Part VI — Crossing Embedding Spaces

One bridge, eight verdicts

Take the linear A→B bridge from Chapter 19. Run every preservation metric on the same held-out set:

cosine preservation (mean cos to native target)      0.74
top-1 preservation (native NN stays NN)               0.61
top-10 preservation                                   0.88
Recall@1  (vs labels, translated query)               0.58   native: 0.63
Recall@10                                             0.71   native: 0.77
MRR                                                   0.64   native: 0.70
nDCG@10                                               0.75   native: 0.81
order preservation (Kendall τ on result lists)        0.66
cluster preservation (ARI vs native clustering)       0.80
hard-negative agreement                               0.41

Ten numbers, not one. A stakeholder who asked “did the translation work?” gets a table, and the table says: coarse structure preserved, fine order degraded, hard distinctions near the source encoder’s own native performance.

This is also where the universal-geometry conjecture (Chapter 18) gets tested rather than asserted. Unpaired translators reach high cosine and top-1 on the coarse metrics; the published evidence stops there. The preservation profile is exactly the instrument for asking the next question: when a translation is aligned well on cosine and retrieval, did negation stay distinct from paraphrase? Did A acquired B stay distinct from B acquired A? Did a calibrated threshold keep its false-accept rate? Chapter 21’s job is to turn “the geometry is universal” from a slogan into a per-property profile.

Which preservation metric answers which question — and what does a good score on one metric fail to guarantee?

The metrics, and what each certifies

• Cosine / MSE to target — reconstruction fidelity. Certifies: translated vectors sit near where native encoding would put them. Does not certify: rankings, because small offsets reorder near-ties.
• Top-1 preservation — does the single nearest native neighbor survive translation? Strict; sensitive to the crowded regions (Chapter 6). A low number here with a high top-10 means “right neighborhood, wrong exact winner.”
• Top-k / neighbor overlap — does the local set survive? The workhorse metric for retrieval and dedup.
• Recall@k, MRR, nDCG@k — task metrics computed with the translated query (or translated corpus) against labels. Certifies end-to-end retrieval quality. Compare against the native target number, not against 1.0 — that is the reference point the translation is trying to reach.
• Order / rank preservation (Kendall τ, Spearman) — does the sequence of results survive, not just the set? Matters when position is consumed (a model reading results in order, a UI ranking).
• Cluster preservation (ARI, NMI, V-measure) — do the same items group together after translation? Certifies clustering, topic organization, routing. Can be high while order preservation is low.
• Hard-negative agreement — do the fine distinctions survive? Almost always the lowest number, and usually close to the source encoder’s own native score on the same hard negatives (Chapter 16). See the note below on what that does and does not bound.
• Relation preservation (per RELATE relation type) — for each labeled relation (paraphrase, entailment, negation, contradiction, relation-swap, temporal-mismatch, partial-support), does the translated space keep the per-relation cosine margin that native B has? A single averaged “hard-negative agreement” hides which kind of distinction was lost. Reported as a vector, one entry per relation. This is the metric that turns the universal-geometry conjecture into a scoped verdict. Measured (row 3.7): for a MiniLM → mpnet bridge the paraphrase-minus-negation cosine gap went from +0.033 native to −0.107 bridged — the map did not just weaken the polarity distinction, it inverted it (pulling negation +0.05 closer and pushing paraphrase −0.09 away), while temporal-mismatch pairs were pulled +0.17 closer. Coarse relations (equivalent, topic-related) held. The universal-geometry conjecture is a claim about the coarse relations only.
• Threshold / calibration transfer — does an operating point calibrated on native B hold for translated-A vectors? Usually no without recalibration; report the score-distribution shift (the change in FAR/FRR at the native operating point) explicitly.

