{"type":"ScholarlyArticle","title":"Error in the Euclidean Preference Model","authors":[{"name":"Luke Thorburn","slug":"luke-thorburn","given":"Luke","family":"Thorburn","affiliation":"King's College London"},{"name":"Maria Polukarov","slug":"maria-polukarov","given":"Maria","family":"Polukarov","affiliation":"King's College London"},{"name":"Carmine Ventre","slug":"carmine-ventre","given":"Carmine","family":"Ventre","affiliation":"King's College London"}],"abstract":"Spatial models of preference, in the form of vector embeddings, are learned by many deep learning and multiagent systems, including recommender systems. Often these models are assumed to approximate a Euclidean structure, where an individual prefers alternatives positioned closer to their \"ideal point\", as measured by the Euclidean metric. However, previous work has shown there are ordinal preference profiles that cannot be represented with this structure if the Euclidean space has two fewer dimensions than there are individuals or alternatives. We extend this result, showing that there are situations in which almost all preference profiles cannot be represented with the Euclidean model, and derive a theoretical lower bound on the expected error when using the Euclidean model to approximate non-Euclidean preference profiles. Our results have implications for the interpretation and use of vector embeddings, because in some cases close approximation of arbitrary, true ordinal relationships can be expected only if the dimensionality of the embeddings is a substantial fraction of the number of entities represented.","doi":"10.24963/ijcai.2023/322","url":"https://ijcai.org/proceedings/2023/322","pdf":"https://ijcai.org/proceedings/2023/0322.pdf","markdown":"https://ijcai.org/proceedings/2023/322.md","bibtex":"https://ijcai.org/proceedings/2023/bibtex/322","volume":{"slug":"2023","title":"Proceedings of the Thirty-Second International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2023","url":"https://ijcai.org/proceedings/2023/","year":2023,"location":"Macao, SAR","dates":"19-25 August 2023","editors":"Edith Elkind","isbn":"978-1-956792-03-4","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2023"},"section":"Game Theory and Economic Paradigms","track":"Main Track","number":322,"pages":{"first":2888,"last":2896},"keywords":["Machine Learning: ML: Learning preferences or rankings","Game Theory and Economic Paradigms: GTEP: Computational social choice","Knowledge Representation and Reasoning: KRR: Preference modelling and preference-based reasoning"],"videos":[],"references":[{"position":1,"raw":"[Almeida and Xexéo, 2019] Felipe Almeida and Geraldo Xexéo. Word Embeddings: A Survey. arXiv preprint arXiv:1901.09069, January 2019.","title":"Felipe Almeida and Geraldo Xexéo","authors":"Xexéo Almeida","year":"2019","venue":null,"doi":null,"url":null,"internal_url":null},{"position":2,"raw":"[Anand and Dey, 2021] Aditya Anand and Palash Dey. Dis- tance restricted manipulation in voting. Theoretical Com- puter Science, 891:149-165, 2021.","title":"Distance restricted manipulation in voting","authors":"Dey Anand, Aditya Anand, Palash Dey","year":"2021","venue":"Theoretical Computer Science","doi":null,"url":null,"internal_url":null},{"position":3,"raw":"[Aydogdu et al., 2017] Aylin Aydogdu, Sean T Mcquade, and Nastassia Pouradier Duteil. Opinion Dynamics on a General Compact Riemannian Manifold. Networks and Heterogeneous Media, 12(3):489 -523, September 2017.","title":"Opinion Dynamics on a General Compact Riemannian Manifold","authors":"Aylin Aydogdu, Sean T Mcquade, Nastassia Pouradier, Duteil","year":"2017","venue":"Networks and Heterogeneous Media","doi":null,"url":null,"internal_url":null},{"position":4,"raw":"[Bennett and Hays, 1960] Joseph F. Bennett and William L. Hays. Multidimensional unfolding: Determining the di- mensionality of ranked