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Closed Forms and Synthetic Twins: Predicting Approximate Nearest Neighbor Recall from Embedding Statistics

Shmuel Herman

arXiv:2609.00364v1cs.IR

TL;DR

Embedding benchmarks evaluate exact retrieval, but production indexes approximate it, and post-hoc corrections can rewrite exact neighbors. This paper predicts index behavior from label-free embedding statistics using formulas and synthetic twins, and tests shaping score margins through training. Its forecasts are validated out of sample, while the task-frame cost of the learned margin lever is measured on a single sparsely judged corpus.

  • Problem

    Encoder benchmarks score exact top-k retrieval, while the effects of approximation and corpus-fitted corrections on that ranking are not captured by those benchmarks.

    Source: §1 · pp. 1–2

  • Method

    The paper maps label-free embedding statistics to index behavior, using closed-form models where possible and cluster-matched twins to simulate navigation indexes.

    Source: §1 · p. 2, §3.4 · p. 10

  • Results

    The geometry-to-recall account was validated out of sample across index families, with each family analyzed using the instrument its behavior required.

    Source: §10 · p. 31

  • Takeaways & Limitations

    Shaping score margins during training can raise recall across indexes without leaving corpus-fitted state in the serving path.

    Source: §8 · p. 29

    The margin lever’s task-frame price was measured on one corpus with sparse labels: 315 judged pairs.

    Source: §6 · p. 21, §9 · p. 30

Abstract

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Embedding models are trained and evaluated as if retrieval were exact; in production they serve behind approximate indexes -- HNSW, IVF, product quantization, or the fixed-dimensional encodings (FDEs) of late-interaction models -- whose behavior the encoder's benchmarks never see: one modern encoder recovers just 14% of its exact top-10 through its raw FDE index. Such failures surface only after an index is built, and the standard patches -- corpus-fitted transforms such as whitening -- must be fitted, stored, and refit as the corpus changes, and can silently rewrite what the encoder returns. This paper shows that index behavior is predictable before anything is built, from label-free statistics of the raw embeddings, through a ladder of instruments matched to what each index family consumes: (1) closed-form moment statistics for the fixed-grid quantizers (PQ, FDE); (2) simulation on a synthetic twin corpus -- cluster statistics made generative, on which any index, composed production systems included, can be built and tested -- for partition indexes; (3) size-extrapolated, lightly calibrated twins for graph indexes at million-document scale. Predictions land within 0.03 of measured recall on an unseen million-document corpus. The same geometry is trainable: targeting the one statistic no post-hoc transform can move -- the score margin -- lifts recall for every index family at once, at a small measured task cost. The result: index choice, correction pricing, and production recall forecast from one cheap measurement pass, on new corpora and new indexes alike; serving without per-corpus transform machinery, suited to continuously changing corpora; and a recall-compute frontier pushed by adapting encoders to geometry rather than coupling them to any single index.

1 Introduction

Embedding benchmarks measure exact retrieval, while production indexes approximate it, and post-hoc corrections can change the exact neighbors they return. The paper predicts index behavior from embedding geometry, prices correction trade-offs, and tests shaping score margins through training.

1 Introduction

  • GTE-ModernColBERT’s raw FDE recall@100 is 0.136, recovering just 14% of exact results—a failure standard encoder benchmarks do not reveal.The paper argues that approximation behavior depends on measurable embedding geometry across FDEs, quantizers, partitions, and graph indexes.Source: §1 · p. 2
  • A label-free statistic-to-behavior map assigns different governing statistics to index families, using synthetic twins where cluster structure prevents closed-form prediction.Moments alone do not determine the cluster and margin statistics needed for navigation-family predictions.Source: §1 · p. 3
  • On a held-out million-document corpus, the synthetic-twin forecast predicts index recall within 0.03.The paper presents this as the size-scaled extension of its label-free prediction framework.Source: §1 · p. 3
  • Corrective transforms trade index efficiency against exact-neighbor fidelity, so a transform helps only when its efficiency gain outweighs its fidelity cost.The balance varies with the index tax: rotations preserve exact rankings, while distortions can rewrite them.Source: §1 · pp. 2, 4
  • A trainable final layer nearly triples score margins and lifts recall for every index family at a measured 3–5-point task price.Unlike corpus-fitted corrections, this margin lever ships with the model and needs no per-corpus refitting.Source: §1 · p. 4

2 Related Work

Prior work studies anisotropy, index-aware training, and retrieval hardness, but the paper positions its contribution as jointly forecasting index behavior and transform costs across index families. It also uses structure-matched twins to attribute behavior and predict absolute recall.

