Geometry as a Missing Axis of Representation Quality: The Variational Geometric Information Bottleneck under Data Scarcity

Abstract:We study latent geometry as an explicit component of representation quality in data-scarce learning. For an encoder (\phi), we define (Q_{\beta,\gamma}(\phi)=I(\phi(X);Y)-\beta\mathcal C(\phi)-\gamma d_{\mathrm{int}}(\phi)), combining task-relevant information with penalties for curvature and intrinsic latent dimension. Thus geometry becomes part of the bottleneck criterion, not only a post hoc diagnostic. Under smooth-manifold, loss-transfer, and estimator-concentration assumptions, we derive non-asymptotic low-label generalization bounds where intrinsic dimension and covering complexity enter explicitly. We characterize the information--geometry frontier and prove empirical-surrogate consistency. The analysis links encoder geometry to learning through latent covering numbers, loss-class entropy, and uniform deviation. We instantiate the theory as \texttt{V-GIB}, adding curvature and dimension penalties to variational bottleneck training. Real low-label benchmarks compare \texttt{V-GIB} with ERM, VIB, and ablations across (1%)--(20%) label fractions. Results show improved performance and reduced geometric complexity in several regimes, especially FashionMNIST and CIFAR-10, while confirming that no fixed regularizer is universally dominant.

Theorem 6 treats repeated curvature probes sharing the same observations as independent, invalidating the stated concentration rate. The theoretical curvature measure also needs to be reconciled with the implemented penalty

Machine Learning (cs.LG)

arXiv:2511.02496 [cs.LG]

(or arXiv:2511.02496v3 [cs.LG] for this version)