Fréchet regression has emerged as a promising approach for regression analysis involving non-Euclidean response variables.

Transfer learning is a powerful paradigm for leveraging knowledge from source domains to enhance learning in a target domain. However, traditional transfer learning approaches often focus on scalar or multivariate data within Euclidean spaces, limiting their applicability to complex data structures such as probability distributions. To address this limitation, we introduce a novel transfer learning framework for regression models whose outputs are probability distributions residing in the Wasserstein space. When the informative subset of transferable source domains is known, we propose an estimator with provable asymptotic convergence rates, quantifying the impact of domain similarity on transfer efficiency. For cases where the informative subset is unknown, we develop a data-driven transfer learning procedure designed to mitigate negative transfer. The proposed methods are supported by rigorous theoretical analysis and are validated through extensive simulations and real-world applications. The code is available at https://github.com/h7nian/WaTL

Fréchet regression has emerged as a promising approach for regression analysis involving non-Euclidean response variables.

Gradient boosting has become a cornerstone of machine learning, enabling base learners such as decision trees to achieve exceptional predictive performance. While existing algorithms primarily handle scalar or Euclidean outputs, increasingly prevalent complex-structured data, such as distributions, networks, and manifold-valued outputs, present challenges for traditional methods. Such non-Euclidean data lack algebraic structures such as addition, subtraction, or scalar multiplication required by standard gradient boosting frameworks. To address these challenges, we introduce Fréchet geodesic boosting (FGBoost), a novel approach tailored for outputs residing in geodesic metric spaces. FGBoost leverages geodesics as proxies for residuals and constructs ensembles in a way that respects the intrinsic geometry of the output space. Through theoretical analysis, extensive simulations, and real-world applications, we demonstrate the strong performance and adaptability of FGBoost, showcasing its potential for modeling complex data.

Fr\'echet regression extends classical regression methods to non-Euclidean metric spaces, enabling the analysis of data relationships on complex structures such as manifolds and graphs. This work establishes a rigorous theoretical analysis for Fr\'echet regression through the lens of comparison geometry which leads to important considerations for its use in practice. The analysis provides key results on the existence, uniqueness, and stability of the Fr\'echet mean, along with statistical guarantees for nonparametric regression, including exponential concentration bounds and convergence rates. Additionally, insights into angle stability reveal the interplay between curvature of the manifold and the behavior of the regression estimator in these non-Euclidean contexts. Empirical experiments validate the theoretical findings, demonstrating the effectiveness of proposed hyperbolic mappings, particularly for data with heteroscedasticity, and highlighting the practical usefulness of these results.

Many problems within personalized medicine and digital health rely on the analysis of continuous-time functional biomarkers and other complex data structures emerging from high-resolution patient monitoring. In this context, this work proposes new optimization-based variable selection methods for multivariate, functional, and even more general outcomes in metrics spaces based on best-subset selection. Our framework applies to several types of regression models, including linear, quantile, or non parametric additive models, and to a broad range of random responses, such as univariate, multivariate Euclidean data, functional, and even random graphs. Our analysis demonstrates that our proposed methodology outperforms state-of-the-art methods in accuracy and, especially, in speed-achieving several orders of magnitude improvement over competitors across various type of statistical responses as the case of mathematical functions. While our framework is general and is not designed for a specific regression and scientific problem, the article is self-contained and focuses on biomedical applications. In the clinical areas, serves as a valuable resource for professionals in biostatistics, statistics, and artificial intelligence interested in variable selection problem in this new technological AI-era.

Fréchet regression, an extension of classical linear regression to general metric spaces, offers a robust framework for modeling complex relationships between variables when the responses lie outside of Euclidean spaces. This approach is especially well suited to high-dimensional datasets, such as vector representations, with particular relevance to fields like imaging, where capturing nonlinear dependencies and the intrinsic data structure is critical for accurate modeling (Fréchet (1948), Petersen and Müller (2019), Bhattacharjee and Müller (2023), Qiu, Yu and Zhu (2024)). A significant consideration in Fréchet regression arises when predicting multiple responses simultaneously, as seen in multi-target or multidimensional problems (Zhang and Zhou (2007), Hyvönen, Jääsaari and Roos (2024)). Unlike traditional regression, where each observation corresponds to a single response, Fréchet regression can be extended to model complex interactions between multiple outputs. This ability to address complex relationships between several responses opens new avenues, particularly in fields such as bioinformatics (Huang et al. (2005)) and image analysis (Lathuilière et al. (2019)), where multidimensional data and interdependencies between responses require adaptive and specialized methodologies. However, to date, the handling of multilabel scenarios within the context of Fréchet regression remains relatively unexplored in the literature, despite its potential significance in addressing complex, multidimensional applications. In this paper, we present an extension of the Global Fréchet regression model, a specific variant of Fréchet regression that generalizes classical multiple linear regression by modeling responses as random objects. This extension enables the explicit modeling of relationships between input variables and multiple responses, thereby addressing the multi-label setting. Our second contribution in this paper addresses the dimensionality challenge in the context of the proposed Fréchet regression extension.

Fr\'echet regression is becoming a mainstay in modern data analysis for analyzing non-traditional data types belonging to general metric spaces. This novel regression method is especially useful in the analysis of complex health data such as continuous monitoring and imaging data. Fr\'echet regression utilizes the pairwise distances between the random objects, which makes the choice of metric crucial in the estimation. In this paper, existing dimension reduction methods for Fr\'echet regression are reviewed, and the effect of metric choice on the estimation of the dimension reduction subspace is explored for the regression between random responses and Euclidean predictors. Extensive numerical studies illustrate how different metrics affect the central and central mean space estimators. Two real applications involving analysis of brain connectivity networks of subjects with and without Parkinson's disease and an analysis of the distributions of glycaemia based on continuous glucose monitoring data are provided, to demonstrate how metric choice can influence findings in real applications.

Advancements in modern science have led to the increasing availability of non-Euclidean data in metric spaces. This paper addresses the challenge of modeling relationships between non-Euclidean responses and multivariate Euclidean predictors. We propose a flexible regression model capable of handling high-dimensional predictors without imposing parametric assumptions. Two primary challenges are addressed: the curse of dimensionality in nonparametric regression and the absence of linear structure in general metric spaces. The former is tackled using deep neural networks, while for the latter we demonstrate the feasibility of mapping the metric space where responses reside to a low-dimensional Euclidean space using manifold learning. We introduce a reverse mapping approach, employing local Fr\'echet regression, to map the low-dimensional manifold representations back to objects in the original metric space. We develop a theoretical framework, investigating the convergence rate of deep neural networks under dependent sub-Gaussian noise with bias. The convergence rate of the proposed regression model is then obtained by expanding the scope of local Fr\'echet regression to accommodate multivariate predictors in the presence of errors in predictors. Simulations and case studies show that the proposed model outperforms existing methods for non-Euclidean responses, focusing on the special cases of probability measures and networks.

This paper explores the field of semi-supervised Fr\'echet regression, driven by the significant costs associated with obtaining non-Euclidean labels. Methodologically, we propose two novel methods: semi-supervised NW Fr\'echet regression and semi-supervised kNN Fr\'echet regression, both based on graph distance acquired from all feature instances. These methods extend the scope of existing semi-supervised Euclidean regression methods. We establish their convergence rates with limited labeled data and large amounts of unlabeled data, taking into account the low-dimensional manifold structure of the feature space. Through comprehensive simulations across diverse settings and applications to real data, we demonstrate the superior performance of our methods over their supervised counterparts. This study addresses existing research gaps and paves the way for further exploration and advancements in the field of semi-supervised Fr\'echet regression.