preference data. Psychometrika, 25(1):27-43, Mar 1960.","title":"Multidimensional unfolding: Determining the dimensionality of ranked preference data","authors":"Joseph F Bennett, William L Hays","year":"1960","venue":"Psychometrika","doi":null,"url":null,"internal_url":null},{"position":5,"raw":"[Bogomolnaia and Laslier, 2007] Anna Bogomolnaia and Jean-Franc ¸ois Laslier. Euclidean preferences. Journal of Mathematical Economics, 43(2):87-98, February 2007.","title":"Euclidean preferences","authors":"Anna Bogomolnaia, Jean-Franc ¸ois Laslier","year":"2007","venue":"Journal of Mathematical Economics","doi":null,"url":null,"internal_url":null},{"position":6,"raw":"[Bukh and Ma, 2014] Boris Bukh and Jie Ma. Longest common subsequences in sets of words. arXiv preprint arXiv:1406.7017, October 2014.","title":"Longest common subsequences in sets of words","authors":"Ma ; Bukh, Boris Bukh, Jie Ma","year":"2014","venue":null,"doi":null,"url":null,"internal_url":null},{"position":7,"raw":"[Christian, 2020] Brian Christian. The Alignment Problem: Machine Learning and Human Values. WW Norton & Company, New York, US, 2020.","title":"The Alignment Problem: Machine Learning and Human Values","authors":"Brian Christian","year":"2020","venue":null,"doi":null,"url":null,"internal_url":null},{"position":8,"raw":"[Eguia, 2011] Jon X. Eguia. Foundations of spatial prefer- ences. Journal of Mathematical Economics, 47(2):200- 205, 2011.","title":"Foundations of spatial preferences","authors":"Jon X Eguia","year":"2011","venue":"Journal of Mathematical Economics","doi":null,"url":null,"internal_url":null},{"position":9,"raw":"[Eguia, 2013] Jon X. Eguia. Challenges to the standard eu- clidean spatial model. In Advances in Political Economy: Institutions, Modelling and Empirical Analysis, pages 169-180. Springer Berlin Heidelberg, Berlin, Heidelberg, 2013.","title":"Challenges to the standard euclidean spatial model","authors":"Jon X Eguia","year":"2013","venue":"Advances in Political Economy: Institutions, Modelling and Empirical Analysis","doi":null,"url":null,"internal_url":null},{"position":10,"raw":"[Elkind and Faliszewski, 2014] Edith Elkind and Piotr Fal- iszewski. Recognizing 1-Euclidean Preferences: An Alter- native Approach. In Ron Lavi, editor, Algorithmic Game Theory, Lecture Notes in Computer Science, pages 146- 157, Berlin, Heidelberg, 2014. Springer. Elkind et al., 2017] Edith Elkind, Martin Lackner, and Do- minik Peters. Structured preferences. In Ulle Endriss, ed- itor, Trends in Computational Social Choice, chapter 10, pages 187-207. AI Access, (online), 2017.","title":"Recognizing 1-Euclidean Preferences: An Alternative Approach","authors":"Faliszewski Elkind, Piotr Faliszewski","year":"2014","venue":"Trends in Computational Social Choice","doi":null,"url":null,"internal_url":null},{"position":11,"raw":"[Egecioglu and Giritligil, 2013] Ömer Egecioglu and Ayc ¸a E. Giritligil. The impartial, anonymous, and neutral culture model: A probability model for sampling public preference structures. The Journal of Mathematical Sociology, 37(4):203-222, 2013.","title":"Ömer Egecioglu and Ayc ¸a E. Giritligil. The impartial, anonymous, and neutral culture model: A probability model for sampling public preference structures","authors":"Giritligil Egecioglu","year":"2013","venue":"The Journal of Mathematical Sociology","doi":null,"url":null,"internal_url":null},{"position":12,"raw":"[Faelens et al., 2021] Lien Faelens, Kristof Hoorelbeke, Ruben Cambier, Jill van Put, Eowyn Van de Putte, Rudi De Raedt, and Ernst H.W. Koster. The relationship be- tween instagram use and indicators of mental health: A systematic review. Computers in Human Behavior Re- ports, 4:100121, 2021.","title":"The relationship between instagram use and indicators of mental health: A systematic review","authors":"Faelens","year":"2021","venue":"Computers