Three Geometries, One Taxonomy

The same 5,595 Google product categories, laid out three ways.Each panel shows the distance it actually returns.Reproducing Nickel & Kiela (2017) on a real product taxonomy, after The Geometry Mistake.

vector search research

Pick a category and its chain of parent categories gets drawn and labeled in all three panels. In the hyperbolic disk that chain runs straight out from the center. In the other two it scatters, because those spaces have run out of room to keep the levels apart. Hover any dot for its name, scroll to zoom, drag to pan. The Test Results tab carries the numbers, including the two approaches we tried that lost.

ctrl + scroll to zoom · drag to pan · double-click resets

Select a category

Click any dot, search above, or pick an example.

Regions plus everything else Solid dots are what you selected, plus its nearest neighbors. Hops are real steps apart in the taxonomy. Lower is better. Distances served live by Qdrant.

The dots are only positioned in 2D so you can look at them. Every distance in the table comes from the full vector, 5 dimensions for the two trained spaces and 384 for the text model, never from where a dot happens to sit.

These are runs against the Google Product Taxonomy, which is 5,595 categories, 7 levels deep, and about 6.4 children per node.

Both geometries got the same data and the same training, and we swept the learning rate separately for each one so neither got a handicap. The only thing that changed was the geometry.

01 Dimensions Needed

Dimensions	Flat	Hyperbolic
2	0.140	0.501
5	0.239	0.905
10	0.354	0.925
20	0.551	0.932
50	0.658	0.934

Hyperbolic is basically done at 5 dimensions. Everything after that is rounding. Flat keeps climbing all the way to 50 and still doesn't get there.

02 Against a Real Embedding

Representation	Dims	MAP
Hyperbolic	5	0.905
BGE-small, full path in the text	384	0.308
BGE-small, name only	384	0.115

Five numbers beat 384. The middle row is the one we didn't expect: we wrote the whole category path into the string, so the model was handed the hierarchy in plain English, and it still only got to 0.308.

03 Why It Wins

Siblings per node	1 to 2	3 to 7	8 to 20	21+
Hyperbolic 5d	0.932	0.909	0.901	0.944
Flat 5d	0.494	0.440	0.368	0.195
Flat 50d	0.593	0.543	0.458	0.464
Text 384d	0.320	0.247	0.182	0.134

This is the table that actually explains the rest of them. Flat space gets worse the wider the tree gets, and the text embedding gets worse faster. Hyperbolic sits around 0.9 whether a category has two siblings or fifty.

04 Where It Loses

Dimensions	Flat	Hyperbolic
5	0.457	0.539
10	0.627	0.532
20	0.806	0.540
50	0.855	0.541

Ask for the one direct parent rather than the whole ancestor set and flat wins from 10 dimensions up. Hyperbolic keeps the entire family line close together, so the real parent is nearly always in the top ten, but it's rarely first. The same thing happened on WordNet, so it's a property of the representation and not of this dataset.

05 Serving It in Qdrant

Two named vectors per category, the 384d text embedding and a 5d hyperbolic one. The hyperbolic vector is small enough to keep in the payload, which means acosh can work out the exact geodesic inside the query itself. Prefetch on meaning, rescore on structure, one request, nothing reranked in your application.

Pipeline	Mean hops	Within 3 hops	Semantic kept
Text only	44.0	0.347	0.795
Hybrid	15.2	0.599	0.752
Hyperbolic only	6.5	0.713	0.655

You get about 69% of the structural improvement and give up roughly 5% of the semantic relevance. Hyperbolic on its own is better structurally but drops 17%, so the blend is the one worth shipping.

Prefetch	Mean hops	Within 3 hops	Semantic	ms
10	43.0	0.357	0.797	11.6
25	23.1	0.507	0.771	15.5
50	16.8	0.573	0.751	22.2
100	13.2	0.618	0.735	34.7
200	13.3	0.635	0.720	61.2
400	13.6	0.651	0.706	113.2

Prefetch 100 is the sweet spot. Go deeper and you're paying double the latency for almost nothing, and mean hops stops improving anyway.

06 The Best Embedding Is the Worst Index

Learning rate	MAP	Norm spread	Recall at prefetch 50
1	0.301	139x	0.965
3	0.724	2,010x	0.789
10	0.905	14,400x	0.498
30	0.762	6,700x	0.383

Turning the learning rate up pushes points towards the edge of the ball. That's what makes the hierarchy come out well, and it's also what breaks prefetch, because the term that blows up lives right there at the boundary. So the setting that wins on the benchmark is the one you'd least want to serve. We only noticed because we trained the thing and then had to index it.

07 Earlier Runs

WordNet mammals	Hyperbolic got 0.949 at 5 dimensions against flat at 0.231. Flat needed 100 dimensions to reach 0.805. This is roughly what the paper reports, so the setup checks out.
WordNet nouns, 82k	Flat matched the paper. Our hyperbolic side came out under-trained because we used a bigger batch to fit it on a laptop, so we stopped the run rather than quote it.
The d+2 transform	It converts hyperbolic ranking into plain dot product, exactly. Brute force scores 0.984, and the same vectors through HNSW score 0.020. The transform leaves some vectors 600x bigger than others and the graph walk has nothing to follow.
Rescue attempts	Magnitude buckets got to 0.767 and clipping to 0.638. Both worked, and both still lost to plain rescoring at 0.868.
Geodesic comparator	0.894 against 0.745, at lower latency. The gap got wider as the data grew rather than narrower: +3, +6, +9, +15.
A trap	The comparator has to be symmetric. The obvious shortcut isn't, and it drops recall from 0.92 to 0.62 without any error at all. Cost us an hour.

08 What This Does Not Prove

Reconstruction trains and scores on the same edges, so it tells you what 5 dimensions can hold. It says nothing about guessing an edge the model never saw. We haven't done the held-out run yet.
The comparator test ran on an HNSW we wrote ourselves at Qdrant's defaults. It isn't Qdrant's engine, because no vector search engine lets you swap the comparator.
Latencies come from local mode, so the ratios are worth something and the milliseconds aren't.
Semantic relevance here just means embedding similarity. Nobody read the results to check.