nucleation

LIT-1808: Unsupervised Hierarchy Matching with Optimal Transport over Hyperbolic Spaces

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Properties5
StatusActive — read in full 2026-10-29 (NOTE-1323); worth reading as a case where the arbitrariness between two learned representations is not a group action at all: embeddings of one hierarchy from different seeds differ non-isometrically, and only a non-linear registration recovers part of the correspondence (THEORY-1136). Its matching uses a symmetric distance and does not preserve the hierarchy's direction.
Version2
Tagsmathematics, representation-learning, anthology-candidate
Cited as “paper” byNOTE-1323 — Alvarez-Melis et al. — Hyperbolic OT hierarchy matching
Cited as “source” byTHEORY-1136 — Poincaré embeddings of one hierarchy from different random seeds differ by more than an isometry of the hyperbolic ball, with branches placed differently and pairwise distances changed, so aligners that allow only a global orthogonal or Möbius map recover almost none of the correspondence, while a non-linear registration by a hyperbolic neural network, trained through a Sinkhorn divergence and started from a pretraining step, recovers a substantial part of it but finds the exact match for under a quarter of items even when the two hierarchies are identical

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Alvarez-Melis, D., Mroueh, Y., & Jaakkola, T. S. (2020). Unsupervised hierarchy matching with optimal transport over hyperbolic spaces. AISTATS 2020, PMLR 108, 1606–1617.

Read in full in NOTE-1323.

Why this is here

A6’s point: entailment, authority and obligation are directed and hierarchical, and symmetric distances do not carry them; the geometry should fit the relations to be preserved. Hyperbolic distance is still symmetric, so it is a partial answer. It bears on normative transport (QUESTION-130). The reading bears this out and narrows it. The hierarchy’s direction enters only through the embedding; the transport cost is the symmetric hyperbolic distance, and the registration is free to break ancestry, so nothing preserves the order on matched items. The paper also does not fit the three-way distinction of LIT-1805: what it registers is a seed-dependent, non-isometric warp, absorbed by one non-linear map of the whole space, which has no frames and no holonomy.

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