in Human Behavior Reports","doi":null,"url":null,"internal_url":null},{"position":13,"raw":"[Gaiha and Gupta, 1977] P. Gaiha and S. K. Gupta. Adjacent Vertices on a Permutohedron. SIAM Journal on Applied Mathematics, 32(2):323-327, March 1977.","title":"Adjacent Vertices on a Permutohedron","authors":"P Gaiha, S K Gupta","year":"1977","venue":"SIAM Journal on Applied Mathematics","doi":null,"url":null,"internal_url":null},{"position":14,"raw":"[Goodfellow et al., 2016] Ian Goodfellow, Yoshua Bengio, and Aaron Courville. Deep Learning. MIT Press, Cam- bridge, MA, USA, 2016. http://www.deeplearningbook. org.","title":"Deep Learning","authors":"Ian Goodfellow, Yoshua Bengio, Aaron Courville","year":"2016","venue":null,"doi":null,"url":"http://www.deeplearningbook.org","internal_url":null},{"position":15,"raw":"[Henry and Mourifié, 2013] Marc Henry and Ismael Mou- rifié. Euclidean revealed preferences: Testing the spa- tial voting model. Journal of Applied Econometrics, 28(4):650-666, 2013.","title":"Euclidean revealed preferences: Testing the spatial voting model","authors":"Mourifié Henry","year":"2013","venue":"Journal of Applied Econometrics","doi":null,"url":null,"internal_url":null},{"position":16,"raw":"[Hinich and Munger, 2008] Melvin J. Hinich and Michael C. Munger. Spatial theory. In Readings in Public Choice and Constitutional Political Economy, pages 295-304.","title":"Spatial theory","authors":"Melvin J Hinich, Michael C Munger","year":null,"venue":"Readings in Public Choice and Constitutional Political Economy","doi":null,"url":null,"internal_url":null},{"position":17,"raw":"[Houdré and Is ¸lak, 2022] Christian Houdré and Ümit Is ¸lak. A Central Limit Theorem for the Length of the Longest Common Subsequences in Random Words. arXiv preprint arXiv:1408.1559, March 2022.","title":"Christian Houdré and Ümit Is ¸lak. A Central Limit Theorem for the Length of the Longest Common Subsequences in Random Words","authors":"Is Houdré, ¸lak","year":"2022","venue":null,"doi":null,"url":null,"internal_url":null},{"position":18,"raw":"[Houdré and Xu, 2018] Christian Houdré and Chen Xu. A Note on the Expected Length of the Longest Common Subsequences of two i.i.d. Random Permutations. arXiv preprint arXiv:1703.07691, June 2018.","title":"A Note on the Expected Length of the Longest Common Subsequences of two i.i.d. Random Permutations","authors":"Xu Houdré, Christian Houdré, Chen Xu","year":"2018","venue":null,"doi":null,"url":null,"internal_url":null},{"position":19,"raw":"[Humphreys and Laver, 2010] Macartan Humphreys and Michael Laver. Spatial models, cognitive metrics, and majority rule equilibria. British Journal of Political Science, 40(1):11-30, 2010.","title":"Spatial models, cognitive metrics, and majority rule equilibria","authors":"Laver Humphreys","year":"2010","venue":"British Journal of Political Science","doi":null,"url":null,"internal_url":null},{"position":20,"raw":"[Huszár et al., 2021] Ferenc Huszár, Sofia Ira Ktena, Conor O'Brien, Luca Belli, Andrew Schlaikjer, and Moritz Hardt. Algorithmic amplification of politics on twit- ter. Proceedings of the National Academy of Sciences, 119(1):e2025334119, 2021.","title":"Algorithmic amplification of politics on twitter","authors":"Huszár","year":"2021","venue":"Proceedings of the National Academy of Sciences","doi":null,"url":null,"internal_url":null},{"position":21,"raw":"[Johnson et al., 2021] Skyler B Johnson, Matthew Parsons, Tanya Dorff, Meena S Moran, John H Ward, Stacey A Cohen, Wallace Akerley, Jessica Bauman, Joleen Hub- bard, Daniel E Spratt, Carma L Bylund, Briony Swire- Thompson, Tracy Onega, Laura D Scherer, Jonathan Tward, and Angela Fagerlin. Cancer Misinformation and Harmful Information on Facebook and Other Social Me- dia: A Brief Report. JNCI: Journal of the National Cancer Institute, 114(7):1036-1039, 07 2021.","title":"Cancer