pith. sign in

arxiv: 2605.31100 · v1 · pith:ROV2BXMGnew · submitted 2026-05-29 · 💻 cs.AI · cs.DB· cs.IR

Vector Linking via Cross-Model Local Isometric Consistency

Ziying Chen , Yang Cao , He Sun , Beining Yang , Tianjian Yang This is my paper

Pith reviewed 2026-06-28 22:34 UTC · model grok-4.3

classification 💻 cs.AI cs.DBcs.IR
keywords vector linkinglocal geometric consistencycross-model embeddingsgeometric hashingbeta-bernoulli aggregationcontrastive encodersembedding correspondenceanchor bootstrapping
0
0 comments X

The pith

Independently trained contrastive encoders preserve short-range distances up to a scale factor, enabling recovery of cross-model object correspondences from a tiny seed set of paired anchors.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that embedding clouds from different encoders share local geometric consistency, with short-range distances preserved approximately up to a scale while long-range distances distort in model-specific ways. This property supports recovery of vector links across partially overlapping datasets using only the vectors themselves. The approach begins with a small seed of known paired anchors and iteratively proposes new links by representing each vector through distances to those anchors, matching in a hash space, and updating confidence via a Beta-Bernoulli posterior. A sympathetic reader would care because the result offers a route to align or integrate separate black-box embedding systems without retraining or data sharing. The work focuses on applications such as vector database integration and cross-model clustering under varying overlap and seed sizes.

Core claim

Independently trained contrastive encoders exhibit local geometric consistency: short-range distances are approximately preserved up to a scale factor, while long-range distances are not due to model-specific distortion. This enables an iterative, reference-based geometric embedding hashing procedure that recovers cross-model object correspondences from a tiny seed set of paired anchors by representing each vector through distances to sampled anchors, proposing candidates via hash-space matching, and aggregating evidence in a Beta-Bernoulli posterior to bootstrap additional high-confidence links.

What carries the argument

Iterative reference-based geometric embedding hashing that represents vectors by distances to sampled paired anchors, matches via hash-space collisions, and aggregates via Beta-Bernoulli posterior to promote new anchors.

If this is right

  • Accurate and robust cross-model linking holds across benchmarks with varying dataset overlap and seed budgets.
  • The procedure remains effective even when anchors come from out-of-domain sources.
  • Vector database integration becomes feasible without access to original training data or model internals.
  • Cross-model clustering can proceed directly from the recovered correspondences.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The local consistency property might be checked first on new encoder pairs to decide whether the linking procedure is likely to succeed before investing in seed collection.
  • Chaining the procedure across more than two models could allow transitive alignment of multiple embedding spaces.
  • The bootstrapping step could be made more stable by incorporating additional geometric invariants beyond distance-to-anchor profiles.

Load-bearing premise

A usable seed set of paired anchors exists and the partial overlap between embedding clouds is sufficient for the Beta-Bernoulli aggregation to bootstrap additional links without excessive error propagation.

What would settle it

Measuring recovered link accuracy that drops to levels indistinguishable from random guessing when the seed set is reduced below a small threshold or when direct checks show absence of local distance preservation in the tested model pair.

Figures

Figures reproduced from arXiv: 2605.31100 by Beining Yang, He Sun, Tianjian Yang, Yang Cao, Ziying Chen.