What the source encoder does and does not bound

It is tempting to call the source encoder’s hard-negative score a “ceiling” the bridge cannot pass. That conflates two different quantities:

• Information content. A deterministic map cannot recover information that is genuinely absent from the source vectors (the data-processing inequality). If the source encoder collapsed X is the capital and X is not the capital to the same point, no T separates them. This is a real limit.
• The source-native score. The source encoder’s cosine score on hard negatives is a weaker quantity. Information can be present in a representation but inaccessible to naive cosine — the probing and concept-erasure literature shows this directly. A supervised bridge (fitted with a contrastive objective on hard-negative triples, not just anchor regression) can linearly reweight axes so a present-but-hidden distinction becomes cosine-accessible in the target — and so, in principle, exceed the source-native score without adding information.

So: report hard-negative agreement against the source-native score as a reference point, not a proven maximum. The true information-theoretic limit is not directly measurable. When a bridge sits at the source-native score, the honest statement is “the bridge did not lose the accessible distinction,” not “the bridge hit a ceiling.” Row 3.6 confirmed the reference point is beatable: a supervised contrastive bridge (MiniLM → mpnet) reached 1.00 structured-hard-negative accuracy on held-out entities against a 0.95 source-native score.

Reading the pattern, not the number

Characteristic patterns and what they mean:

Baselines you must include

• Native target — the reference point (not a proven maximum; see the source-encoder note above). Every metric reported as (bridge, native).
• Random map — a bridge with a random W. If your bridge is not far above this, it learned nothing.
• Identity / no map — feeding space-A vectors directly to space B (Chapter 16’s ~0.02 cosine). The floor.
• Round-trip — A→B→A. How much is lost going and coming back? Bounds composition (Chapter 20).

Demonstration: comparing two bridges by pattern

MEASURED on RELATE v0.1, Wave 3 rows 3.4, 3.6, 3.7, 3.9 — artifacts under experiments/embeddings-from-first-principles/wave3/artifacts/. all-mpnet-base-v2 → bge-large-en-v1.5, anchors from split_entity:train, evaluated on held-out entities.

The full preservation profile, three linear/nonlinear bridges, all fitted on the same anchors:

property                          Procrustes   ridge   MLP (2x512)
coordinate reconstruction cos        0.40       0.79      0.75
10-NN neighborhood overlap           0.73       0.68      0.67
retrieval nDCG@10 ratio              0.94       0.77      0.76
rank-triplet agreement               0.73       0.79      0.73
calibration threshold transfer       0.98       0.83      0.83
relation-order correlation           0.91       0.83      0.89
hard-negative margin ratio           0.61       0.98      0.95

MEASURED: there is no bridge that wins every column. Procrustes (rotation only) keeps retrieval, calibration, and relation ordering best but discards absolute position (reconstruction 0.40) and loses hard-negative margin. Ridge reconstructs position and keeps the hard-negative margin but scrambles more retrieval near-ties. The MLP does not lead on anything — it lands between the two linear maps at the cost of a training run and seed variance. The right bridge is the one whose strong columns match the consumer, and the preservation profile — not one headline cosine — is what tells you which that is.

Two more measurements the profile depends on:

• The “source ceiling” is beatable. A supervised bridge (contrastive on hard-negative triples, all-MiniLM-L6-v2 → all-mpnet-base-v2) reached 1.00 structured-hard-negative accuracy on held-out entities against a source-native score of 0.95 (row 3.6). The source-native score is a reference point, not a proven maximum — it can be exceeded when the distinction is present in the source vectors but not cosine-accessible.
• Round trips lose a little each way. mpnet → bge → mpnet (ridge both directions) returns to cosine 0.74 vs a single forward hop’s 0.79 — composition is modestly lossy, and every extra hop compounds it (row 3.9).

What this chapter establishes and what it does not

Establishes: eight-plus preservation metrics and what each certifies and fails to certify; that a bridge produces a profile, not a verdict; the mandatory baselines (native reference, random floor, identity floor, round-trip); that a deterministic map cannot recover genuinely-absent information but the source-native score is a reference point, not a proven maximum; that calibration rarely transfers.

Does not establish: a single “preservation score” (there isn’t one), or that a bridge passing retrieval metrics is safe for all uses (it is scoped, per Chapter 20). It establishes how to read the profile and choose a bridge by the metric that matches the consumer.

Lab 21: the preservation profile

PROPOSED, not executed.

