nucleation

THEORY-825: Hyperbolic graph embedding models trained on task losses that never mention the hierarchy do not place a tree's root nearest the origin, and their distance from the origin orders hierarchical levels only partly

2 min read

Properties4
StatusProposed
Tagsrepresentation-learning, network-science, anthology-candidate
SourceLIT-871 — Hyperbolic Representation Learning: Revisiting and Advancing
ExtendsTHEORY-421 — In HGCN embeddings of two synthetic 3-ary trees trained for link prediction or node classification, distance from the origin is only a partial reading of hierarchical level: the root is not innermost (distance 1.8–3.3 against a minimum of 1.2–2.4) and level order is recovered for 69–75% of node pairs; a loss built only on pairwise distances leaves the origin arbitrary

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Source

Yang, Zhou, Ying, Chen & King (2023), LIT-871 (ICML 2023), read in full in NOTE-669. The general claim is the paper’s: after Figure 2 it says that “other hyperbolic models have demonstrated comparable results”, without showing them.

What was actually shown

What supports it is THEORY-421: for HGCN trained for link prediction or node classification on two synthetic 3-ary trees of eight levels, the root sits at distance 1.8–3.3 from the origin against a minimum of 1.2–2.4, and distance from the origin orders 69–75% of 5,000 random node pairs by level (Figure 2; Table 9). One architecture, two trees, no variance. The general thesis would have been in trouble if HGCN had put the root at the minimum and ordered levels nearly completely; it did not.

A reason to expect it more widely is the record’s own: a loss built only on pairwise hyperbolic distances is unchanged by any isometry, so it does not fix where the origin lies relative to the embedding (THEORY-421, re-derived in the promotion audit). Losses that read features at the origin’s tangent space, as HGCN’s classifier does, are not covered by that argument.

What this does not say

  • That other hyperbolic models were measured. The paper asserts it; none is shown. Poincaré embeddings, hyperbolic attention networks and entailment-cone models (THEORY-189), which encode hierarchy by angle as well as radius, are untested here.
  • That recentring fails. How much of the gap is only the arbitrary origin, which recentring on the centroid would remove, is not separated in the paper. The promotion condition asks for exactly that.
  • That the paper’s fix recovers hierarchy. Its outward-push loss gains most on the least tree-like benchmarks, and levels are never measured on the real graphs.

Connections

  • THEORY-421 is the measurement this generalizes, and is Active.
  • THEORY-196 gives the near-isometric tree embedding the geometry allows; this thesis says task training does not realise it by itself.
  • QUESTION-025: a space suited to a hierarchy does not show that a trained representation in it holds the hierarchy.

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