Elastic Representations via Hyperbolic Geometry

Learned representations (or embeddings) are the foundation of modern machine learning systems, yet they are typically trained as fixed-size embeddings, without accounting for varying downstream resource or task constraints. Matryoshka Representation Learning (MRL) addresses this limitation by learning nested representations, but requires costly end-to-end retraining and does not support post hoc expansion, while Contrastive Sparse Representations (CSR) reduce training overhead via lightweight adaptors but operate at a 4x larger dimensionality. Additionally, all prior works restrict their framework towards compression only. In this work, we introduce \textbf{H}yperbolic \textbf{E}lastic \textbf{R}epresentation \textbf{L}earning (HERL), a framework for learning dimension-adaptive embeddings (compression and expansion) through a lightweight adaptor without retraining the base encoder. Our key idea is to exploit the exponential capacity of hyperbolic geometry by learning a non-linear mapping from the encoder space to hyperbolic space. We train a simple MLP to downsample the representations before projecting them onto a Lorentzian and learn adaptive representations that respect the Lorentzian geometry. Our approach is model-agnostic and effective in both supervised and unsupervised settings. Through extensive experiments across image and text modalities, we show that HERL preserves downstream performance across a wide range of embedding sizes, exhibits smooth interpolation between dimensions, and achieves these benefits at a fraction of the computational cost.