Embedding anisotropy

  • Anisotropy research measures exact-scoring or task quality, whereas this paper also models how embedding geometry affects the approximation layer and separates semantic shifts from index effects.For example, whitening can help dense retrieval in one reference frame while rewriting 62% of true neighbors in this paper’s measurement.Source: §2 · p. 5

Hardness and local structure

  • Prior hardness work emphasizes graph and partition methods, observational measures, or worst-case constructions; this paper also covers fixed-grid quantization and validates held-out forecasts.Its twin hierarchy compares which levels of geometric structure matter rather than relying only on regression.Source: §2 · pp. 5–6

Index-aware training and index-side learning

  • Index-aware training couples representations to a specific index and corpus, while this paper identifies score margins as a transferable, label-free target that lifts every index family.The paper presents this statistic as a candidate target for existing co-training machinery.Source: §2 · p. 6

Positioning

  • The paper positions its framework as unifying label-free, out-of-sample forecasts across fixed-grid, partition, and graph indexes with transform pricing and synthetic-twin attribution.It claims no known prior work combines these four elements.Source: §2 · p. 6

3 The Prediction Framework: Instruments and the Assembled Ladder

The framework fixes a label-free reference ranking, measures statistics at resolutions matched to index families, and uses formulas or synthetic twins to forecast behavior. It assembles these instruments into a pre-build ladder and labels claims by evidential strength.

3.1 Ground truth: the encoder’s contract

  • The framework uses the original embedding space’s exact top-k as a fixed, label-free ground truth for indexes and blind transforms; learned encoder changes also require a task-relevance check.The task frame checks whether learned changes preserve retrieval of human-relevant documents, which the label-free frame cannot establish.Source: §3.1 · p. 7

3.2 Fidelity, efficiency, and the exchange bound

  • At matched depth, end-to-end recall is bounded by r ≥ p + s − 1, combining transform fidelity with index efficiency.The bound is exact as a set-theoretic lower bound and is tight in untransformed spaces.Source: §3.2 · p. 8

3.3 The master measurement

  • A label-free measurement pass records moments at index-relevant resolutions plus margins, cluster structure, and local-structure statistics that moments do not determine.Closed forms map selected statistics to fixed-grid behavior; cluster and margin statistics support predictions where moments alone are insufficient.Source: §3.3 · pp. 8–9

3.4 The twin hierarchy

  • Cluster-matched twins preserve real cluster weights, centers, and within-cluster covariances; when their measured cluster panel matches within 2%, behavior differences indicate structure beyond clusters.The twin hierarchy also serves as an attribution instrument and a zero-fitted-parameter simulator, with size-ladder and counterfactual modes.Source: §3.4 · pp. 10–11

3.5 Claim labels

  • Predictive claims are labeled proved, derived, or measured according to their basis and validation requirements.Derived claims are tested on held-out pairs and demoted if residuals show structure.Source: §3.5 · p. 12

3.6 Experimental setup

  • Experiments cover 17 dense and 4 late-interaction encoders, several corpora and index families, and a 1,001,595-document scale test.The setup includes PQ, OPQ, IVF, HNSW, and spec-faithful MUVERA FDE configurations.Source: §3.6 · p. 12

3.7 The ladder, assembled

  • The assembled ladder measures raw embeddings, reads fixed-grid verdicts from closed forms, prices transforms, mints twins for navigation, then simulates intended indexes.For large-scale HNSW, it uses size extrapolation and calibration when a full-size twin cannot meet the matching tolerance; optional counterfactuals explore unseen geometry.Source: §3.7 · pp. 12–13

4 Per-Family Analysis

The analysis maps each index family to the embedding statistics that govern its behavior, using closed forms for fixed-grid indexes and twin simulation for navigation indexes. It finds that graph recall levels require simulation, while fixed-grid behavior can be tied to specific geometric statistics.

