Data-Informed Geometric Space Selection

Abstract

Geometric representation learning (e.g., hyperbolic and spherical geometry) has proven to be efficacious in solving many intricate machine learning tasks. The fundamental challenge of geometric representation learning lies in aligning the inherent geometric bias with the underlying structure of the data, which is a rarely explored topic in the literature. Existing methods heavily rely on heuristic assumptions on the data structure to decide the type of geometry to be adopted, which often leads to suboptimal performance. This work aims to automate the alignment process via a data-informed strategy such that we optimize model performance with minimal overhead. Specifically, a sparse gating mechanism is employed to enable each input data point $\mathit{p}$ to select $K$ geometric spaces from a given candidate geometric space pool with $N$ ($K<N$) spaces of different geometry. The selected $K$ spaces are then tightly integrated to formulate a Cartesian product space, which is leveraged to process this input data $\mathit{p}$. In doing so, each input data is processed by the spaces it selected with maximum specialization. We empirically show that this method can effectively align data and spaces without human interventions and further boost performance on real-world tasks, demonstrating its potential in eliciting the expressive power of geometric representations and practical usability.