Figure 1
Figure 1. Figure 1: Consistency (linear correlation) VS. vector distances: The x-axis shows the pairwise distance in the reference space (Mis￾tral), while the y-axis reports the Pearson correlation (ρ) of these distances with their counterparts in the target space (OpenAI) [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: The geometric embedding hashing (GEH) framework Framework. The framework, denoted by GEH (Geometric Embedding Hashing) and shown in [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Impact of view construction and distance encoding: on SciDocs with Mistral VS. OpenAI, we compared the precision (left) and recall (right) of view strategies (FPS, Random), each with Kernelized or Raw distances.Shaded areas show variance (±1 std). mt=1 and At,1=Lt−1 (Section 3). • − Bootstrapping: run a single iteration on the S; no anchor-pool growth (Section 4) [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Integrated vector database retrieval performance: over SciFact, Recall@100 (left) and NDCG@100 (right) vs. over￾lap ratio α, where the overlap contains no benchmark answers. Mistral and OpenAI are the theoretical upper limit of retrieval quality where we embed all objects with one single model. pairs only; and (iii) Union retrieval without cross-space mapping (directly taking the union of two databases). A… view at source ↗
Figure 7
Figure 7. Figure 7: Distance consistency across embedding spaces. Each subplot shows Pearson correlation ρ between pairwise distances in the reference space and their counterparts in the target space, binned by the reference distance. (a–f) Six contrastive encoder pairs. (g) Mistral→OpenAI on SciFact for a sweep of OpenAI dimensionalities. (h) Mistral→OpenAI on two clustering benchmarks. (i–l) Non-contrastive comparison: each… view at source ↗
Figure 8
Figure 8. Figure 8: Cross-embedding retrieval consistency analysis:SciFact , Each panel reports the mean ± 1 std of the per-query Jaccard index between top-k retrieval results from two embedding spaces over 100 random queries. Here, pj represents the component of the anchor a ′ j orthogonal to the query v. We assume the data lies on a submanifold M ⊂ S D−1 of intrinsic dimension d ≪ D. For the hash to be stable (locally injec… view at source ↗
Figure 9
Figure 9. Figure 9: Posterior/Precision vs. anchor proximity. For Mistral ↔OpenAI linking at α = 0.2 overlap with |S| = 15 seeds, we bin predicted links by their minimum distance to the anchors that voted for them (30 quantile bins) and plot per-bin empirical precision and mean posterior confidence. where Mt := |Ut|, N := max{|E1|, |E2|}. We define MNN ratiot := Mt/N and terminate bootstrapping if any of the following holds: … view at source ↗
Figure 10
Figure 10. Figure 10: Sensitivity to view scheduling and CSLS hyperparameters. F1 on SciFact and NFCorpus for Mistral ↔OpenAI linking with overlap ratio α = 0.3 and |S| = 15 seeds. We vary (left) the logarithmic growth constant c in sf(g) = 1 + c log g, (middle) the CSLS neighborhood size kCSLS, and (right) the base per-view anchor fraction ρ0. The shaded gray region denotes the near-optimal range achieving at least 97% of the… view at source ↗
Figure 11
Figure 11. Figure 11: Out-of-domain reference transfer (additional settings):Accuracy (left) and recall (right) on five target datasets (columns) when seeds are drawn from an out-of-domain reference dataset (rows). Each panel varies the number of seeds n and target overlap o. The main text reports the case n=30, o=0.3 ( [PITH_FULL_IMAGE:figures/full_fig_p030_11.png] view at source ↗
read the original abstract

We study Vector Linking: given two embedding clouds produced by different black-box encoders over partially overlapping datasets, recover cross-model object correspondences using only vectors. Empirically and theoretically, we show that independently trained contrastive encoders exhibit local geometric consistency: short-range distances are approximately preserved up to a scale factor, while long-range distances are not due to model-specific distortion. Building on this, we propose an iterative, reference-based geometric embedding hashing that recovers vector links from a tiny seed set of paired anchors. It represents each vector by distances to sampled paired anchors, proposes candidate links via hash-space matching, and aggregates evidence across views in a Beta-Bernoulli posterior to bootstrap high-confidence links as new anchors. Experiments across multiple benchmarks and embedding model pairs demonstrate accurate and robust linking under varying overlap, seed budgets, and out-of-domain anchors, with applications to vector database integration and cross-model clustering. Code is available at https://github.com/DBgroup-Edinburgh/VecLinking.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper claims that independently trained contrastive encoders exhibit local geometric consistency (short-range distances preserved up to a scale factor, long-range distances distorted in a model-specific way). Building on this, it introduces an iterative reference-based geometric embedding hashing procedure that starts from a tiny seed set of paired anchors, represents vectors by distances to anchors, proposes links via hash matching, and aggregates evidence with a Beta-Bernoulli posterior to bootstrap additional high-confidence links. Experiments across benchmarks, model pairs, overlap levels, and seed budgets are reported, with public code provided.

