{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:13:04Z","timestamp":1760235184296,"version":"build-2065373602"},"reference-count":40,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,7,27]],"date-time":"2021-07-27T00:00:00Z","timestamp":1627344000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["GRK 1994"],"award-info":[{"award-number":["GRK 1994"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Ordinal embedding is the task of computing a meaningful multidimensional representation of objects, for which only qualitative constraints on their distance functions are known. In particular, we consider comparisons of the form \u201cWhich object from the pair (j,k) is more similar to object i?\u201d. In this paper, we generalize this framework to the case where the ordinal constraints are not given at the level of individual points, but at the level of sets, and propose a distributional triplet embedding approach in a scalable learning framework. We show that the query complexity of our approach is on par with the single-item approach. Without having access to features of the items to be embedded, we show the applicability of our model on toy datasets for the task of reconstruction and demonstrate the validity of the obtained embeddings in experiments on synthetic and real-world datasets.<\/jats:p>","DOI":"10.3390\/e23080964","type":"journal-article","created":{"date-parts":[[2021,7,27]],"date-time":"2021-07-27T12:18:31Z","timestamp":1627388311000},"page":"964","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Learning Ordinal Embedding from Sets"],"prefix":"10.3390","volume":"23","author":[{"given":"A\u00efssatou","family":"Diallo","sequence":"first","affiliation":[{"name":"Research Training Group AIPHES, Technische Universit\u00e4t Darmstadt, 64289 Darmstadt, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1207-0159","authenticated-orcid":false,"given":"Johannes","family":"F\u00fcrnkranz","sequence":"additional","affiliation":[{"name":"Computational Data Analytics, FAW, Johannes Kepler University Linz, 4040 Linz, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1007\/BF02289630","article-title":"The analysis of proximities: Multidimensional scaling with an unknown distance function. I","volume":"27","author":"Shepard","year":"1962","journal-title":"Psychometrika"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1016\/0022-2496(66)90017-4","article-title":"Metric structures in ordinal data","volume":"3","author":"Shepard","year":"1966","journal-title":"J. Math. Psychol."},{"key":"ref_3","unstructured":"Agarwal, S., Wills, J., Cayton, L., Lanckriet, G., Kriegman, D., and Belongie, S. (2007, January 21\u201324). Generalized non-metric multidimensional scaling. Proceedings of the 11th International Conference on Artificial Intelligence and Statistics (AISTATS), San Juan, PR, USA."},{"key":"ref_4","unstructured":"McFee, B. (2012). More Like This: Machine Learning Approaches to Music Similarity. [Ph.D. Thesis, University of California]."},{"key":"ref_5","unstructured":"Tamuz, O., Liu, C., Belongie, S., Shamir, O., and Kalai, A. (July, January 28). Adaptively Learning the Crowd Kernel. Proceedings of the 28th International Conference on Machine Learning (ICML), Bellevue, WA, USA."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Van Der Maaten, L., and Weinberger, K. (2012, January 23\u201326). Stochastic triplet embedding. Proceedings of the IEEE International Workshop on Machine Learning for Signal Processing, Santander, Spain.","DOI":"10.1109\/MLSP.2012.6349720"},{"key":"ref_7","unstructured":"Terada, Y., and von Luxburg, U. (2014, January 21\u201326). Local ordinal embedding. Proceedings of the 31st International Conference on Machine Learning (ICML), Beijing, China."},{"key":"ref_8","unstructured":"Kleindessner, M., and von Luxburg, U. (2017, January 4\u20139). Kernel functions based on triplet comparisons. Proceedings of the Advances in Neural Information Processing Systems 30, Long Beach, CA, USA."