Normalizing Flow to Augmented Posterior: Conditional Density Estimation with Interpretable Dimension Reduction for High Dimensional Data

TL;DR

This paper introduces AP-CDE, a novel normalizing flow-based method for high-dimensional conditional density estimation that incorporates interpretable dimension reduction, improving interpretability and separation of factors in complex data like images.

Contribution

The paper extends normalizing flow neural networks to high-dimensional responses with a simple modification, enabling interpretable dimension reduction in conditional density estimation.

Findings

AP-CDE effectively separates x-related variations from other factors in image data.

The method improves interpretability over traditional NF models.

Experiments demonstrate better factor separation and interpretability.

Abstract

The conditional density characterizes the distribution of a response variable $y$ given other predictor $x$, and plays a key role in many statistical tasks, including classification and outlier detection. Although there has been abundant work on the problem of Conditional Density Estimation (CDE) for a low-dimensional response in the presence of a high-dimensional predictor, little work has been done for a high-dimensional response such as images. The promising performance of normalizing flow (NF) neural networks in unconditional density estimation acts a motivating starting point. In this work, we extend NF neural networks when external $x$ is present. Specifically, they use the NF to parameterize a one-to-one transform between a high-dimensional $y$ and a latent $z$ that comprises two components \([z_P,z_N]\). The $z_P$ component is a low-dimensional subvector obtained from the posterior distribution of an elementary predictive model for $x$, such as logistic/linear regression. The $z_N$ component is a high-dimensional independent Gaussian vector, which explains the variations in $y$ not or less related to $x$. Unlike existing CDE methods, the proposed approach, coined Augmented Posterior CDE (AP-CDE), only requires a simple modification on the common normalizing flow framework, while significantly improving the interpretation of the latent component, since $z_P$ represents a supervised dimension reduction. In image analytics applications, AP-CDE shows good separation of $x$-related variations due to factors such as lighting condition and subject id, from the other random variations. Further, the experiments show that an unconditional NF neural network, based on an unsupervised model of $z$, such as Gaussian mixture, fails to generate interpretable results.