Misinformation and Harmful Information on Facebook and Other Social Media: A Brief Report","authors":"Matthew Skyler B Johnson, Tanya Parsons, Dorff, John H Meena S Moran, Stacey A Ward, Wallace Cohen, Jessica Akerley, Joleen Bauman, Daniel E Hubbard, Carma L Spratt, Briony Bylund, Tracy Swire-Thompson, Laura D Onega, Jonathan Scherer, Angela Tward, Fagerlin","year":"2021","venue":"JNCI: Journal of the National Cancer Institute","doi":null,"url":null,"internal_url":null},{"position":22,"raw":"[Jurafsky and Martin, 2022] Daniel Jurafsky and James H. Martin. Speech and language processing. https://web. stanford.edu/ ∼ jurafsky/slp3/ed3book jan122022.pdf, 2022. Accessed: 2022-08-09.","title":"Speech and language processing","authors":"Daniel Jurafsky, James H Martin","year":"2022","venue":null,"doi":null,"url":"https://web.stanford.edu/∼jurafsky/slp3/ed3bookjan122022.pdf","internal_url":null},{"position":23,"raw":"[Kumar and Vassilvitskii, 2010] Ravi Kumar and Sergei Vassilvitskii. Generalized distances between rankings. In Proceedings of the 19th International Conference on World Wide Web, WWW '10, page 571-580, New York, NY, USA, 2010. Association for Computing Machinery.","title":"Generalized distances between rankings","authors":"Ravi Kumar, Sergei Vassilvitskii","year":"2010","venue":"Proceedings of the 19th International Conference on World Wide Web, WWW '10","doi":null,"url":null,"internal_url":null},{"position":24,"raw":"[Luaces et al., 2015] Oscar Luaces, Jorge Díez, Thorsten Joachims, and Antonio Bahamonde. Mapping preferences into Euclidean space. Expert Systems with Applications, 42(22):8588-8596, December 2015.","title":"Mapping preferences into Euclidean space","authors":"Luaces","year":"2015","venue":"Expert Systems with Applications","doi":null,"url":null,"internal_url":null},{"position":25,"raw":"[Miller, 2015] Nicholas R. Miller. The spatial model of so- cial choice and voting, chapter 10, pages 163-181. Ed- ward Elgar Publishing, Cheltenham, UK, 2015.","title":"The spatial model of social choice and voting","authors":"R Nicholas, Miller","year":"2015","venue":null,"doi":null,"url":null,"internal_url":null},{"position":26,"raw":"[Nagy and Anderson, 1984] William E. Nagy and Richard C. Anderson. How many words are there in printed school english? Reading Research Quarterly, 19(3):304-330, 1984.","title":"How many words are there in printed school english","authors":"William E Nagy, Richard C Anderson","year":"1984","venue":"Reading Research Quarterly","doi":null,"url":null,"internal_url":null},{"position":27,"raw":"[Neal, 2008] Peter Neal. The Generalised Coupon Collector Problem. Journal of Applied Probability, 45(3):621-629, 2008.","title":"The Generalised Coupon Collector Problem","authors":"Peter Neal","year":"2008","venue":"Journal of Applied Probability","doi":null,"url":null,"internal_url":null},{"position":28,"raw":"[Obraztsova and Elkind, 2012] Svetlana Obraztsova and Edith Elkind. Optimal manipulation of voting rules. In Proceedings of the Twenty-Sixth AAAI Conference on Artificial Intelligence, AAAI'12, page 2141-2147, Toronto, Ontario, Canada, 2012. AAAI Press.","title":"Optimal manipulation of voting rules","authors":"Elkind Obraztsova, Svetlana Obraztsova, Edith Elkind","year":"2012","venue":"Proceedings of the Twenty-Sixth AAAI Conference on Artificial Intelligence, AAAI'12","doi":null,"url":null,"internal_url":null},{"position":29,"raw":"[Obraztsova et al., 2013] Svetlana Obraztsova, Edith Elkind, Piotr Faliszewski, and Arkadii Slinko. On swap-distance geometry of voting rules. In Proceedings of the 2013 International Conference on Autonomous Agents and Multi-Agent Systems, AAMAS '13, page 383-390, Rich- land, SC, 2013. International Foundation for Autonomous Agents and Multiagent Systems.","title":"On swap-distance geometry of voting rules","authors":"Svetlana Obraztsova, Edith