On Linear Interpolation in the Latent Space of Deep Generative Models

TL;DR

This paper investigates the geometry of latent spaces in deep generative models, revealing that linear interpolations often deviate from geodesics, and proposes methods to compare and improve interpolation quality.

Contribution

It introduces a geometric framework to analyze and compare linear interpolations in latent spaces, emphasizing the importance of the output space and extending the pull-back metric.

Findings

Linear interpolations can significantly deviate from geodesics in latent space.

Shorter interpolation curves can be found using the pull-back metric.

Choosing the appropriate output space affects the quality of interpolations.

Abstract

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.