# On Explicit Curvature Regularization in Deep Generative Models

arXiv: https://arxiv.org/abs/2309.10237 · HTML: https://mlsanity.com/paper/2309.10237/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 31 references that are in the map.
- First posted on arXiv 19 Sep 2023.

## Builds on (most cited first, 4 of 4)

- [NTU RGB+D: A Large Scale Dataset for 3D Human Activity Analysis](https://mlsanity.com/paper/1604.02808/) (2016)
- [Optimizing the Latent Space of Generative Networks](https://mlsanity.com/paper/1707.05776/) (2017)
- [Flows for simultaneous manifold learning and density estimation](https://mlsanity.com/paper/2003.13913/) (2020)
- [Learning Flat Latent Manifolds with VAEs](https://mlsanity.com/paper/2002.04881/) (2020)

## Built on by (most cited first, 13 of 13)

- [Metric Flow Matching for Smooth Interpolations on the Data Manifold](https://mlsanity.com/paper/2405.14780/) (2024)
- [MMP++: Motion Manifold Primitives with Parametric Curve Models](https://mlsanity.com/paper/2310.17072/) (2023)
- [SymNoise: Advancing Language Model Fine-tuning with Symmetric Noise](https://mlsanity.com/paper/2312.01523/) (2023)
- [Gamma-VAE: Curvature regularized variational autoencoders for uncovering emergent low dimensional geometric structure in high dimensional data](https://mlsanity.com/paper/2403.01078/) (2024)
- [Motion Manifold Flow Primitives for Task-Conditioned Trajectory Generation under Complex Task-Motion Dependencies](https://mlsanity.com/paper/2407.19681/) (2024)
- [Riemannian Generative Decoder](https://mlsanity.com/paper/2506.19133/) (2025)
- [(Deep) Generative Geodesics](https://mlsanity.com/paper/2407.11244/) (2024)
- [Differentiable Motion Manifold Primitives for Reactive Motion Generation under Kinodynamic Constraints](https://mlsanity.com/paper/2410.12193/) (2024)
- [Bayesian Distributional Models of Executive Functioning](https://mlsanity.com/paper/2510.00387/) (2025)
- [Assessing local deformation and computing scalar curvature with nonlinear conformal regularization of decoders](https://mlsanity.com/paper/2508.20413/) (2025)
- [LEGO: Latent-space Exploration for Geometry-aware Optimization of Humanoid Kinematic Design](https://mlsanity.com/paper/2604.08636/) (2026)
- [Geometric Decoupling: Diagnosing the Structural Instability of Latent](https://mlsanity.com/paper/2604.18804/) (2026)
- [Understanding and Improving Noisy Embedding Techniques in Instruction Finetuning](https://mlsanity.com/paper/2605.23171/) (2026)

## Abstract (arXiv)

We propose a family of curvature-based regularization terms for deep generative model learning. Explicit coordinate-invariant formulas for both intrinsic and extrinsic curvature measures are derived for the case of arbitrary data manifolds embedded in higher-dimensional Euclidean space. Because computing the curvature is a highly computation-intensive process involving the evaluation of second-order derivatives, efficient formulas are derived for approximately evaluating intrinsic and extrinsic curvatures. Comparative studies are conducted that compare the relative efficacy of intrinsic versus extrinsic curvature-based regularization measures, as well as performance comparisons against existing autoencoder training methods. Experiments involving noisy motion capture data confirm that curvature-based methods outperform existing autoencoder regularization methods, with intrinsic curvature measures slightly more effective than extrinsic curvature measures.