A Hyperbolic Perspective on Hierarchical Structure in Object-Centric Scene Representations

TL;DR

This paper demonstrates that projecting object-centric scene representations into hyperbolic space reveals latent hierarchical structures not visible in Euclidean space, with implications for improved scene understanding.

Contribution

It introduces a post-hoc hyperbolic projection pipeline for slot attention embeddings, uncovering hierarchical scene structures and analyzing the effects of hyperbolic curvature.

Findings

Hyperbolic projection reveals scene-object hierarchy in slot representations.

Coarse slots occupy greater hyperbolic depth than fine slots.

Different curvatures affect hierarchical separation and retrieval performance.

Abstract

Slot attention has emerged as a powerful framework for unsupervised object-centric learning, decomposing visual scenes into a small set of compact vector representations called \emph{slots}, each capturing a distinct region or object. However, these slots are learned in Euclidean space, which provides no geometric inductive bias for the hierarchical relationships that naturally structure visual scenes. In this work, we propose a simple post-hoc pipeline to project Euclidean slot embeddings onto the Lorentz hyperboloid of hyperbolic space, without modifying the underlying training pipeline. We construct five-level visual hierarchies directly from slot attention masks and analyse whether hyperbolic geometry reveals latent hierarchical structure that remains invisible in Euclidean space. Integrating our pipeline with SPOT (images), VideoSAUR (video), and SlotContrast (video), We find that hyperbolic projection exposes a consistent scene-level to object-level organisation, where coarse slots occupy greater manifold depth than fine slots, which is absent in Euclidean space. We further identify a "curvature--task tradeoff": low curvature ($c{=}0.2$) matches or outperforms Euclidean on parent slot retrieval, while moderate curvature ($c{=}0.5$) achieves better inter-level separation. Together, these findings suggest that slot representations already encode latent hierarchy that hyperbolic geometry reveals, motivating end-to-end hyperbolic training as a natural next step. Code and models are available at \href{https://github.com/NeeluMadan/HHS}{github.com/NeeluMadan/HHS}.