Significance. If the local-consistency observation holds, the method offers a practical route to cross-model vector linking and database integration without joint training or full dataset overlap. The availability of public code together with multi-benchmark experiments supplies reproducible empirical support and allows direct falsification of the robustness claims under varying seed budgets and out-of-domain anchors.

minor comments (3)
  1. [Abstract / Introduction] The abstract states both empirical and theoretical support for local consistency; the theoretical argument should be expanded with an explicit statement of the assumptions under which short-range isometry holds (e.g., properties of the contrastive loss or embedding dimension).
  2. [Method / Experiments] The Beta-Bernoulli prior parameters are listed among the free parameters; a sensitivity plot or table showing linking accuracy as a function of these hyperparameters would strengthen the robustness claims.
  3. [Figures] Figure captions and axis labels should explicitly state the overlap fraction and seed size used in each panel so that the reported accuracy numbers can be interpreted without cross-referencing the text.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The description of the method and claims is accurate.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's core claim rests on an empirical observation of local geometric consistency (short-range distances preserved up to scale) across independently trained contrastive encoders, demonstrated via multi-benchmark experiments rather than any derivation that reduces to fitted parameters or self-referential definitions. The iterative geometric embedding hashing and Beta-Bernoulli aggregation constitute a statistical bootstrapping procedure from an external seed set; this does not equate the output to its inputs by construction, nor does it rely on load-bearing self-citations, uniqueness theorems imported from the authors, or ansatzes smuggled via prior work. The seed/overlap requirement is treated explicitly as an empirical parameter. No quoted equations or steps in the provided material exhibit the enumerated circular patterns.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The approach rests on the domain assumption of local isometric consistency in contrastive embeddings and introduces a small number of tunable elements in the hashing and aggregation steps; no new physical entities are postulated.

free parameters (2)
  • number of sampled paired anchors
    Controls the dimensionality of the hash representation; chosen per experiment.
  • Beta-Bernoulli prior parameters
    Hyperparameters of the posterior used to decide when to promote candidate links to anchors.
axioms (1)
  • domain assumption Short-range distances in independently trained contrastive embeddings are approximately preserved up to a global scale factor.
    Invoked in the abstract as the foundation for the geometric hashing step.

pith-pipeline@v0.9.1-grok · 5701 in / 1309 out tokens · 16629 ms · 2026-06-28T22:34:57.510743+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

43 extracted references · 28 canonical work pages · 9 internal anchors

  1. [1]

    URL http://dx.doi.org/ 10.18653/v1/P18-1073

    doi: 10.18653/v1/p18-1073. URL http://dx.doi.org/ 10.18653/v1/P18-1073. Besl, P. and McKay, N. D. A method for registration of 3-d shapes.IEEE Transactions on Pattern Analysis and Machine Intelligence, 14(2):239–256,

    work page doi:10.18653/v1/p18-1073

  2. [2]

    Enriching Word Vectors with Subword Information

    doi: 10.1109/34.121791. Bojanowski, P., Grave, E., Joulin, A., and Mikolov, T. En- riching word vectors with subword information.arXiv preprint arXiv:1607.04606,

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1109/34.121791

  3. [3]

    A full-text learning to rank dataset for medical information retrieval

    Boteva, V ., Gholipour, D., Sokolov, A., and Riezler, S. A full-text learning to rank dataset for medical information retrieval. InAdvances in Information Retrieval: 38th European Conference on IR Research, ECIR 2016, Padua, Italy, March 20–23,

    2016

  4. [4]

