symbolic regression by substituting one new variable for the combination, effectively reducing the number of variables. However, finding valid substitutions is challenging. Here, we address this challenge by searching over the expression space of small substitutions and testing for validity. The validity test is reduced to a test of functional dependence. The resulting iterative dimension reduction procedure can be used with any symbolic regression approach. We show that it reliably identifies valid substitutions and significantly boosts the performance of different types of state-of-the-art symbolic regression algorithms.