# Latent SDEs on Homogeneous Spaces

arXiv: https://arxiv.org/abs/2306.16248 · HTML: https://mlsanity.com/paper/2306.16248/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 65 references that are in the map.
- First posted on arXiv 28 Jun 2023.

## Builds on (most cited first, 30 of 32)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Simplified State Space Layers for Sequence Modeling](https://mlsanity.com/paper/2208.04933/) (2022)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [Multi-Time Attention Networks for Irregularly Sampled Time Series](https://mlsanity.com/paper/2101.10318/) (2021)
- [Cyclical Annealing Schedule: A Simple Approach to Mitigating KL Vanishing](https://mlsanity.com/paper/1903.10145/) (2019)
- [Latent ODEs for Irregularly-Sampled Time Series](https://mlsanity.com/paper/1907.03907/) (2019)
- [Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit](https://mlsanity.com/paper/1905.09883/) (2019)
- [Neural SDEs as Infinite-Dimensional GANs](https://mlsanity.com/paper/2102.03657/) (2021)
- [Dissecting Neural ODEs](https://mlsanity.com/paper/2002.08071/) (2020)
- [Neural Jump Stochastic Differential Equations](https://mlsanity.com/paper/1905.10403/) (2019)
- [Neural Rough Differential Equations for Long Time Series](https://mlsanity.com/paper/2009.08295/) (2020)
- [Set Functions for Time Series](https://mlsanity.com/paper/1909.12064/) (2019)
- [Modeling Irregular Time Series with Continuous Recurrent Units](https://mlsanity.com/paper/2111.11344/) (2021)
- [Neural SDE: Stabilizing Neural ODE Networks with Stochastic Noise](https://mlsanity.com/paper/1906.02355/) (2019)
- [Theoretical guarantees for sampling and inference in generative models with latent diffusions](https://mlsanity.com/paper/1903.01608/) (2019)
- [Hyperspherical Prototype Networks](https://mlsanity.com/paper/1901.10514/) (2019)
- [Gaussian Process Prior Variational Autoencoders](https://mlsanity.com/paper/1810.11738/) (2018)
- [Efficient and Accurate Gradients for Neural SDEs](https://mlsanity.com/paper/2105.13493/) (2021)
- [Recurrent Kalman Networks: Factorized Inference in High-Dimensional Deep Feature Spaces](https://mlsanity.com/paper/1905.07357/) (2019)
- [Verifying the Union of Manifolds Hypothesis for Image Data](https://mlsanity.com/paper/2207.02862/) (2022)
- [Deep Hyperspherical Learning](https://mlsanity.com/paper/1711.03189/) (2017)
- [Variational Autoencoder with Learned Latent Structure](https://mlsanity.com/paper/2006.10597/) (2020)
- [ODE2VAE: Deep generative second order ODEs with Bayesian neural networks](https://mlsanity.com/paper/1905.10994/) (2019)
- [Learning with Hyperspherical Uniformity](https://mlsanity.com/paper/2103.01649/) (2021)
- [The Power Spherical distribution](https://mlsanity.com/paper/2006.04437/) (2020)

## Built on by (most cited first, 16 of 16)

- [Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series](https://mlsanity.com/paper/2410.05602/) (2024)
- [In-Context Learning of Stochastic Differential Equations with Foundation Inference Models](https://mlsanity.com/paper/2502.19049/) (2025)
- [SING: SDE Inference via Natural Gradients](https://mlsanity.com/paper/2506.17796/) (2025)
- [Kuramoto Orientation Diffusion Models](https://mlsanity.com/paper/2509.15328/) (2025)
- [Generative Modeling of Neural Dynamics via Latent Stochastic Differential Equations](https://mlsanity.com/paper/2412.12112/) (2024)
- [Stochastic Optimal Control for Continuous-Time fMRI Representation Learning](https://mlsanity.com/paper/2502.04892/) (2025)
- [Explicit and Effectively Symmetric Schemes for Neural SDEs on Lie Groups](https://mlsanity.com/paper/2509.20599/) (2025)
- [HGAN-SDEs: Learning Neural Stochastic Differential Equations with Hermite-Guided Adversarial Training](https://mlsanity.com/paper/2512.20272/) (2025)
- [Anomaly Detection for Sparse and Irregular Multivariate Time Series with Latent SDEs](https://mlsanity.com/paper/2606.18898/) (2026)
- [Time-changed normalizing flows for accurate SDE modeling](https://mlsanity.com/paper/2312.14698/) (2023)
- [Neural Persistence Dynamics](https://mlsanity.com/paper/2405.15732/) (2024)
- [Improving the Noise Estimation of Latent Neural Stochastic Differential Equations](https://mlsanity.com/paper/2412.17499/) (2024)
- [The Stochastic Occupation Kernel (SOCK) Method for Learning Stochastic Differential Equations](https://mlsanity.com/paper/2505.11622/) (2025)
- [Physically-Constrained Mamba-SDE for Remaining Useful Life Prediction under Irregular Observations](https://mlsanity.com/paper/2606.01894/) (2026)
- [Liquid Gated Attention](https://mlsanity.com/paper/2608.30695/) (2026)
- [Continuous-Time Trajectory Generation from Discrete Observations with Stochasticity](https://mlsanity.com/paper/2609.32026/) (2026)

## Abstract (arXiv)

We consider the problem of variational Bayesian inference in a latent variable model where a (possibly complex) observed stochastic process is governed by the solution of a latent stochastic differential equation (SDE). Motivated by the challenges that arise when trying to learn an (almost arbitrary) latent neural SDE from data, such as efficient gradient computation, we take a step back and study a specific subclass instead. In our case, the SDE evolves on a homogeneous latent space and is induced by stochastic dynamics of the corresponding (matrix) Lie group. In learning problems, SDEs on the unit n-sphere are arguably the most relevant incarnation of this setup. Notably, for variational inference, the sphere not only facilitates using a truly uninformative prior, but we also obtain a particularly simple and intuitive expression for the Kullback-Leibler divergence between the approximate posterior and prior process in the evidence lower bound. Experiments demonstrate that a latent SDE of the proposed type can be learned efficiently by means of an existing one-step geometric Euler-Maruyama scheme. Despite restricting ourselves to a less rich class of SDEs, we achieve competitive or even state-of-the-art results on various time series interpolation/classification problems.