Abstract
Hyperbolic embeddings are widely used in deep learning to represent hierarchical data, and a substantial body of work reports empirical advantages over Euclidean representations. In recent years, however, a critical literature has accumulated suggesting that some of these advantages reflect poorly tuned baselines or implicit regularization rather than geometry itself. The present paper examines this question through a series of controlled experiments designed to separate the regimes in which hyperbolicity is expected to help from those in which it is not.On the theoretical side, we record a simple invariance: any diffeomorphism, including the Poincar\'e exponential map, preserves the closure of a dense function class in the uniform topology on compact subsets of the latent space. This indicates that a purely capacity-based advantage of hyperbolic layers over Euclidean layers of matched parameter count is not available through reparametrization alone, and that any observed empirical difference on flat data must be sought in optimization-theoretic or regularization-based mechanisms rather than in approximation theory.On the experimental side, we report four findings. First, a positive control on a synthetic perfect binary tree shows that a hyperbolic encoder can preserve the tree metric strictly better than a Euclidean encoder of matched capacity, with the advantage growing monotonically with tree depth: from roughly three hundredths of a point of Spearman correlation at depth five to roughly nine hundredths at depth seven, with p-values below one part in ten thousand. Second, a positive control on real biological data---the seed trie of the Huesken siRNA library, a balanced four-ary tree of depth seven constructed from the seven-nucleotide seed regions of two thousand three hundred and sixty one siRNAs---shows the same qualitative advantage: roughly five to nine hundredths of a point of Spearman correlation over a capacity-matched Euclidean encoder, again with p-values below one part in ten thousand. Third, on the secondary-structure tree of a real mRNA (PCSK9, of length three thousand six hundred and thirty seven nucleotides and tree diameter three hundred and twenty seven), the hyperbolic encoder either fails numerically without target rescaling or, once target distances are rescaled to match the well-conditioned regime, ties with a Euclidean encoder of matched capacity. Fourth, on the siRNA efficacy prediction task itself (PCSK9, one hundred and one samples; Huesken, two thousand three hundred and sixty one samples), the hyperbolic layer shows no advantage specific to its geometry: the modest gain observed on Huesken structural features is reproduced by a non-hyperbolic radial projection into the unit ball, and is not reproduced by the element-wise hyperbolic tangent or by clipping.We read these observations as evidence that hyperbolic layers are a specialized tool: they help on balanced trees of moderate depth, whether synthetic or real, but not on long thin biological hierarchies and not on effectively flat tasks such as siRNA efficacy prediction. All four trees studied here are exact tree metrics and therefore have vanishing Gromov hyperbolicity; the relevant diagnostic for whether hyperbolicity helps is instead the aspect ratio of the tree, defined as the ratio of its depth to its diameter. This refines the standard picture and suggests a set of practical controls---radial, capacity, shape, and stability---that any claim of a geometric advantage should report.