# Converting High-Dimensional Regression to High-Dimensional Conditional Density Estimation

## TL;DR

FlexCode is a nonparametric method that transforms high-dimensional conditional density estimation into a regression problem, enabling efficient and flexible density estimation in complex, real-world datasets.

## Contribution

The paper introduces FlexCode, a novel nonparametric approach that reformulates CDE as a regression problem, allowing for scalable and adaptable high-dimensional density estimation.

## Key findings

- FlexCode effectively estimates complex conditional densities in high dimensions.
- The method adapts to various data structures and types, including functional and mixed data.
- Empirical results outperform traditional CDE methods on real-world datasets.

## Abstract

There is a growing demand for nonparametric conditional density estimators (CDEs) in fields such as astronomy and economics. In astronomy, for example, one can dramatically improve estimates of the parameters that dictate the evolution of the Universe by working with full conditional densities instead of regression (i.e., conditional mean) estimates. More generally, standard regression falls short in any prediction problem where the distribution of the response is more complex with multi-modality, asymmetry or heteroscedastic noise. Nevertheless, much of the work on high-dimensional inference concerns regression and classification only, whereas research on density estimation has lagged behind. Here we propose FlexCode, a fully nonparametric approach to conditional density estimation that reformulates CDE as a non-parametric orthogonal series problem where the expansion coefficients are estimated by regression. By taking such an approach, one can efficiently estimate conditional densities and not just expectations in high dimensions by drawing upon the success in high-dimensional regression. Depending on the choice of regression procedure, our method can adapt to a variety of challenging high-dimensional settings with different structures in the data (e.g., a large number of irrelevant components and nonlinear manifold structure) as well as different data types (e.g., functional data, mixed data types and sample sets). We study the theoretical and empirical performance of our proposed method, and we compare our approach with traditional conditional density estimators on simulated as well as real-world data, such as photometric galaxy data, Twitter data, and line-of-sight velocities in a galaxy cluster.