    Cohan, A., Feldman, S., Beltagy, I., Downey, D., and Weld, D

    URL https: //arxiv.org/abs/2303.00721. Cohan, A., Feldman, S., Beltagy, I., Downey, D., and Weld, D. S. Specter: Document-level representation learning using citation-informed transformers.arXiv preprint arXiv:2004.07180,

    work page arXiv 2004

  5. [5]

    Enevoldsen, K., Chung, I., Kerboua, I., Kardos, M., Mathur, A., Stap, D., Gala, J., Siblini, W., Krzemi´nski, D., Winata, G. I., Sturua, S., Utpala, S., Ciancone, M., Schaeffer, M., Sequeira, G., Misra, D., Dhakal, S., Rystrøm, J., Solo- matin, R., ¨Omer C ¸a˘gatan, Kundu, A., Bernstorff, M., Xiao, S., Sukhlecha, A., Pahwa, B., Po ´swiata, R., GV , K. K.,...

    work page arXiv

  6. [6]

    URL https://arxiv

    doi: 10.48550/arXiv.2502.13595. URL https://arxiv. org/abs/2502.13595. Fischler, M. A. and Bolles, R. C. Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography.Commun. ACM, 24(6):381–395, June

    work page doi:10.48550/arxiv.2502.13595

  7. [7]

    URL https://doi.org/10

    1145/358669.358692. URL https://doi.org/10. 1145/358669.358692. Ganin, Y ., Ustinova, E., Ajakan, H., Germain, P., Larochelle, H., Laviolette, F., Marchand, M., and Lempitsky, V . Domain-adversarial training of neural networks,

    work page arXiv

  8. [8]

    Domain-Adversarial Training of Neural Networks

    URLhttps://arxiv.org/abs/1505.07818. Geigle, G., Reimers, N., R¨uckl´e, A., and Gurevych, I. Tweac: Transformer with extendable qa agent classifiers,

    work page internal anchor Pith review Pith/arXiv arXiv

  9. [9]

    Gonzalez, T

    URLhttps://arxiv.org/abs/2104.07081. Gonzalez, T. F. Clustering to minimize the maximum intercluster distance.Theoretical Computer Sci- ence, 38:293–306,

    work page arXiv

  10. [10]

    doi: https://doi.org/10.1016/0304-3975(85)90224-5

    ISSN 0304-3975. doi: https://doi.org/10.1016/0304-3975(85)90224-5. URL https://www.sciencedirect.com/ science/article/pii/0304397585902245. Grave, E., Joulin, A., and Berthet, Q. Unsupervised align- ment of embeddings with wasserstein procrustes. In Chaudhuri, K. and Sugiyama, M. (eds.),Proceedings of the Twenty-Second International Conference on Artifici...

    work page doi:10.1016/0304-3975(85)90224-5

  11. [11]

    CyCADA: Cycle-Consistent Adversarial Domain Adaptation

    URL https:// arxiv.org/abs/1711.03213. Hu, W., Bansal, R., Cao, K., Rao, N., Subbian, K., and Leskovec, J. Learning backward compatible embeddings. InKDD,

    work page internal anchor Pith review Pith/arXiv arXiv

  12. [12]

    Mistral 7B

    Jiang, A. Q., Sablayrolles, A., Mensch, A., Bamford, C., Chaplot, D. S., de las Casas, D., Bressand, F., Lengyel, G., Lample, G., Saulnier, L., Lavaud, L. R., Lachaux, M.- A., Stock, P., Le Scao, T., Lavril, T., Wang, T., Lacroix, T., and El Sayed, W. Mistral 7b.arXiv preprint arXiv:2310.06825,

    work page internal anchor Pith review Pith/arXiv arXiv

  13. [13]

    doi: 10.48550/arXiv.2310. 06825. URL https://arxiv.org/abs/2310. 06825. Joulin, A., Bojanowski, P., Mikolov, T., J´egou, H., and Grave, E. Loss in translation: Learning bilingual word mapping with a retrieval criterion. In Riloff, E., Chiang, D., Hock- enmaier, J., and Tsujii, J. (eds.),Proceedings of the 2018 Conference on Empirical Methods in Natural La...