4.1 Exact scoring is invariant (proved)

  • In the ρ-cone model, exact inner-product/MaxSim ranking is independent of the common-mean component, so exact benchmarks cannot reveal the index behaviors studied here.Centering and re-normalizing real pipelines can perturb rankings, creating a fidelity cost.Source: §4.1 · p. 13

4.2 FDE / SimHash (derived + theorem; measured on real encoders)

  • On GTE-ModernColBERT, raw FDE recall@100 is 0.136 and centering raises it to 0.468, while whitening adds no efficiency over centering and costs more fidelity.The theorem links FDE degeneration to anisotropy; across measured encoders, raw recall falls and centering recovery rises with token MCS, with centering harmful for low-MCS encoders.Source: §4.2 · pp. 13–14

4.3 PQ: the misalignment factor (derived)

  • On BGE-large/MS MARCO, OPQ raises candidate recall from 0.575 to 0.817 at fidelity 1.0; its covariance-derived misalignment factor identifies cross-subvector correlation as the PQ tax.A rotation leaves score margins unchanged; reducing the misalignment factor from 4.43 to 1.80 accompanies an efficiency rise from 0.324 to 0.481, and one curve fits 37 states with R2 = 0.87.Source: §4.3 · pp. 15–16

4.4 IVF: routing as gap survival (derived form; constants fitted)

  • IVF efficiency follows a gap-survival form involving routing and score margins; its in-family fit reaches R2 = 0.98 across 32 real and twin states.The routing-margin effect is causal in a targeted probe, while absolute recall levels for unseen encoders are assigned to twin simulation; transfer to PLAID or ScaNN remains an untested expectation.Source: §4.4 · p. 16, Figure 6 · p. 17

4.5 HNSW: score margins over cluster structure (derived by twin simulation)

  • HNSW cluster twins predict absolute recall with MAE 0.039 across 17 encoders, after cluster structure recovers about 80% of the moments-only gap.Moments alone under-predict recall by 0.26; the consistent signed remainder is about −0.08 across both navigation families.Source: §4.5 · pp. 17–18

5 The Trade-off Calculus

The trade-off calculus combines predicted index efficiency with the fidelity a transform spends, yielding an end-to-end recall verdict before index construction. Its measured examples show that the preferred correction depends on the index tax, while fidelity loss can be predicted from score-gap perturbations.

5 The Trade-off Calculus

  • The calculus combines transform fidelity and index efficiency predictions through the bound r ≥ p + s − 1 to judge end-to-end recall before building an index.Fixed-grid efficiency comes from closed forms, while navigation efficiency comes from twin simulation.Source: §5 · p. 19

5.1 One pricing rule, at both extremes of the tax

  • On dense/PQ data, whitening’s best sweep reaches 0.534 versus 0.575 raw, while OPQ reaches 0.817; on late-interaction/FDE data, centering spends 7.6% fidelity for a 3.4× efficiency gain.The operating rule compares efficiency gained with fidelity spent; rotations have zero fidelity cost.Source: §5.1 · p. 19, Figure 8 · p. 20

5.2 Fidelity is predictable (derived under assumptions; promoted from measured on the held-out calibration of Sects. 7.2–7.3)

  • On 34 fresh dense pairs, the gap model predicts centering fidelity with MAE 0.008; fidelity loss tracks gap noise rather than the average gap shift.Near-neighbor perturbations are coherent, so the naive independent-perturbation model over-predicts fidelity loss by 3–4×.Source: §5.2 · p. 19

6 Can the Correction Be Learned? Probes and the Margin Lever

Controlled probes test whether geometry corrections should be learned or applied post hoc. A label-free margin-sharpening layer raises recall across index families, while its task-relevance cost and the navigation probes’ differing effects qualify that gain.

Fixed-grid probes

  • Dose-corrected learned rotations reach MF 4.18 versus OPQ’s 1.92, while stale OPQ stays within 0.004 of refitting across disjoint shards.The learned rotations’ failure to match the specialized solver is consistent with the fixed-grid prediction map.Source: §6 · p. 20

Margin sharpening

  • Nearly 3× score margins lift efficiency for PQ (+0.15), IVF (+0.14), and HNSW (+0.10–0.16) at base fidelity 1.0.The rank-only adapter is a trainable final encoder layer; rotations cannot create margins, and distorting transforms change them only by changing the top-k.Source: §6 · p. 21

