# Geometry-Aware Hierarchical Bayesian Learning on Manifolds

arXiv: https://arxiv.org/abs/2111.00184 · HTML: https://mlsanity.com/paper/2111.00184/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 65 references that are in the map.
- First posted on arXiv 30 Oct 2021.

## Builds on (most cited first, 19 of 19)

- [PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space](https://mlsanity.com/paper/1706.02413/) (2017)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [Deep Kernel Learning](https://mlsanity.com/paper/1511.02222/) (2015)
- [Doubly Stochastic Variational Inference for Deep Gaussian Processes](https://mlsanity.com/paper/1705.08933/) (2017)
- [Deep Convolutional Networks as shallow Gaussian Processes](https://mlsanity.com/paper/1808.05587/) (2018)
- [Pointwise Convolutional Neural Networks](https://mlsanity.com/paper/1712.05245/) (2017)
- [Hilbert Space Methods for Reduced-Rank Gaussian Process Regression](https://mlsanity.com/paper/1401.5508/) (2014)
- [Mat'ern Gaussian processes on Riemannian manifolds](https://mlsanity.com/paper/2006.10160/) (2020)
- [MCMC for Variationally Sparse Gaussian Processes](https://mlsanity.com/paper/1506.04000/) (2015)
- [Convolutional Gaussian Processes](https://mlsanity.com/paper/1709.01894/) (2017)
- [Spectral Mixture Kernels for Multi-Output Gaussian Processes](https://mlsanity.com/paper/1709.01298/) (2017)
- [Deep convolutional Gaussian processes](https://mlsanity.com/paper/1810.03052/) (2018)
- [Bayesian Image Classification with Deep Convolutional Gaussian Processes](https://mlsanity.com/paper/1902.05888/) (2019)
- [Graph Convolutional Gaussian Processes](https://mlsanity.com/paper/1905.05739/) (2019)
- [Extrinsic Gaussian processes for regression and classification on manifolds](https://mlsanity.com/paper/1706.08757/) (2017)
- [Gaussian Process Landmarking on Manifolds](https://mlsanity.com/paper/1802.03479/) (2018)
- [Facial Keypoints Detection](https://mlsanity.com/paper/1710.05279/) (2017)

## Built on by (most cited first, 1 of 1)

- [Improved Prediction of Beta-Amyloid and Tau Burden Using Hippocampal Surface Multivariate Morphometry Statistics and Sparse Coding](https://mlsanity.com/paper/2211.05235/) (2022)

## Abstract (arXiv)

Bayesian learning with Gaussian processes demonstrates encouraging regression and classification performances in solving computer vision tasks. However, Bayesian methods on 3D manifold-valued vision data, such as meshes and point clouds, are seldom studied. One of the primary challenges is how to effectively and efficiently aggregate geometric features from the irregular inputs. In this paper, we propose a hierarchical Bayesian learning model to address this challenge. We initially introduce a kernel with the properties of geometry-awareness and intra-kernel convolution. This enables geometrically reasonable inferences on manifolds without using any specific hand-crafted feature descriptors. Then, we use a Gaussian process regression to organize the inputs and finally implement a hierarchical Bayesian network for the feature aggregation. Furthermore, we incorporate the feature learning of neural networks with the feature aggregation of Bayesian models to investigate the feasibility of jointly learning on manifolds. Experimental results not only show that our method outperforms existing Bayesian methods on manifolds but also demonstrate the prospect of coupling neural networks with Bayesian networks.