# On Linear Interpolation in the Latent Space of Deep Generative Models

arXiv: https://arxiv.org/abs/2105.03663 · HTML: https://mlsanity.com/paper/2105.03663/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 14 references that are in the map.
- First posted on arXiv 8 May 2021.

## Builds on (most cited first, 9 of 9)

- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Analyzing and Improving the Image Quality of StyleGAN](https://mlsanity.com/paper/1912.04958/) (2019)
- [Interpreting the Latent Space of GANs for Semantic Face Editing](https://mlsanity.com/paper/1907.10786/) (2019)
- [Unsupervised Discovery of Interpretable Directions in the GAN Latent Space](https://mlsanity.com/paper/2002.03754/) (2020)
- [Latent Space Oddity: on the Curvature of Deep Generative Models](https://mlsanity.com/paper/1710.11379/) (2017)
- [Inverting Deep Generative models, One layer at a time](https://mlsanity.com/paper/1906.07437/) (2019)
- [Fast and Robust Shortest Paths on Manifolds Learned from Data](https://mlsanity.com/paper/1901.07229/) (2019)
- [Geodesic Clustering in Deep Generative Models](https://mlsanity.com/paper/1809.04747/) (2018)
- [Geodesics in fibered latent spaces: A geometric approach to learning correspondences between conditions](https://mlsanity.com/paper/2005.07852/) (2020)

## Built on by (most cited first, 8 of 8)

- [Action-Minimization Meets Generative Modeling: Efficient Transition Path Sampling with the Onsager-Machlup Functional](https://mlsanity.com/paper/2504.18506/) (2025)
- [Geometry-Aware Generative Autoencoders for Warped Riemannian Metric Learning and Generative Modeling on Data Manifolds](https://mlsanity.com/paper/2410.12779/) (2024)
- [Exploring the Latent Space of Autoencoders with Interventional Assays](https://mlsanity.com/paper/2106.16091/) (2021)
- [Self Supervised Networks for Learning Latent Space Representations of Human Body Scans and Motions](https://mlsanity.com/paper/2411.03475/) (2024)
- [Optimized latent-code selection for explainable conditional text-to-image GANs](https://mlsanity.com/paper/2204.12678/) (2022)
- [Generation of Geodesics with Actor-Critic Reinforcement Learning to Predict Midpoints](https://mlsanity.com/paper/2407.01991/) (2024)
- [(Deep) Generative Geodesics](https://mlsanity.com/paper/2407.11244/) (2024)
- [Understanding Variational Autoencoders with Intrinsic Dimension and Information Imbalance](https://mlsanity.com/paper/2411.01978/) (2024)

## Abstract (arXiv)

The underlying geometrical structure of the latent space in deep generative models is in most cases not Euclidean, which may lead to biases when comparing interpolation capabilities of two models. Smoothness and plausibility of linear interpolations in latent space are associated with the quality of the underlying generative model. In this paper, we show that not all such interpolations are comparable as they can deviate arbitrarily from the shortest interpolation curve given by the geodesic. This deviation is revealed by computing curve lengths with the pull-back metric of the generative model, finding shorter curves than the straight line between endpoints, and measuring a non-zero relative length improvement on this straight line. This leads to a strategy to compare linear interpolations across two generative models. We also show the effect and importance of choosing an appropriate output space for computing shorter curves. For this computation we derive an extension of the pull-back metric.