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Jamieson, K.G., and Nowak, R.D. (2011, January 28\u201330). Low-dimensional embedding using adaptively selected ordinal data. Proceedings of the 49th Annual Allerton Conference on Communication, Control, and Computing, Monticello, IL, USA.","DOI":"10.1109\/Allerton.2011.6120287"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Diallo, A., and F\u00fcrnkranz, J. (2021). Elliptical Ordinal Embedding. arXiv.","DOI":"10.1007\/978-3-030-88942-5_25"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1145\/3130332.3130334","article-title":"Accurately interpreting clickthrough data as implicit feedback","volume":"51","author":"Joachims","year":"2017","journal-title":"ACM SIGIR Forum"},{"key":"ref_12","unstructured":"Muzellec, B., and Cuturi, M. (2018, January 3\u20138). Generalizing point embeddings using the Wasserstein space of elliptical distributions. Proceedings of the Advances in Neural Information Processing System 31, Montr\u00e9al, QC, Canada."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1016\/0024-3795(82)90112-4","article-title":"The distance between two random vectors with given dispersion matrices","volume":"48","author":"Olkin","year":"1982","journal-title":"Linear Algebra Appl."},{"key":"ref_14","unstructured":"Frogner, C., Zhang, C., Mobahi, H., Araya, M., and Poggio, T.A. (2015, January 2\u20135). Learning with a Wasserstein loss. Proceedings of the Advances in Neural Information Processing Systems 28, Montr\u00e9al, QB, Canada."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2663","DOI":"10.1088\/0305-4470\/32\/14\/007","article-title":"Explicit formulae for the Bures metric","volume":"32","author":"Dittmann","year":"1999","journal-title":"J. Phys. Math. Gen."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1214\/aos\/1176343842","article-title":"Minimum Hellinger distance estimates for parametric models","volume":"5","author":"Beran","year":"1977","journal-title":"Ann. Stat."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"LeCun, Y., Chopra, S., Hadsell, R., Ranzato, M., and Huang, F.J. (2006). A Tutorial on Energy-Based Learning. \u201cPredicting Structured Data, MIT Press.","DOI":"10.7551\/mitpress\/7443.003.0014"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Hoffer, E., and Ailon, N. (2015, January 7\u20139). Deep metric learning using triplet network. Proceedings of the 3rd International Conference on Similarity-Based Pattern Recognition (ICLR Workshop Track), San Diego, CA, USA.","DOI":"10.1007\/978-3-319-24261-3_7"},{"key":"ref_19","unstructured":"Haghiri, S., Vankadara, L.C., and von Luxburg, U. (2019). Large scale representation learning from triplet comparisons. arXiv."},{"key":"ref_20","unstructured":"Chen, B., and Ankenman, J. (2006). The Mathematics of Poker, ConJelCo LLC., Jerrod."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1017\/S1351324910000306","article-title":"Natural language processing with python","volume":"17","author":"Xue","year":"2011","journal-title":"Nat. Lang. Eng."},{"key":"ref_22","unstructured":"Jain, L., Jamieson, K.G., and Nowak, R. (2016, January 5\u201310). Finite Sample Prediction and Recovery Bounds for Ordinal Embedding. Proceedings of the Advances in Neural Information Processing Systems 29, Barcelona, Spain."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Heikinheimo, H., and Ukkonen, A. (2013, January 7\u20139). The crowd-median algorithm. Proceedings of the 1st AAAI Conference on Human Computation and Crowdsourcing (HCOMP), Palm Springs, CA, USA.","DOI":"10.1609\/hcomp.v1i1.13079"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Ukkonen, A., Derakhshan, B., and Heikinheimo, H. (2015, January 8\u201311). Crowdsourced nonparametric density estimation using relative distances. Proceedings of the 3rd AAAI Conference on Human Computation and (HCOMP), San Diego, CA, USA.","DOI":"10.1609\/hcomp.v3i1.13232"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Ukkonen, A. (2017, January 18\u201321). Crowdsourced correlation clustering with relative distance comparisons. Proceedings of the IEEE International Conference on Data Mining (ICDM), New Orleans, LA, USA.","DOI":"10.1109\/ICDM.2017.148"},{"key":"ref_26","unstructured":"Guyon, I., Luxburg, U.V., Bengio, S., Wallach, H., Fergus, R., Vishwanathan, S., and Garnett, R. (2017). Deep Sets. Advances in Neural Information Processing Systems 30, Curran Associates, Inc."