Elkind, Piotr Faliszewski, Arkadii Slinko","year":"2013","venue":"Proceedings of the 2013 International Conference on Autonomous Agents and Multi-Agent Systems, AAMAS '13","doi":null,"url":null,"internal_url":null},{"position":30,"raw":"[Pan and Ding, 2019] Shimei Pan and Tao Ding. Social media-based user embedding: A literature review. arXiv preprint arXiv:1907.00725, 2019.","title":"Shimei Pan and Tao Ding. Social media-based user embedding: A literature review","authors":"Ding Pan","year":"2019","venue":null,"doi":null,"url":null,"internal_url":null},{"position":31,"raw":"[Peters, 2017] Dominik Peters. Recognising multidimen- sional euclidean preferences. In Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence, AAAI'17, page 642-648, San Francisco, CA, USA, 2017. AAAI Press.","title":"Recognising multidimensional euclidean preferences","authors":"Dominik Peters","year":"2017","venue":"Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence, AAAI'17","doi":null,"url":null,"internal_url":null},{"position":32,"raw":"[Rodríguez, 2011] Gonzalo Rivero Rodríguez. Integrality and separability in multidimensional voting models: Ideol- ogy and nationalism in spanish regional elections. Techni- cal Report 265, Juan March Institute Center for Advanced Study in the Social Sciences, 2011.","title":"Integrality and separability in multidimensional voting models: Ideology and nationalism in spanish regional elections","authors":"Gonzalo Rivero, Rodríguez","year":"2011","venue":null,"doi":null,"url":null,"internal_url":null},{"position":33,"raw":"[Satuluri et al., 2020] Venu Satuluri, Yao Wu, Xun Zheng, Yilei Qian, Brian Wichers, Qieyun Dai, Gui Ming Tang, Jerry Jiang, and Jimmy Lin. Simclusters: Community- based representations for heterogeneous recommendations at twitter. In Proceedings of the 26th ACM SIGKDD In- ternational Conference on Knowledge Discovery & Data Mining, KDD '20, page 3183-3193, New York, NY, USA, 2020. Association for Computing Machinery.","title":"Simclusters: Communitybased representations for heterogeneous recommendations at twitter","authors":"Venu Satuluri, Yao Wu, Xun Zheng, Yilei Qian, Brian Wichers, Qieyun Dai, Gui, Ming Tang, Jerry Jiang, Jimmy Lin","year":"2020","venue":"Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, KDD '20","doi":null,"url":null,"internal_url":null},{"position":34,"raw":"[Small et al., 2021] Christopher Small, Michael Bjorkegren, Timo Erkkilä, Lynette Shaw, and Colin Megill. Polis: Scaling deliberation by mapping high dimensional opinion spaces. Recerca. Revista de Pensament i Anàlisi, 26(2):1- 26, 2021.","title":"Polis: Scaling deliberation by mapping high dimensional opinion spaces","authors":"Christopher Small, Michael Bjorkegren, Timo Erkkilä, Lynette Shaw, Colin Megill","year":"2021","venue":"Recerca. Revista de Pensament i Anàlisi","doi":null,"url":null,"internal_url":null},{"position":35,"raw":"[Twitter, 2022] Twitter. Community Notes Guide: Note Ranking. https://twitter.github.io/communitynotes/ ranking-notes/, 2022. Accessed: 2023-01-17.","title":"Twitter. Community Notes Guide: Note Ranking","authors":null,"year":"2022","venue":null,"doi":null,"url":"https://twitter.github.io/communitynotes/ranking-notes/,2022","internal_url":null},{"position":36,"raw":"[Tydrichová, 2023] Magdaléna Tydrichová. Structural and algorithmic aspects of preference domain restrictions in collective decision making: Contributions to the study of single-peaked and Euclidean preferences. PhD thesis, Sor- bonne Université, March 2023.","title":"Magdaléna Tydrichová. Structural and algorithmic aspects of preference domain restrictions in collective decision making: Contributions to the study of single-peaked and Euclidean preferences","authors":"Tydrichová","year":"2023","venue":null,"doi":null,"url":null,"internal_url":null}],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}