    work page doi:10.48550/arxiv.2310 2018

  14. [14]

    Lamdan, Y

    doi: 10.18653/V1/D18-1330. Lamdan, Y . and Wolfson, H. Geometric hashing: A general and efficient model-based recognition scheme. In[1988 Proceedings] Second International Conference on Com- puter Vision, pp. 238–249,

    work page doi:10.18653/v1/d18-1330 1988

  15. [15]

    doi: 10.1109/CCV .1988. 589995. Lample, G., Conneau, A., Ranzato, M., Denoyer, L., and J´egou, H. Word translation without parallel data. In ICLR,

    work page doi:10.1109/ccv 1988

  16. [16]

    Towards General Text Embeddings with Multi-stage Contrastive Learning

    Li, Z., Zhang, X., Zhang, Y ., Long, D., Xie, P., and Zhang, M. Towards general text embeddings with multi-stage contrastive learning.arXiv preprint arXiv:2308.03281,

    work page internal anchor Pith review Pith/arXiv arXiv

  17. [17]

    Deep multilingual correlation for improved word embed- dings

    Lu, A., Wang, W., Bansal, M., Gimpel, K., and Livescu, K. Deep multilingual correlation for improved word embed- dings. In Mihalcea, R., Chai, J., and Sarkar, A. (eds.), Proceedings of the 2015 Conference of the North Amer- ican Chapter of the Association for Computational Lin- guistics: Human Language Technologies, pp. 250–256, Denver, Colorado, May–June

    2015

  18. [18]

    doi: 10.3115/v1/N15-1028

    Association for Com- putational Linguistics. doi: 10.3115/v1/N15-1028. URL https://aclanthology.org/N15-1028/. Maia, M., Handschuh, S., Freitas, A., Davis, B., McDermott, R., Zarrouk, M., and Balahur, A. Www’18 open chal- lenge: financial opinion mining and question answering. InCompanion proceedings of the the web conference 2018, pp. 1941–1942,

    work page doi:10.3115/v1/n15-1028 2018

  19. [19]

    Exploiting Similarities among Languages for Machine Translation

    Mikolov, T., Le, Q. V ., and Sutskever, I. Exploiting simi- larities among languages for machine translation.CoRR, abs/1309.4168,

    work page internal anchor Pith review Pith/arXiv arXiv

  20. [20]

    text-embedding-3-small (model documenta- tion)

    OpenAI. text-embedding-3-small (model documenta- tion). https://platform.openai.com/docs/ models/text-embedding-3-small . Accessed: 2026-01-26. Otsu, N. A threshold selection method from gray-level histograms.IEEE Transactions on Systems, Man, and Cybernetics, 9(1):62–66,

    2026

  21. [21]

    doi: 10.1109/TSMC.1979. 4310076. Pennington, J., Socher, R., and Manning, C. GloVe: Global vectors for word representation. In Moschitti, A., Pang, B., and Daelemans, W. (eds.),Proceedings of the 2014 Conference on Empirical Methods in Natural Lan- guage Processing (EMNLP), pp. 1532–1543, Doha, Qatar, October

    work page doi:10.1109/tsmc.1979 1979

  22. [22]

    doi: 10.3115/v1/D14-1162

    Association for Computational Linguis- tics. doi: 10.3115/v1/D14-1162. URL https:// aclanthology.org/D14-1162/. Petersen, K. B., Pedersen, M. S., et al. The matrix cookbook. Technical University of Denmark, 7(15):510,

    work page doi:10.3115/v1/d14-1162

  23. [23]

    Smith, S

    URLhttps://arxiv.org/abs/2003.11942. Smith, S. L., Turban, D. H. P., Hamblin, S., and Hammerla, N. Y . Offline bilingual word vectors, orthogonal transfor- mations and the inverted softmax. In5th International Conference on Learning Representations, ICLR 2017, Toulon, France, April 24-26, 2017, Conference Track Pro- ceedings. OpenReview.net,

    work page arXiv 2003

  24. [24]