},{"key":"ref_27","unstructured":"Qi, C.R., Su, H., Mo, K., and Guibas, L.J. (2017, January 21\u201326). Pointnet: Deep learning on point sets for 3d classification and segmentation. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Honolulu, HI, USA."},{"key":"ref_28","unstructured":"Cortes, C., Lawrence, N., Lee, D., Sugiyama, M., and Garnett, R. (2015). Pointer Networks. Advances in Neural Information Processing Systems 28, Curran Associates, Inc."},{"key":"ref_29","unstructured":"Vinyals, O., Bengio, S., and Kudlur, M. (2016, January 2\u20134). Order Matters: Sequence to sequence for sets. Proceedings of the 4th International Conference on Learning Representations (ICLR), San Juan, PR, USA."},{"key":"ref_30","unstructured":"Zhang, Y., Hare, J., and Pr\u00fcgel-Bennett, A. (May, January 26). FSPool: Learning Set Representations with Featurewise Sort Pooling. Proceedings of the International Conference on Learning Representations (ICLR), Addis Ababa, Ethiopia."},{"key":"ref_31","unstructured":"Zhang, Y., Hare, J., and Pr\u00fcgel-Bennett, A. (2019, January 8\u201314). Deep Set Prediction Networks. Proceedings of the Advances in Neural Information Processing Systems 32, Vancouver, BC, Canada."},{"key":"ref_32","unstructured":"Zhang, Y., Hare, J., and Pr\u00fcgel-Bennett, A. (2019, January 6\u20139). Learning Representations of Sets through Optimized Permutations. Proceedings of the International Conference on Learning Representations (ICLR), New Orleans, LA, USA."},{"key":"ref_33","unstructured":"Diallo, A., Zopf, M., and F\u00fcrnkranz, J. (September, January 31). Permutation Learning via Lehmer Codes. Proceedings of the 24th European Conference on Artificial Intelligence (ECAI), Santiago de Compostela, Spain."},{"key":"ref_34","unstructured":"Mena, G., Belanger, D., Linderman, S., and Snoek, J. (May, January 30). Learning Latent Permutations with Gumbel-Sinkhorn Networks. Proceedings of the International Conference on Learning Representations (ICLR), Vancouver, BC, Canada."},{"key":"ref_35","unstructured":"Weston, J., Chopra, S., and Bordes, A. (2014). Memory networks. arXiv."},{"key":"ref_36","unstructured":"Lee, J., Lee, Y., Kim, J., Kosiorek, A., Choi, S., and Teh, Y.W. (2019, January 10\u201315). Set transformer: A framework for attention-based permutation-invariant neural networks. Proceedings of the International Conference on Machine Learning, PMLR, Long Beach, CA, USA."},{"key":"ref_37","unstructured":"Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A.N., Kaiser, \u0141., and Polosukhin, I. (2017, January 4\u20139). Attention is all you need. Proceedings of the Advances in Neural Information Processing Systems, Long Beach, CA, USA."},{"key":"ref_38","unstructured":"Murphy, R.L., Srinivasan, B., Rao, V., and Ribeiro, B. (2019, January 6\u20139). Janossy Pooling: Learning Deep Permutation-Invariant Functions for Variable-Size Inputs. Proceedings of the International Conference on Learning Representations, New Orleans, LA, USA."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1007\/s11263-019-01217-w","article-title":"Robust attentional aggregation of deep feature sets for multi-view 3D reconstruction","volume":"128","author":"Yang","year":"2020","journal-title":"Int. J. Comput. Vis."},{"key":"ref_40","unstructured":"Skianis, K., Nikolentzos, G., Limnios, S., and Vazirgiannis, M. (2020, January 26\u201328). Rep the set: Neural networks for learning set representations. Proceedings of the International Conference on Artificial Intelligence and Statistics. PMLR, Palermo, Italy."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/8\/964\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:35:40Z","timestamp":1760164540000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/8\/964"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,27]]},"references-count":40,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2021,8]]}},"alternative-id":["e23080964"],"URL":"https:\/\/doi.org\/10.3390\/e23080964","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2021,7,27]]}}}