Variational Geometric Information Bottleneck: Learning the Shape of Understanding
This work addresses the challenge of creating data-efficient and interpretable representations for machine learning, though it is incremental as it builds on existing information bottleneck and geometric methods.
The paper tackles the problem of learning representations that balance informativeness and geometric simplicity by proposing the Variational Geometric Information Bottleneck (V-GIB), which unifies mutual-information compression with curvature regularization, resulting in robust performance and substantial gains in interpretive efficiency across datasets like CIFAR-10.
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.