    BEIR: A Heterogenous Benchmark for Zero-shot Evaluation of Information Retrieval Models

    Thakur, N., Reimers, N., R ¨uckl´e, A., Srivastava, A., and Gurevych, I. Beir: A heterogenous benchmark for zero- shot evaluation of information retrieval models.arXiv preprint arXiv:2104.08663,

    work page internal anchor Pith review Pith/arXiv arXiv

  25. [25]

    FEVER: a large-scale dataset for Fact Extraction and VERification

    URL https://arxiv.org/ abs/1803.05355. van den Oord, A., Li, Y ., and Vinyals, O. Repre- sentation learning with contrastive predictive coding. arXiv:1807.03748,

    work page internal anchor Pith review Pith/arXiv arXiv

  26. [26]

    L., van Zuylen, M., Cohan, A., and Hajishirzi, H

    Wadden, D., Lin, S., Lo, K., Wang, L. L., van Zuylen, M., Cohan, A., and Hajishirzi, H. Fact or fiction: Verifying scientific claims. InProceedings of the 2020 Conference on Empirical Methods in Natu- ral Language Processing (EMNLP), pp. 7534–7550, Online, November

    2020

  27. [27]

    doi: 10.18653/v1/2020.emnlp-main

    Association for Computa- tional Linguistics. doi: 10.18653/v1/2020.emnlp-main

    work page doi:10.18653/v1/2020.emnlp-main 2020

  28. [28]

    emnlp-main.609/

    URL https://aclanthology.org/2020. emnlp-main.609/. Wang, C. and Mahadevan, S. Heterogeneous domain adap- tation using manifold alignment. InProceedings of the Twenty-Second International Joint Conference on Arti- ficial Intelligence - Volume Volume Two, IJCAI’11, pp. 1541–1546. AAAI Press,

    2020

  29. [29]

    Visual Domain Adaptation with Manifold Embedded Distribution Alignment

    URL https://arxiv. org/abs/1807.07258. Wang, T. and Isola, P. Understanding contrastive represen- tation learning through alignment and uniformity on the hypersphere. InICML,

    work page internal anchor Pith review Pith/arXiv arXiv

  30. [30]

    Normalized word embedding and orthogonal transform for bilingual word translation

    Xing, C., Wang, D., Liu, C., and Lin, Y . Normalized word embedding and orthogonal transform for bilingual word translation. In Mihalcea, R., Chai, J., and Sarkar, A. (eds.), Proceedings of the 2015 Conference of the North Amer- ican Chapter of the Association for Computational Lin- guistics: Human Language Technologies, pp. 1006–1011, Denver, Colorado, May–June

    2015

  31. [31]

    doi: 10.3115/v1/N15-1104

    Association for Com- putational Linguistics. doi: 10.3115/v1/N15-1104. URL https://aclanthology.org/N15-1104/. Yang, B., Cao, Y ., and Ren, Y . Integrating vector databases across embedding models. InSIGMOD,

    work page doi:10.3115/v1/n15-1104

  32. [32]

    org/abs/2001.07715

    URL https://arxiv. org/abs/2001.07715. Yang, J., Li, H., Campbell, D., and Jia, Y . Go-icp: A globally optimal solution to 3d icp point-set reg- istration.IEEE Transactions on Pattern Analysis and Machine Intelligence, 38(11):2241–2254, Novem- ber

    work page arXiv 2001

  33. [33]

    doi: 10.1109/tpami.2015

    ISSN 2160-9292. doi: 10.1109/tpami.2015. 2513405. URL http://dx.doi.org/10.1109/ TPAMI.2015.2513405. Zhang, Y ., Li, M., Long, D., Zhang, X., Lin, H., Yang, B., Xie, P., Yang, A., Liu, D., Lin, J., Huang, F., and Zhou, J. Qwen3 embedding: Advancing text embedding and reranking through foundation models.arXiv preprint arXiv:2506.05176,

    work page doi:10.1109/tpami.2015 2015

  34. [34]

