Deep Quantile Regression for Uncertainty Estimation in Unsupervised and Supervised Lesion Detection

Abstract

Despite impressive state-of-the-art performance on a wide variety of machine learning tasks in multiple applications, deep learning methods can produce over-confident predictions, particularly with limited training data. Therefore, quantifying uncertainty is particularly important in critical applications such as lesion detection and clinical diagnosis, where a realistic assessment of uncertainty is essential in determining surgical margins, disease status and appropriate treatment. In this work, we propose a novel approach that uses quantile regression for quantifying aleatoric uncertainty in both supervised and unsupervised lesion detection problems. The resulting confidence intervals can be used for lesion detection and segmentation. In the unsupervised settings, we combine quantile regression with the Variational AutoEncoder (VAE). The VAE is trained on lesion-free data, so when presented with an image with a lesion, it tends to reconstruct a lesion-free version of the image. To detect the lesion, we then compare the input (lesion) and output (lesion-free) images. Here we address the problem of quantifying uncertainty in the images that are reconstructed by the VAE as the basis for principled outlier or lesion detection. The VAE models the output as a conditionally independent Gaussian characterized by its mean and variance. Unfortunately, joint optimization of both mean and variance in the VAE leads to the well-known problem of shrinkage or underestimation of variance. Here we describe an alternative Quantile-Regression VAE (QR-VAE) that avoids this variance shrinkage problem by directly estimating conditional quantiles for the input image. Using the estimated quantiles, we compute the conditional mean and variance for input image from which we then detect outliers by thresholding at a false-discovery-rate corrected p-value. In the supervised setting, we develop binary quantile regression (BQR) for the supervised lesion segmentation task. We show how BQR can be used to capture uncertainty in lesion boundaries in a manner that characterizes expert disagreement.