Minimax Rates for Hyperbolic Hierarchical Learning

We prove an exponential separation in sample complexity between Euclidean and hyperbolic representations for learning on hierarchical data under standard Lipschitz regularization. For depth-R hierarchies with branching factor m, we first establish a geometric obstruction for Euclidean space: any bounded-radius embedding forces volumetric collapse, mapping exponentially many tree-distant points to nearby locations. This necessitates Lipschitz constants scaling as exp(Omega(R)) to realize even simple hierarchical targets, yielding exponential sample complexity under capacity control. We then show this obstruction vanishes in hyperbolic space: constant-distortion hyperbolic embeddings admit O(1)-Lipschitz realizability, enabling learning with n = O(mR log m) samples. A matching Omega(mR log m) lower bound via Fano's inequality establishes that hyperbolic representations achieve the information-theoretic optimum. We also show a geometry-independent bottleneck: any rank-k prediction space captures only O(k) canonical hierarchical contrasts.