In multi-output regression, we identify a previously neglected challenge that arises from the inability of training distribution to cover all combinations of input features, leading to combinatorial distribution shift (CDS). To the best of our knowledge, this is the first work to formally define and address this problem. We tackle it through a novel tensor decomposition perspective, proposing the Functional t-Singular Value Decomposition (Ft-SVD) theorem which extends the classical tensor SVD to infinite and continuous feature domains, providing a natural tool for representing and analyzing multi-output functions. Within the Ft-SVD framework, we formulate the multi-output regression problem under CDS as a low-rank tensor estimation problem under the missing not at random (MNAR) setting, and introduce a series of assumptions about the true functions, training and testing distributions, and spectral properties of the ground-truth embeddings, making the problem more tractable.To address the challenges posed by CDS in multi-output regression, we develop a tailored Double-Stage Empirical Risk Minimization (ERM-DS) algorithm that leverages the spectral properties of the embeddings and uses specific hypothesis classes in each frequency component to better capture the varying spectral decay patterns. We provide rigorous theoretical analyses that establish performance guarantees for the ERM-DS algorithm. This work lays a preliminary theoretical foundation for multi-output regression under CDS.

Malaria remains a major public health concern in Ethiopia, particularly in the Amhara Region, where seasonal and unpredictable transmission patterns make prevention and control challenging. Accurately forecasting malaria outbreaks is essential for effective resource allocation and timely interventions. This study proposes a hybrid predictive modeling framework that combines time-series forecasting, multi-output regression, and conventional regression-based prediction to forecast the incidence of malaria. Environmental variables, past malaria case data, and demographic information from Amhara Region health centers were used to train and validate the models. The multi-output regression approach enables the simultaneous prediction of multiple outcomes, including Plasmodium species-specific cases, temporal trends, and spatial variations, whereas the hybrid framework captures both seasonal patterns and correlations among predictors. The proposed model exhibits higher prediction accuracy than single-method approaches, exposing hidden patterns and providing valuable information to public health authorities. This study provides a valid and repeatable malaria incidence prediction framework that can support evidence-based decision-making, targeted interventions, and resource optimization in endemic areas.

We provide rigorous theoretical analyses that establish performance guarantees for the ERM-DS algorithm.

In multi-output regression, we identify a previously neglected challenge that arises from the inability of training distribution to cover all combinations of input features, leading to combinatorial distribution shift (CDS). To the best of our knowledge, this is the first work to formally define and address this problem. We tackle it through a novel tensor decomposition perspective, proposing the Functional t-Singular Value Decomposition (Ft-SVD) theorem which extends the classical tensor SVD to infinite and continuous feature domains, providing a natural tool for representing and analyzing multi-output functions. Within the Ft-SVD framework, we formulate the multi-output regression problem under CDS as a low-rank tensor estimation problem under the missing not at random (MNAR) setting, and introduce a series of assumptions about the true functions, training and testing distributions, and spectral properties of the ground-truth embeddings, making the problem more tractable.To address the challenges posed by CDS in multi-output regression, we develop a tailored Double-Stage Empirical Risk Minimization (ERM-DS) algorithm that leverages the spectral properties of the embeddings and uses specific hypothesis classes in each frequency component to better capture the varying spectral decay patterns. We provide rigorous theoretical analyses that establish performance guarantees for the ERM-DS algorithm.

Conformal Prediction (CP) is a principled framework for quantifying uncertainty in blackbox learning models, by constructing prediction sets with finite-sample coverage guarantees. Traditional approaches rely on scalar nonconformity scores, which fail to fully exploit the geometric structure of multivariate outputs, such as in multi-output regression or multiclass classification. Recent methods addressing this limitation impose predefined convex shapes for the prediction sets, potentially misaligning with the intrinsic data geometry. We introduce a novel CP procedure handling multivariate score functions through the lens of optimal transport. Specifically, we leverage Monge-Kantorovich vector ranks and quantiles to construct prediction region with flexible, potentially non-convex shapes, better suited to the complex uncertainty patterns encountered in multivariate learning tasks. We prove that our approach ensures finite-sample, distribution-free coverage properties, similar to typical CP methods. We then adapt our method for multi-output regression and multiclass classification, and also propose simple adjustments to generate adaptive prediction regions with asymptotic conditional coverage guarantees. Finally, we evaluate our method on practical regression and classification problems, illustrating its advantages in terms of (conditional) coverage and efficiency.

This paper describes our participation in the MentalRiskES task at IberLEF 2023. The task involved predicting the likelihood of an individual experiencing depression based on their social media activity. The dataset consisted of conversations from 175 Telegram users, each labeled according to their evidence of suffering from the disorder. We used a combination of traditional machine learning and deep learning techniques to solve four predictive subtasks: binary classification, simple regression, multiclass classification, and multi-output regression. We approached this by training a model to solve the multi-output regression case and then transforming the predictions to work for the other three subtasks. We compare the performance of two modeling approaches: fine-tuning a BERT-based model directly for the task or using its embeddings as inputs to a linear regressor, with the latter yielding better results.

We present a sparse approximation approach for dependent output Gaussian processes (GP). Employing a latent function framework, we apply the convolution process formalism to establish dependencies between output variables, where each latent function is represented as a GP. Based on these latent functions, we establish an approximation scheme using a conditional independence assumption between the output processes, leading to an approximation of the full covariance which is determined by the locations at which the latent functions are evaluated. We show results of the proposed methodology for synthetic data and real world applications on pollution prediction and a sensor network.

This paper presents a computationally efficient variant of gradient boosting for multi-class classification and multi-output regression tasks. Standard gradient boosting uses a 1-vs-all strategy for classifications tasks with more than two classes. This strategy translates in that one tree per class and iteration has to be trained. In this work, we propose the use of multi-output regressors as base models to handle the multi-class problem as a single task. In addition, the proposed modification allows the model to learn multi-output regression problems. An extensive comparison with other multi-ouptut based gradient boosting methods is carried out in terms of generalization and computational efficiency. The proposed method showed the best trade-off between generalization ability and training and predictions speeds.

DN), thus, limiting their use to moderately sized data sets. To enable posterior inference in GPs on large-scale problems, Inspired by recent advances in the field of expertbased recent work (see e.g. Liu et al. [2020] for a detailed approximations of Gaussian processes (GPs), review) mainly resorts to global approximations to the posterior, we present an expert-based approach to large-scale e.g., using inducing points, or local approximations multi-output regression using single-output GP that aim to distribute the computation of the posterior distribution experts. Employing a deeply structured mixture onto local experts. Unfortunately, most of these of single-output GPs encoded via a probabilistic approaches only focus on single-output regression, i.e., the circuit allows us to capture correlations between dependent variable is univariate, and in the case of local multiple output dimensions accurately. By recursively approximations, do not easily extend to multi-output regression partitioning the covariate space and the output tasks, see Bruinsma et al. [2020] for a detailed space, posterior inference in our model reduces to discussion on recent techniques on multi-output GPs.