Setup. A fitted bridge, held-out anchors, labeled retrieval eval with a hard-negative subset, a clustering labeling.

Task.

1. Compute all preservation metrics, each as (bridge, native).
2. Compute the three baselines (random map, identity, round-trip).
3. Classify your result into one of the characteristic patterns above.
4. Given your actual consumer, pick the deciding metric and state whether the bridge passes.

Optional extension: fit a supervised bridge (contrastive objective on hard-negative triples) and check whether its hard-negative agreement exceeds the source-native score — if it does, “ceiling” was the wrong word.

Success criterion. A filled profile, a named pattern, and a pass/fail against the one metric your consumer actually depends on — with the native reference and random floor shown so the numbers have scale.

Companion component: the preservation profile report

preservation_profile(bridge, eval_sets):
  metrics:      { name: {bridge, native, random, round_trip} }
  relation_preservation:   { paraphrase: .., entailment: .., negation: .., contradiction: ..,
                             relation_swap: .., temporal_mismatch: .., partial_support: .. }
                           # per-relation margin retained vs native — the universal-geometry probe
  calibration_preservation: { far_delta, frr_delta } at the native operating point
  pattern:      <one of the characteristic patterns>
  source_native_score: hard_negative_agreement of the source encoder (reference, not a ceiling)
  null_map_baseline: same metrics for T(x)=x   # if close to the bridge, the spaces were pre-aligned
  consumer_metric: <the deciding metric for the registered consumer>
  verdict:      PASS | FAIL  (against consumer_metric only)
  notes:        "calibration does NOT transfer; recalibrate per scope"

The Observatory generates this whenever a bridge is evaluated and uses verdict plus consumer_metric to populate the bridge’s usable_for (Chapter 20). A bridge gets relation_tasks on its usable_for list only when relation_preservation clears a bar for the relations the consumer cares about — by default it does not.

Failure modes

• Reporting one preservation number. There is no single one; report the profile.
• Comparing to 1.0 instead of native. The bridge is trying to reach the target encoder’s native performance, not perfection.
• No random-map baseline. You cannot tell “learned a lot” from “learned a little.”
• Assuming retrieval preservation implies calibration preservation. It does not; the score distribution shifted.
• Treating the source-native score as a hard ceiling. A deterministic map cannot add information, but a supervised map can expose a present-but-cosine-hidden distinction — so a bridge below the source-native score has likely lost something, while a bridge at or above it has not hit a wall.
• Choosing a bridge by cosine when the consumer reads ranked lists. Use order preservation.

What this chapter established

• Eight-plus preservation metrics — cosine, top-1, top-k, Recall@k, MRR, nDCG, order, cluster, hard-negative, per-relation preservation, calibration transfer — and what each certifies and misses.
• A bridge yields a profile, not a verdict; characteristic patterns map to consumer suitability.
• Per-relation preservation turns the universal-geometry conjecture into a scoped verdict: a translation aligned well on cosine and retrieval can still lose negation, relation direction, or calibration, and the profile says which.
• Mandatory baselines: native reference, random floor, identity floor, round-trip, null map (T(x)=x — a good null-map score means the spaces were pre-aligned and the bridge’s numbers are inflated).
• A deterministic map cannot recover genuinely-absent information; the source-native score is a reference point, not a proven ceiling (a supervised map can expose a present-but-hidden distinction). Calibration rarely transfers.
• On RELATE (row 3.4), no bridge won every column: Procrustes kept retrieval, calibration, and relation ordering best while discarding absolute position; ridge reconstructed position and kept the hard-negative margin but scrambled more retrieval near-ties; the MLP led on nothing. The right bridge is the one whose strong columns match the consumer — the profile, not one headline cosine, is what says which.
• The preservation profile report, feeding the bridge’s usable_for.

Next

Part VI translated between spaces of similar size and measured what survived. Part VII asks the same question of two more transformations: what survives when a document is compressed into a summary (Chapter 22), and what survives when a semantic edit is applied as a geometric operator (Chapter 23)? Both answer with a preservation profile.