# Learning Weighted Submanifolds with Variational Autoencoders and Riemannian Variational Autoencoders

arXiv: https://arxiv.org/abs/1911.08147 · HTML: https://mlsanity.com/paper/1911.08147/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Nov 2019.

## Built on by (most cited first, 14 of 14)

- [Data Augmentation in High Dimensional Low Sample Size Setting Using a Geometry-Based Variational Autoencoder](https://mlsanity.com/paper/2105.00026/) (2021)
- [Manifold Integrated Gradients: Riemannian Geometry for Feature Attribution](https://mlsanity.com/paper/2405.09800/) (2024)
- [Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structures](https://mlsanity.com/paper/2407.09468/) (2024)
- [Multi-manifold Attention for Vision Transformers](https://mlsanity.com/paper/2207.08569/) (2022)
- [RDA-INR: Riemannian Diffeomorphic Autoencoding via Implicit Neural Representations](https://mlsanity.com/paper/2305.12854/) (2023)
- [Riemann2: Learning Riemannian Submanifolds from Riemannian Data](https://mlsanity.com/paper/2503.05540/) (2025)
- [Recent Advances in Deep Learning Techniques for Face Recognition](https://mlsanity.com/paper/2103.10492/) (2021)
- [Data-driven Nonlinear Parametric Model Order Reduction Framework using Deep Hierarchical Variational Autoencoder](https://mlsanity.com/paper/2307.06816/) (2023)
- [Staying the course: Locating equilibria of dynamical systems on Riemannian manifolds defined by point-clouds](https://mlsanity.com/paper/2204.10413/) (2022)
- [Riemannian Geometry-Preserving Variational Autoencoder for MI-BCI Data Augmentation](https://mlsanity.com/paper/2603.10563/) (2026)
- [Testing geometric representation hypotheses from simulated place cell recordings](https://mlsanity.com/paper/2211.09096/) (2022)
- [Aligning the Unseen in Attributed Graphs: Interplay between Graph Geometry and Node Attributes Manifold](https://mlsanity.com/paper/2601.22806/) (2026)
- [Geometry-Preserving Neural Architectures on Manifolds with Boundary](https://mlsanity.com/paper/2602.03082/) (2026)
- [An Elastic Shape Variational Autoencoder for Skeleton Pose Trajectories](https://mlsanity.com/paper/2605.09231/) (2026)

## Abstract (arXiv)

Manifold-valued data naturally arises in medical imaging. In cognitive neuroscience, for instance, brain connectomes base the analysis of coactivation patterns between different brain regions on the analysis of the correlations of their functional Magnetic Resonance Imaging (fMRI) time series - an object thus constrained by construction to belong to the manifold of symmetric positive definite matrices. One of the challenges that naturally arises consists of finding a lower-dimensional subspace for representing such manifold-valued data. Traditional techniques, like principal component analysis, are ill-adapted to tackle non-Euclidean spaces and may fail to achieve a lower-dimensional representation of the data - thus potentially pointing to the absence of lower-dimensional representation of the data. However, these techniques are restricted in that: (i) they do not leverage the assumption that the connectomes belong on a pre-specified manifold, therefore discarding information; (ii) they can only fit a linear subspace to the data. In this paper, we are interested in variants to learn potentially highly curved submanifolds of manifold-valued data. Motivated by the brain connectomes example, we investigate a latent variable generative model, which has the added benefit of providing us with uncertainty estimates - a crucial quantity in the medical applications we are considering. While latent variable models have been proposed to learn linear and nonlinear spaces for Euclidean data, or geodesic subspaces for manifold data, no intrinsic latent variable model exists to learn nongeodesic subspaces for manifold data. This paper fills this gap and formulates a Riemannian variational autoencoder with an intrinsic generative model of manifold-valued data. We evaluate its performances on synthetic and real datasets by introducing the formalism of weighted Riemannian submanifolds.