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

arXiv: https://arxiv.org/abs/2109.09374 · HTML: https://mlsanity.com/paper/2109.09374/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 50 references that are in the map.
- First posted on arXiv 20 Sep 2021.

## Builds on (most cited first, 24 of 24)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Conformalized Quantile Regression](https://mlsanity.com/paper/1905.03222/) (2019)
- [Autoencoding beyond pixels using a learned similarity metric](https://mlsanity.com/paper/1512.09300/) (2015)
- [A Probabilistic U-Net for Segmentation of Ambiguous Images](https://mlsanity.com/paper/1806.05034/) (2018)
- [Deep Autoencoding Models for Unsupervised Anomaly Segmentation in Brain MR Images](https://mlsanity.com/paper/1804.04488/) (2018)
- [Stochastic Segmentation Networks: Modelling Spatially Correlated Aleatoric Uncertainty](https://mlsanity.com/paper/2006.06015/) (2020)
- [Unsupervised Detection of Lesions in Brain MRI using constrained adversarial auto-encoders](https://mlsanity.com/paper/1806.04972/) (2018)
- [A comparison of some conformal quantile regression methods](https://mlsanity.com/paper/1909.05433/) (2019)
- [Unsupervised Lesion Detection via Image Restoration with a Normative Prior](https://mlsanity.com/paper/2005.00031/) (2020)
- [Reliable training and estimation of variance networks](https://mlsanity.com/paper/1906.03260/) (2019)
- [Leveraging Uncertainty Estimates for Predicting Segmentation Quality](https://mlsanity.com/paper/1807.00502/) (2018)
- [Supervised Uncertainty Quantification for Segmentation with Multiple Annotations](https://mlsanity.com/paper/1907.01949/) (2019)
- [Is segmentation uncertainty useful?](https://mlsanity.com/paper/2103.16265/) (2021)
- [Automated Variational Inference in Probabilistic Programming](https://mlsanity.com/paper/1301.1299/) (2013)
- [Beyond expectation: Deep joint mean and quantile regression for spatio-temporal problems](https://mlsanity.com/paper/1808.08798/) (2018)
- [Spatially Varying Label Smoothing: Capturing Uncertainty from Expert Annotations](https://mlsanity.com/paper/2104.05788/) (2021)
- [Leveraging the Exact Likelihood of Deep Latent Variable Models](https://mlsanity.com/paper/1802.04826/) (2018)
- [Modelling the Distribution of 3D Brain MRI using a 2D Slice VAE](https://mlsanity.com/paper/2007.04780/) (2020)
- [Robust Variational Autoencoder](https://mlsanity.com/paper/1905.09961/) (2019)
- [Validating uncertainty in medical image translation](https://mlsanity.com/paper/2002.04639/) (2020)
- [Variational Variance: Simple, Reliable, Calibrated Heteroscedastic Noise Variance Parameterization](https://mlsanity.com/paper/2006.04910/) (2020)

## Built on by (most cited first, 4 of 4)

- [Distributional Learning of Variational AutoEncoder: Application to Synthetic Data Generation](https://mlsanity.com/paper/2302.11294/) (2023)
- [Beta quantile regression for robust estimation of uncertainty in the presence of outliers](https://mlsanity.com/paper/2309.07374/) (2023)
- [Balanced Marginal and Joint Distributional Learning via Mixture Cramer-Wold Distance](https://mlsanity.com/paper/2312.03307/) (2023)
- [Distribution-Aware Conformal Prediction: A Framework for generating efficient prediction intervals for time series](https://mlsanity.com/paper/2605.26569/) (2026)

## Abstract (arXiv)

Despite impressive state-of-the-art performance on a wide variety of machine learning tasks, 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 setting, we combine quantile regression with the Variational AutoEncoder (VAE). 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 the 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.