Learning complex distributions is a fundamental challenge in contemporary applications. Generative models, such as diffusion models, have demonstrated remarkable success in overcoming many limitations of traditional statistical methods. Shen and Meinshausen (2024) introduced engression, a generative approach based on scoring rules that maps noise (and covariates, if available) directly to data. While effective, engression struggles with highly complex distributions, such as those encountered in image data. In this work, we extend engression to improve its capability in learning complex distributions. We propose a framework that defines a general forward process transitioning from the target distribution to a known distribution (e.g., Gaussian) and then learns a reverse Markov process using multiple engression models. This reverse process reconstructs the target distribution step by step. Our approach supports general forward processes, allows for dimension reduction, and naturally discretizes the generative process. As a special case, when using a diffusion-based forward process, our framework offers a method to discretize the training and inference of diffusion models efficiently. Empirical evaluations on simulated and climate data validate our theoretical insights, demonstrating the effectiveness of our approach in capturing complex distributions.

Extrapolation is crucial in many statistical and machine learning applications, as it is common to encounter test data outside the training support. However, extrapolation is a considerable challenge for nonlinear models. Conventional models typically struggle in this regard: while tree ensembles provide a constant prediction beyond the support, neural network predictions tend to become uncontrollable. This work aims at providing a nonlinear regression methodology whose reliability does not break down immediately at the boundary of the training support. Our primary contribution is a new method called `engression' which, at its core, is a distributional regression technique for pre-additive noise models, where the noise is added to the covariates before applying a nonlinear transformation. Our experimental results indicate that this model is typically suitable for many real data sets. We show that engression can successfully perform extrapolation under some assumptions such as a strictly monotone function class, whereas traditional regression approaches such as least-squares regression and quantile regression fall short under the same assumptions. We establish the advantages of engression over existing approaches in terms of extrapolation, showing that engression consistently provides a meaningful improvement. Our empirical results, from both simulated and real data, validate these findings, highlighting the effectiveness of the engression method. The software implementations of engression are available in both R and Python.