    Zimmermann, R

    URL https: //arxiv.org/abs/2506.20923. Zimmermann, R. S., Sharma, Y ., Schneider, S., Bethge, M., and Brendel, W. Contrastive learning inverts the data generating process. InICML,

    work page arXiv

  35. [35]

    Since D(expx)(0) = Id TxM (e.g.,(Lee, 2003)), we have Dg(0) = Df(x)

    By the chain rule, Dg(0) =Df(x)◦D(exp x)(0). Since D(expx)(0) = Id TxM (e.g.,(Lee, 2003)), we have Dg(0) = Df(x) . In an orthonormal basis of TxM, Df(x) is represented by Jf(x) and Gf(x) =J f(x)⊤Jf(x). Let v=v(x, x +) and assume ∥v∥ ≤r

    2003

  36. [36]

    The leading term is governed by the Jacobian Jf(x), hence by the induced metric Gf(x) =J f(x)⊤Jf(x)

    The intuition of the proof is that, for a nearby point y around x, the encoder admits a first-order Taylor approximation along the unique short geodesic from x to y. The leading term is governed by the Jacobian Jf(x), hence by the induced metric Gf(x) =J f(x)⊤Jf(x). Local encoder optimality forces Gf(x) to be a scalar multiple of the identity, which makes...

    2003

  37. [37]

    The number of partitions is mt =⌈|L t−1|/dmax⌉, where dmax = max(dE1 , dE2), which ensures each view has sufficient anchors for a dimensionally well-posed local neighborhood

    Otherwise, at iterationt, we replace FPS with a k-means partition of Lt−1 in E1 to ensure diversity of views. The number of partitions is mt =⌈|L t−1|/dmax⌉, where dmax = max(dE1 , dE2), which ensures each view has sufficient anchors for a dimensionally well-posed local neighborhood. Each anchor in Lt−1 is included in its ρ nearest clusters ( ρ=2 in our e...

    2021

  38. [38]

    The queries are short factual claims, and the corpus introductory sections of Wikipedia pages

    is a fact-verification dataset. The queries are short factual claims, and the corpus introductory sections of Wikipedia pages. Table 5 provides dataset statistics, including query counts, corpus sizes, and the average number of relevant documents per query. C.1.2. EMBEDDINGMODELS We generate embeddings using a mix of proprietary API services and open-weig...

    2023

  39. [39]

    We optimizeWwith Adam (learning rate10 −3) for 100 epochs. Canonical Correlation Analysis (CCA).We standardize each space independently, fit CCA on the seed pairs to learn one linear projection per space that maximizes correlation between projected seed embeddings. Multi-Layer Perceptron (MLP).We train a single-hidden-layer MLP mapping from the source emb...

    2018

  40. [40]

    by sweeping the seed budget n∈ {15,20,30} and the target overlap ratio α∈ {0.15,0.2,0.3} . Overall, most reference–target pairs preserve strong precision and recall under OOD seeding; the few degraded cases align with our Theorem 1, which predicts that links supported primarily by long-range anchors are less reliable. D. Details of Section 6 D.1. Implemen...

    2025

  41. [41]

    2048 50 65.49 18 / 299 StackExchangeClustering.v2 (Geigle et al.,

    2048

  42. [42]

    2048 121 57.51 19 / 148 return the integrated database T(D 1)∪D

    2048

  43. [43]

    We use a FAISS GPU index with inner-product search overℓ2-normalized embeddings (equivalently cosine similarity)

    , whererel i is the graded relevance of the item at rankiandIDCG@kis the DCG of the ideal ranking. We use a FAISS GPU index with inner-product search overℓ2-normalized embeddings (equivalently cosine similarity). D.1.2. GLOBALCROSS-MODELCLUSTERING We evaluate cross-model clustering using two clustering benchmarks from MTEB (Enevoldsen et al., 2025). Both ...

    2025