Task cost

  • Exact qrel recall@10 falls from 0.992 to 0.930, despite 0.9997 top-10 preservation within the N=100 candidate set and unchanged recall@100 of 0.993.The measurement used 300 judged queries and 315 relevance pairs; relevant documents remained among exact candidates but moved out of top ranks.Source: §6 · p. 21

Navigation probes

  • The navigability probe raises routing margin from 0.061 to 1.692, but fidelity falls to 0.621; IVF rises to 0.799 while HNSW stays flat near 0.91.By contrast, margin sharpening raises HNSW recall to 0.994 at efSearch 16; Fig. 9 shows the probes largely move different statistics.Source: §6 · pp. 21–22, Figure 9 · p. 22

FDE probe

  • The token-level FDE probe loses to centering: fidelity is 0.27–0.45 versus 0.924, and candidate recall is 0.10–0.17 versus 0.468.The probe removes token cone similarity, but the tight cone makes aligned per-token remapping scramble relative geometry that uniform centering preserves.Source: §6 · p. 22

7 The Forecast Protocol and Results

The evaluation tests closed-form forecasts, FDE rules, and synthetic twins on held-out settings, then checks transfer at million-document scale. Fixed-grid forecasts transfer; IVF levels transfer at 1M, while HNSW level prediction needs size extrapolation and anchoring.

7.1 Evaluation protocol

  • All results are out-of-sample, and across every family and state—including 116 HNSW rows—the exchange bound is violated zero times.The second round uses fresh encoders and datasets; the prospective FDE corpus was selected after rules, thresholds, and seeds were fixed.Source: §7.1 · pp. 22–23

7.2 Fixed-grid forecasts transfer; navigation formulas fail

  • In round one, the misalignment factor predicts whether OPQ helps on 20 of 22 pairs, while IVF ordering improves from ρ 0.42 to 0.76 when cluster excess is included.Fidelity-centering MAE is 0.019; absolute PQ levels miss by MAE 0.19, and HNSW intervals hold for only 11 of 22 pairs.Source: §7.2 · p. 23

7.3 The revised models on fresh encoders and datasets

  • On 34 fresh dense pairs, PQ achieves MAE 0.089 and rank 0.900, with the OPQ-helps sign correct on all 34; IVF levels reach MAE 0.105, while HNSW levels fail again.For HNSW, absolute intervals hold for 14 of 34 pairs, although encoder ordering within each corpus reaches ρ = 0.706 and 0.757.Source: §7.3 · p. 23

7.4 The FDE rules, across encoders and prospectively

  • Both FDE rank rules hold on all 12 ordered pairs; prospectively, MCS orders native FDE self-recall on five of six pairs and predicts all four centering signs.On unseen HotpotQA, the prospective ordering has ρ = −0.800; the fixed MCS threshold predicts the centering signs with macro mean absolute effect 0.073, including ColBERTv2’s −0.023 harm.Source: §7.4 · pp. 23–24

7.5 Twin simulation closes the navigation families

  • Twin simulations achieve HNSW absolute-recall MAE 0.039 and IVF within-corpus-level MAE 0.034 across 17 held-out encoders at 25–50k scales.For BGE-large/MS MARCO, cluster structure carries about 80% of both navigation families’ beyond-moment behavior; real corpora are slightly harder than their cluster-ideal twins.Source: §7.5 · p. 25
  • Query-distribution mismatch explains residual twin error: using real queries closes HNSW MAE to 0.012, while a Gaussian query model fails at 0.207.Reweighting mixture components by query traffic gives MAE 0.060 versus the default document-proxy’s 0.040; queries sit closer to their cluster centroids at cosine 0.659 than documents do at 0.708.Source: §7.5 · pp. 25–26

7.6 The million-document scale test

  • At 1M, the real-query IVF twin has MAE 0.027; HNSW’s size ladder reaches median anchored MAE 0.027, while unanchored runs miss the 0.05 target.At 1M, other full-size twin configurations have reproducible offsets, and fixed k=64 exceeds matching tolerance for three of four encoders.Source: §7.6 · p. 27

7.7 Robustness

  • Across 39 reported quantities, median seed deviation is 0.000, the 90th percentile is 0.004, and the maximum is 0.012.HNSW geometry effects preserve encoder ordering across efSearch 16–128, with rank agreement 0.82–0.96.Source: §7.7 · p. 28

8 Discussion

The discussion maps fixed-grid indexes to formula-ready statistics and navigation indexes to cluster organization reproduced by a synthetic twin. It argues for training-time margin shaping where post-hoc corrections are weak, while noting the task-relevance cost and the scope of the evidence.

