{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,18]],"date-time":"2026-08-18T13:45:13Z","timestamp":1787060713047,"version":"3.56.0"},"reference-count":32,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,5,21]],"date-time":"2025-05-21T00:00:00Z","timestamp":1747785600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003359","name":"Generalitat Valenciana (Spain)","doi-asserted-by":"publisher","award":["PROMETEO 2024 CIPROM\/2023\/32"],"award-info":[{"award-number":["PROMETEO 2024 CIPROM\/2023\/32"]}],"id":[{"id":"10.13039\/501100003359","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We present a new framework to study the stability of semantic projections based on word embeddings. Roughly speaking, semantic projections are indices taking values in the interval [0,1] that measure how terms share contextual meaning with the words of a given universe. Since there are many ways to define such projections, it is important to establish a procedure for verifying whether a group of them behaves similarly. Moreover, when fixing one particular projection, it is important to assess whether the average projections remain consistent when replacing the original universe with a similar one describing the same semantic environment. The aim of this paper is to address the lack of formal tools for assessing the stability of semantic projections (that is, their invariance under formal changes which preserve the underlying semantic context) across alternative but semantically related universes in word embedding models. To address these problems, we employ a combination of statistical and AI methods, including correlation analysis, clustering, chi-squared distance measures, weighted approximations, and Lipschitz-based estimators. The methodology provides theoretical guarantees under mild mathematical assumptions, ensuring bounded errors in projection estimations based on the assumption of Lipschitz continuity. We demonstrate the practical applicability of our approach through two case studies involving agricultural terminology across multiple data sources (DOAJ, Scholar, Google, and Arxiv). Our results show that semantic stability can be quantitatively evaluated and that the careful modeling of projection functions and universes is crucial for robust semantic analysis in NLP.<\/jats:p>","DOI":"10.3390\/axioms14050389","type":"journal-article","created":{"date-parts":[[2025,5,21]],"date-time":"2025-05-21T09:58:22Z","timestamp":1747821502000},"page":"389","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Measuring Semantic Stability: Statistical Estimation of Semantic Projections via Word Embeddings"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2544-8875","authenticated-orcid":false,"given":"Roger","family":"Arnau","sequence":"first","affiliation":[{"name":"Instituto Universitario de Matem\u00e1tica Pura y Aplicada, Universitat Polit\u00e8cnica de Val\u00e8ncia, Camino de Vera s\/n, 46022 Valencia, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0005-6489-3279","authenticated-orcid":false,"given":"Ana","family":"Coronado Ferrer","sequence":"additional","affiliation":[{"name":"Departamento de TICS, Florida Universitaria, 46470 Catarroja, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0001-2328-2173","authenticated-orcid":false,"given":"\u00c1lvaro","family":"Gonz\u00e1lez Cort\u00e9s","sequence":"additional","affiliation":[{"name":"Instituto Universitario de Matem\u00e1tica Pura y Aplicada, Universitat Polit\u00e8cnica de Val\u00e8ncia, Camino de Vera s\/n, 46022 Valencia, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0004-1245-9289","authenticated-orcid":false,"given":"Claudia","family":"S\u00e1nchez Arnau","sequence":"additional","affiliation":[{"name":"Instituto Universitario de Matem\u00e1tica Pura y Aplicada, Universitat Polit\u00e8cnica de Val\u00e8ncia, Camino de Vera s\/n, 46022 Valencia, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8854-3154","authenticated-orcid":false,"given":"Enrique A.","family":"S\u00e1nchez P\u00e9rez","sequence":"additional","affiliation":[{"name":"Instituto Universitario de Matem\u00e1tica Pura y Aplicada, Universitat Polit\u00e8cnica de Val\u00e8ncia, Camino de Vera s\/n, 46022 Valencia, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,21]]},"reference":[{"key":"ref_1","first-page":"2005","article-title":"Word Embeddings: A Comprehensive Survey","volume":"28","author":"Pak","year":"2024","journal-title":"Comput. Sist."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"975","DOI":"10.1038\/s41562-022-01316-8","article-title":"Semantic projection recovers rich human knowledge of multiple object features from word embeddings","volume":"6","author":"Grand","year":"2022","journal-title":"Nat. Hum. Behav."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3961","DOI":"10.3934\/math.2025185","article-title":"Mathematical features of semantic projections and word embeddings for automatic linguistic analysis","volume":"10","year":"2025","journal-title":"AIMS Math."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Fern\u00e1ndez de C\u00f3rdoba, P., Reyes P\u00e9rez, C.A., S\u00e1nchez Arnau, C., and S\u00e1nchez P\u00e9rez, E.A. (2025). Set-Word Embeddings and Semantic Indices: A New Contextual Model for Empirical Language Analysis. Computers, 14.","DOI":"10.3390\/computers14010030"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1146\/annurev-linguistics-030514-125254","article-title":"Distributional models of word meaning","volume":"4","author":"Lenci","year":"2018","journal-title":"Annu. Rev. Linguist."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1146\/annurev-linguistics-011619-030303","article-title":"Distributional semantics and linguistic theory","volume":"6","author":"Boleda","year":"2020","journal-title":"Annu. Rev. Linguist."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"635","DOI":"10.1002\/lnco.362","article-title":"Vector space models of word meaning and phrase meaning: A survey","volume":"6","author":"Erk","year":"2012","journal-title":"Lang. Linguist. Compass"},{"key":"ref_8","first-page":"3111","article-title":"Distributed representations of words and phrases and their compositionality","volume":"26","author":"Mikolov","year":"2013","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Pennington, J., Socher, R., and Manning, C. (2014, January 25\u201329). GloVe: Global Vectors for Word Representation. Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP), Doha, Qatar.","DOI":"10.3115\/v1\/D14-1162"},{"key":"ref_10","first-page":"2177","article-title":"Neural word embedding as implicit matrix factorization","volume":"27","author":"Levy","year":"2014","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Wulff, D.U., and Mata, R. (2025). Semantic Embeddings Reveal and Address Taxonomic Incommensurability in Psychological Measurement. Nat. Hum. Behav., 1\u201311.","DOI":"10.1038\/s41562-024-02089-y"},{"key":"ref_12","unstructured":"Mikolov, T., Yih, W.t., and Zweig, G. (2013, January 9\u201314). Linguistic regularities in continuous space word representations. Proceedings of the 2013 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Atlanta, GA, USA."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"4176","DOI":"10.1073\/pnas.1814779116","article-title":"Emergence of analogy from relation learning","volume":"116","author":"Lu","year":"2019","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_14","unstructured":"Baroni, M., and Zamparelli, R. (2010, January 9\u201311). Nouns are vectors, adjectives are matrices: Representing adjective-noun constructions in semantic space. Proceedings of the 2010 Conference on Empirical Methods in Natural Language Processing, Cambridge, MA, USA."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Lappin, S., and Fox, C. (2015). Vector space models of lexical meaning. The Handbook of Contemporary Semantics, Blackwell.","DOI":"10.1002\/9781118882139"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1753","DOI":"10.1177\/00491241221140142","article-title":"Theoretical Foundations and Limits of Word Embeddings: What Types of Meaning Can They Capture?","volume":"53","year":"2024","journal-title":"Sociol. Methods Res."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Boutyline, A., and Arseniev-Koehler, A. (2025). Meaning in Hyperspace: Word Embeddings as Tools for Cultural Measurement. Annu. Rev. Sociol., 51.","DOI":"10.1146\/annurev-soc-090324-024027"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1016\/j.patcog.2017.04.022","article-title":"Cross-view semantic projection learning for person re-identification","volume":"75","author":"Dai","year":"2018","journal-title":"Pattern Recognit."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Xian, Y., Choudhury, S., He, Y., Schiele, B., and Akata, Z. (2019, January 15\u201320). Semantic projection network for zero- and few-label semantic segmentation. Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition, Long Beach, CA, USA.","DOI":"10.1109\/CVPR.2019.00845"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1080\/01969727208542910","article-title":"A Fuzzy-Set-Theoretic Interpretation of Linguistic Hedges","volume":"2","author":"Zadeh","year":"1972","journal-title":"J. Cybern."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Cobza\u015f, S., Miculescu, R., and Nicolae, A. (2019). Lipschitz Functions, Springer.","DOI":"10.1007\/978-3-030-16489-8"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1016\/j.patrec.2018.12.007","article-title":"A note on the triangle inequality for the Jaccard distance","volume":"120","author":"Kosub","year":"2019","journal-title":"Pattern Recognit. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Deza, M.M., and Deza, E. (2009). Encyclopedia of Distances, Springer. [1st ed.].","DOI":"10.1007\/978-3-642-00234-2"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Gardner, A., Kanno, J., Duncan, C.A., and Selmic, R. (2014, January 23\u201328). Measuring distance between unordered sets of different sizes. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Columbus, OH, USA.","DOI":"10.1109\/CVPR.2014.25"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Manetti, A., Ferrer-Sapena, A., S\u00e1nchez-P\u00e9rez, E.A., and Lara-Navarra, P. (2021). Design Trend Forecasting by Combining Conceptual Analysis and Semantic Projections: New Tools for Open Innovation. J. Open Innov. Technol. Mark. Complex., 7.","DOI":"10.3390\/joitmc7010092"},{"key":"ref_26","unstructured":"(2025, April 03). Directory of Open Access Journals (DOAJ). Available online: https:\/\/www.doaj.org\/."},{"key":"ref_27","unstructured":"(2025, April 05). Google Scholar. Available online: https:\/\/scholar.google.com\/."},{"key":"ref_28","unstructured":"(2025, April 01). Google. Available online: https:\/\/www.google.com\/."},{"key":"ref_29","unstructured":"arXiv (2025, April 07). arXiv e-Print Archive. Available online: https:\/\/arxiv.org\/."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Arnau, R., Calabuig, J.M., Gonz\u00e1lez, A., and S\u00e1nchez P\u00e9rez, E.A. (2024). Moduli of Continuity in Metric Models and Extension of Livability Indices. Axioms, 13.","DOI":"10.3390\/axioms13030192"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1","DOI":"10.15388\/namc.2022.27.27493","article-title":"Index spaces and standard indices in metric modelling","volume":"27","author":"Erdogan","year":"2022","journal-title":"Nonlinear Anal. Model. Control"},{"key":"ref_32","unstructured":"Pennington, J., Socher, R., and Manning, C.D. (2024, December 17). GloVe: Global Vectors for Word Representation. Pre-Trained Embeddings, 50 Dimensions, Trained on 6B Tokens (Wikipedia + Gigaword). Available online: http:\/\/nlp.stanford.edu\/data\/glove.6B.zip."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/5\/389\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:36:38Z","timestamp":1760031398000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/5\/389"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,21]]},"references-count":32,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2025,5]]}},"alternative-id":["axioms14050389"],"URL":"https:\/\/doi.org\/10.3390\/axioms14050389","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,5,21]]}}}