• Bernoulli
  • Volume 23, Number 3 (2017), 1663-1693.

Some theory for ordinal embedding

Ery Arias-Castro

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Motivated by recent work on ordinal embedding (In Proceedings of the 27th Conference on Learning Theory (2014) 40–67), we derive large sample consistency results and rates of convergence for the problem of embedding points based on triple or quadruple distance comparisons. We also consider a variant of this problem where only local comparisons are provided. Finally, inspired by (In Communication, Control, and Computing (Allerton), 2011 49th Annual Allerton Conference on (2011) 1077–1084 IEEE), we bound the number of such comparisons needed to achieve consistency.

Article information

Bernoulli, Volume 23, Number 3 (2017), 1663-1693.

Received: January 2015
Revised: July 2015
First available in Project Euclid: 17 March 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

dissimilarity comparisons landmark multidimensional scaling non-metric multidimensional scaling (MDS) ordinal embedding


Arias-Castro, Ery. Some theory for ordinal embedding. Bernoulli 23 (2017), no. 3, 1663--1693. doi:10.3150/15-BEJ792.

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  • [1] Agarwal, S., Wills, J., Cayton, L., Lanckriet, G., Kriegman, D.J. and Belongie, S. (2007). Generalized non-metric multidimensional scaling. In International Conference on Artificial Intelligence and Statistics 11–18.
  • [2] Alestalo, P., Trotsenko, D.A. and Väisälä, J. (2001). Isometric approximation. Israel J. Math. 125 61–82.
  • [3] Aumann, R.J. and Kruskal, J.B. (1958). The coefficients in an allocation problem. Naval Res. Logist. Quart. 5 111–123.
  • [4] Borg, I. and Groenen, P.J.F. (2005). Modern Multidimensional Scaling: Theory and Applications, 2nd ed. Springer Series in Statistics. New York: Springer.
  • [5] Burago, D., Burago, Y. and Ivanov, S. (2001). A Course in Metric Geometry. Graduate Studies in Mathematics 33. Providence, RI: Amer. Math. Soc..
  • [6] Cuevas, A., Fraiman, R. and Pateiro-López, B. (2012). On statistical properties of sets fulfilling rolling-type conditions. Adv. in Appl. Probab. 44 311–329.
  • [7] Davenport, M.A. (2013). Lost without a compass: Nonmetric triangulation and landmark multidimensional scaling. In Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), 2013 IEEE 5th International Workshop on 13–16. IEEE.
  • [8] De Silva, V. and Tenenbaum, J.B. (2004). Sparse multidimensional scaling using landmark points. Technical report, Stanford Univ.
  • [9] Ellis, D.P., Whitman, B., Berenzweig, A. and Lawrence, S. (2002). The quest for ground truth in musical artist similarity. In Proceedings of the International Symposium on Music Information Retrieval (ISMIR) 170–177.
  • [10] Federer, H. (1959). Curvature measures. Trans. Amer. Math. Soc. 93 418–491.
  • [11] Horn, R.A. and Johnson, C.R. (1990). Matrix Analysis. Cambridge: Cambridge Univ. Press.
  • [12] Jamieson, K.G. and Nowak, R.D. (2011). Low-dimensional embedding using adaptively selected ordinal data. In Communication, Control, and Computing (Allerton), 2011 49th Annual Allerton Conference on 1077–1084. IEEE.
  • [13] Kelley, J.L. (1975). General Topology. New York: Springer.
  • [14] Kleindessner, M. and Von Luxburg, U. (2014). Uniqueness of ordinal embedding. In Proceedings of the 27th Conference on Learning Theory 40–67.
  • [15] Kruskal, J.B. (1964). Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika 29 1–27.
  • [16] McFee, B. and Lanckriet, G. (2011). Learning multi-modal similarity. J. Mach. Learn. Res. 12 491–523.
  • [17] Nhat, V.D.M., Vo, N., Challa, S. and Lee, S. (2008). Nonmetric MDS for sensor localization. In 3rd International Symposium on Wireless Pervasive Computing (ISWPC) 396–400.
  • [18] Shepard, R.N. (1962). The analysis of proximities: Multidimensional scaling with an unknown distance function. I. Psychometrika 27 125–140.
  • [19] Shepard, R.N. (1962). The analysis of proximities: Multidimensional scaling with an unknown distance function. II. Psychometrika 27 219–246.
  • [20] Shepard, R.N. (1966). Metric structures in ordinal data. J. Math. Psych. 3 287–315.
  • [21] Sikorska, J. and Szostok, T. (2004). On mappings preserving equilateral triangles. J. Geom. 80 209–218.
  • [22] Suppes, P. and Winet, M. (1955). An axiomatization of utility based on the notion of utility differences. Management Sci. 1 259–270.
  • [23] Terada, Y. and Von Luxburg, U. (2014). Local ordinal embedding. In Proceedings of the 31st International Conference on Machine Learning (ICML-14) 847–855.
  • [24] Vestfrid, I.A. (2003). Linear approximation of approximately linear functions. Aequationes Math. 66 37–77.
  • [25] Von Luxburg, U. and Alamgir, M. (2013). Density estimation from unweighted k-nearest neighbor graphs: A roadmap. In Advances in Neural Information Processing Systems 225–233.
  • [26] Waldmann, S. (2014). Topology: An Introduction. Cham: Springer.
  • [27] Young, F.W. and Hamer, R.M., eds. (1987). Multidimensional Scaling: History, Theory, and Applications. Lawrence Erlbaum Associates.