A two-mechanism map

  • Fixed-grid indexes close through formula-ready statistics, whereas partition and graph indexes consume cluster organization that formulas summarize imperfectly but twins reproduce.The discussion assigns mean direction to hashing, covariance-block alignment to quantization, and margins to tolerance; navigation indexes instead close through simulation.Source: §8 · p. 28

The map covers the deployed landscape

  • A twin can host any index, including composed or not-yet-invented systems, for pre-build testing; signs and levers transfer across production stages, but constants require fitting.The discussion describes IVF-PQ, ScaNN, PLAID, and MUVERA as compositions of analyzed mechanisms.Source: §8 · pp. 28–29

Where corrections belong

  • A trainable final layer raises margins 2.8× at essentially unchanged exact neighbors and improves recall across index budgets without per-corpus serving state.The margin lever has a measured 3–5-point served-recall cost at PQ and IVF budgets; full encoder training and whether joint objectives avoid that price remain open questions.Source: §8 · p. 29

Operational implications

  • For changing corpora, a margin-trained encoder avoids per-corpus correction artifacts, while the measurement-and-twin procedure makes verdicts available before index construction.The statistics pass takes seconds per encoder and twin simulation minutes, compared with minutes for one full 1M HNSW build and hours for the measured grid.Source: §8 · p. 30

9 Limitations

The paper’s evidence and predictions have important scope limits, particularly for million-document results, graph-index levels, and task-frame measurements.

9 Limitations

  • The full measurement grid stops at 50,000 documents, while million-document evidence covers one corpus and four dense encoders.The beyond-cluster residual changes sign between 25k and 1M documents, and matching tolerance is unmet at 1M.Source: §9 · p. 30
  • At 1M documents, HNSW recall error is ≈0.06 unanchored and 0.026–0.027 with one anchored measurement; full-size level claims are scoped to 25–50k.A scale-aware twin that grows its cluster count is beyond the paper’s scope.Source: §9 · p. 30
  • The full twin decomposition is direct on one corpus; replication across 40 pairs is indirect, and the simulator is validated on a second corpus.The 1M IVF test used the real-query twin.Source: §9 · p. 30
  • FDE breadth covers three fresh encoders and supports ordering-grade claims only; an absolute FDE level model remains open.Scalar and binary quantization are not analyzed, though their designated statistics are recorded.Source: §9 · p. 30
  • The margin lever’s task-frame price and the navigability probe are measured on one corpus, with 315 judged pairs for the margin measurement.Learned probes are bounded by the training budget, and a ranking-supervised token objective remains untested.Source: §9 · p. 30
  • The cone model is idealized and Theorem 1’s constants are loose; the beyond-cluster residual and closed-form navigation-family derivations remain open.HNSW ordering and the anchored scheme are post-hoc analyses; unanchored closed-form HNSW levels are a validated negative.Source: §9 · p. 30

10 Conclusion

The conclusion presents a statistics-and-twin workflow for forecasting approximate retrieval, then training encoders for index-relevant margins. It reports out-of-sample validation and a reproducible artifact for the paper’s numbers and figures.

10 Conclusion

  • The audit matches each index family with the statistics or synthetic-twin instrument needed to predict its behavior.It extends stage by stage to composed systems such as IVF-PQ, ScaNN, and PLAID, and can simulate new indexes on statistical replicas.Source: §10 · p. 31
  • Forecasts were tested on encoders and corpora withheld from development, and each index family was evaluated with its required instrument.The workflow is to predict behavior and correction pricing, train margins, and deploy without per-corpus correction machinery.Source: §10 · p. 31
  • All paper numbers and figures regenerate from the supplementary artifact, which includes measurement and simulation code, configuration, and per-run manifests.Embeddings rebuild from public models and datasets; one Python environment suffices, with CPU covering everything except encoding.Source: §10 · p. 31

A Formal Statements and Proof Sketches

The formal results connect embedding geometry to approximate-index behavior, deriving where FDEs, quantizers, partitions, and transforms gain or lose retrieval fidelity. They also specify conditions under which those results apply.

