Variational Geometric Information Bottleneck: Learning the Shape of Understanding

TL;DR

This paper introduces a geometric information bottleneck framework that balances informativeness and geometric simplicity, leading to more efficient and interpretable representations with theoretical guarantees and practical benefits.

Contribution

It formalizes understanding as a trade-off between information and geometric complexity, deriving bounds and introducing a variational estimator for geometry-aware learning.

Findings

Generalization error scales with intrinsic dimension.

Curvature controls approximation stability.

Curvature-aware encoders perform well with limited data.

Abstract

We propose a unified information-geometric framework that formalizes understanding in learning as a trade-off between informativeness and geometric simplicity. An encoder phi is evaluated by U(phi) = I(phi(X); Y) - beta * C(phi), where C(phi) penalizes curvature and intrinsic dimensionality, enforcing smooth, low-complexity manifolds. Under mild manifold and regularity assumptions, we derive non-asymptotic bounds showing that generalization error scales with intrinsic dimension while curvature controls approximation stability, directly linking geometry to sample efficiency. To operationalize this theory, we introduce the Variational Geometric Information Bottleneck (V-GIB), a variational estimator that unifies mutual-information compression and curvature regularization through tractable geometric proxies such as the Hutchinson trace, Jacobian norms, and local PCA. Experiments across synthetic manifolds, few-shot settings, and real-world datasets (Fashion-MNIST, CIFAR-10) reveal a robust information-geometry Pareto frontier, stable estimators, and substantial gains in interpretive efficiency. Fractional-data experiments on CIFAR-10 confirm that curvature-aware encoders maintain predictive power under data scarcity, validating the predicted efficiency-curvature law. Overall, V-GIB provides a principled and measurable route to representations that are geometrically coherent, data-efficient, and aligned with human-understandable structure.