# Distributional loss for convolutional neural network regression and application to GNSS multi-path estimation

arXiv: https://arxiv.org/abs/2206.01473 · HTML: https://mlsanity.com/paper/2206.01473/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 49 references that are in the map.
- First posted on arXiv 3 Jun 2022.

## Builds on (most cited first, 7 of 7)

- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [DeepPose: Human Pose Estimation via Deep Neural Networks](https://mlsanity.com/paper/1312.4659/) (2013)
- [Deep Label Distribution Learning with Label Ambiguity](https://mlsanity.com/paper/1611.01731/) (2016)
- [Improving Regression Performance with Distributional Losses](https://mlsanity.com/paper/1806.04613/) (2018)
- [A Comprehensive Analysis of Deep Regression](https://mlsanity.com/paper/1803.08450/) (2018)
- [A GPS spoofing detection and classification correlator-based technique using the LASSO](https://mlsanity.com/paper/2004.13900/) (2020)

## Built on by (most cited first, 2 of 2)

- [Neural Regression For Scale-Varying Targets](https://mlsanity.com/paper/2211.07447/) (2022)
- [Diff-GNSS: Diffusion-based Pseudorange Error Estimation](https://mlsanity.com/paper/2509.17397/) (2025)

## Abstract (arXiv)

Convolutional Neural Network (CNN) have been widely used in image classification. Over the years, they have also benefited from various enhancements and they are now considered as state of the art techniques for image like data. However, when they are used for regression to estimate some function value from images, fewer recommendations are available. In this study, a novel CNN regression model is proposed. It combines convolutional neural layers to extract high level features representations from images with a soft labelling technique. More specifically, as the deep regression task is challenging, the idea is to account for some uncertainty in the targets that are seen as distributions around their mean. The estimations are carried out by the model in the form of distributions. Building from earlier work, a specific histogram loss function based on the Kullback-Leibler (KL) divergence is applied during training. The model takes advantage of the CNN feature representation and is able to carry out estimation from multi-channel input images. To assess and illustrate the technique, the model is applied to Global Navigation Satellite System (GNSS) multi-path estimation where multi-path signal parameters have to be estimated from correlator output images from the I and Q channels. The multi-path signal delay, magnitude, Doppler shift frequency and phase parameters are estimated from synthetically generated datasets of satellite signals. Experiments are conducted under various receiving conditions and various input images resolutions to test the estimation performances quality and robustness. The results show that the proposed soft labelling CNN technique using distributional loss outperforms classical CNN regression under all conditions. Furthermore, the extra learning performance achieved by the model allows the reduction of input image resolution from 80x80 down to 40x40 or sometimes 20x20.