Ranking invariance

  • In the shared-mean cone model, exact inner-product and MaxSim rankings are independent of the common mean, but centering followed by re-normalization can perturb rankings.The invariance holds without re-normalization; real pipelines that re-normalize incur measurable fidelity cost.Source: A Formal Statements and Proof Sketches · pp. 31–32

FDE anisotropy

  • Sufficient anisotropy can make MUVERA-style FDEs collapse token representations to mean-pooled scores and invert an exact MaxSim ranking.The theorem’s all-pairs collision bound is loose at measured token MCS ≈0.9; it establishes the mechanism, not measured collision rates.Source: A Formal Statements and Proof Sketches · pp. 32–33

FDE guarantee

  • An additive-ε FDE guarantee on full-space scores becomes an ε/(1 − MCS) certificate on ranking-relevant residual scores.At fixed residual-score error, parameters scaling as ε^-2 inflate by (1 − MCS)^-2.Source: A Formal Statements and Proof Sketches · p. 33

Residual schemes

  • Translation leaves residual distances unchanged, while whitening can expand low-variance directions and harm both coarse assignment and residual coding.Cone concentration reduces the effective dimensionality that coarse partitions must cover.Source: A Formal Statements and Proof Sketches · p. 33

Hubness invariance

  • Centering and orthonormal rotation preserve L2 hubness, removing hubness as a lever for those transforms.Hubness’s empirical relationship to graph recall is inconsistent as a predictor.Source: A Formal Statements and Proof Sketches · p. 33

PQ misalignment

  • High-rate PQ distortion reflects both block-determinant imbalance and cross-block covariance; orthonormal rotation can rebalance determinants without labels.The rotation-invariant distortion floor is reached when covariance is block-diagonal; approximate attainment is associated with OPQ.Source: A Formal Statements and Proof Sketches · pp. 33–34

Transform fidelity

  • Under Gaussian score-gap perturbations, transformed top-k survival is Φ((γi + shift)/σ), with extreme-distortion predictions potentially optimistic.The model uses the native boundary item as a proxy; near-neighbor perturbations are coherent.Source: A Formal Statements and Proof Sketches · p. 34

Partition routing

  • For IVF, routing-margin log has a positive partial effect on logit efficiency, while in-cell ranking contributes a score-margin term.The cell model’s validity scales with cluster excess, and fitted constants limit coefficient-level interpretation.Source: A Formal Statements and Proof Sketches · p. 34

B Experimental Details

The experiments span dense and late-interaction encoders, several corpora, and configured index families, using label-free measurements, synthetic sweeps, and trained probes. They also compare twin-based forecasts with alternatives and record diagnostic compute costs.

Corpora and encoders

  • The dense evaluation uses 17 sentence-transformers models across MS MARCO shards and three BEIR corpora; late-interaction evaluation uses four encoders on MS MARCO and SciFact.Late-interaction embeddings are stored at token level with ragged storage.Source: B Experimental Details · p. 35

Index configurations

  • Configured indexes include PQ, OPQ, IVF, HNSW, and MUVERA-style FDEs, evaluated at candidate depth N=100 and k=10.The reported configurations include PQ with M=16 and 8 bits, IVF with nlist 1024 and nprobe 16, and HNSW with M=32.Source: B Experimental Details · p. 35

Measurements and probes

  • Each dataset–transform–index run measures embedding statistics, score margins, fidelity, efficiency, cluster and local-structure panels, and qrel recall where labels exist.Probe training uses frozen encoders, label-free exact-neighbor pseudo-positives, geometry objectives, and an alignment term.Source: B Experimental Details · p. 35

Synthetic causal sweeps

  • Synthetic sweeps vary token anisotropy while preserving exact rankings, contrasting coarse topic-overlap relevance with fine-grained signal-token relevance.The fine-grained setup tests whether mean pooling drowns out a document-specific signal that exact MaxSim can resolve.Source: B Experimental Details · p. 35

Forecast comparison

  • On full-size twins, document-proxy queries yield MAE 0.040 for HNSW and 0.033 for IVF.The document-proxy query source improves on the cluster-traffic mixture and ridged-Gaussian query sources in the reported ladder.Source: B Experimental Details · p. 36

Operational costs

  • At one million documents, the twin pipeline takes tens of minutes per encoder on the recorded workstation, while building HNSW takes 131–157 seconds.These are diagnostic single-machine costs, not serving benchmarks.Source: B Experimental Details · p. 36
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