Multidimensional scaling on metric measure spaces. (English) Zbl 1441.62934

Summary: Multidimensional scaling (MDS) is a popular technique for mapping a finite metric space into a low-dimensional Euclidean space in a way that best preserves pairwise distances. We overview the theory of classical MDS, along with its optimality properties and goodness of fit. Further, we present a notion of MDS on infinite metric measure spaces that generalizes these optimality properties. As a consequence we can study the MDS embeddings of the geodesic circle \(S^1\) into \(\mathbb{R}^m\) for all \(m\), and ask questions about the MDS embeddings of the geodesic \(n\)-spheres \(S^n\) into \(\mathbb{R}^m\). Finally, we address questions on convergence of MDS. For instance, if a sequence of metric measure spaces converges to a fixed metric measure space \(X\), then in what sense do the MDS embeddings of these spaces converge to the MDS embedding of \(X\)?


62R20 Statistics on metric spaces
62H25 Factor analysis and principal components; correspondence analysis
51F99 Metric geometry
15A18 Eigenvalues, singular values, and eigenvectors
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
Full Text: DOI arXiv Euclid


[1] Y. Bengio, O. Delalleau, N. L. Roux, J.-F. Paiement, P. Vincent, and M. Ouimet, “Learning eigenfunctions links spectral embedding and kernel PCA”, Neural computation 16:10 (2004), 2197-2219. · Zbl 1085.68120 · doi:10.1162/0899766041732396
[2] L. M. Blumenthal, Theory and applications of distance geometry, 2nd edition, Chelsea Publishing Co., New York, 1970. · Zbl 0208.24801
[3] M. Blumstein and H. Kvinge, “Multi-dimensional scaling on groups”, preprint, 2018.
[4] E. Bogomolny, O. Bohigas, and C. Schmit, “Spectral properties of distance matrices”, J. Phys. A 36:12 (2003), 3595-3616. · Zbl 1057.15027 · doi:10.1088/0305-4470/36/12/341
[5] A. Buja, D. F. Swayne, M. L. Littman, N. Dean, H. Hofmann, and L. Chen, “Data visualization with multidimensional scaling”, J. Comput. Graph. Statist. 17:2 (2008), 444-472. · Zbl 1388.62182 · doi:10.1198/106186008X318440
[6] T. F. Cox and M. A. A. Cox, Multidimensional scaling, 2nd ed., Monographs on Statistics and Applied Probability 88, Chapman & Hall, London, 2000. · Zbl 1004.91067
[7] J. Dattorro, Convex optimization and Euclidean distance geometry, Meboo, Palo Alto, CA, 2013. · Zbl 1142.15014 · doi:10.1016/j.laa.2007.12.008
[8] P. Diaconis, S. Goel, and S. Holmes, “Horseshoes in multidimensional scaling and local kernel methods”, Ann. Appl. Stat. 2:3 (2008), 777-807. · Zbl 1149.62316 · doi:10.1214/08-AOAS165
[9] P. J. F. Groenen and I. Borg, “Past, present, and future of multidimensional scaling”, pp. 95-117 in Visualization and verbalization of data, edited by J. Blasius and M. Greenacre, CRC Press, Boca Raton, FL, 2014.
[10] L. Kassab, Multidimensional scaling: infinite metric measure spaces, master’s thesis, Colorado State University, 2019,.
[11] V. Koltchinskii and E. Giné, “Random matrix approximation of spectra of integral operators”, Bernoulli 6:1 (2000), 113-167. · Zbl 0949.60078 · doi:10.2307/3318636
[12] T. Kühn, “Eigenvalues of integral operators generated by positive definite Hölder continuous kernels on metric compacta”, Nederl. Akad. Wetensch. Indag. Math. 49:1 (1987), 51-61. · Zbl 0622.47031
[13] J. de Leeuw and W. Heiser, “Theory of multidimensional scaling”, pp. 285-316 in Classification, pattern recognition and reduction of dimensionality, edited by P. R. Krishnaiah and L. N. Kanal, Handbook of Statist. 2, North-Holland, Amsterdam, 1982. · Zbl 0511.62076
[14] K. V. Mardia, “Some properties of classical multi-dimensional scaling”, Comm. Statist. A-Theory Methods 7:13 (1978), 1233-1241. · Zbl 0403.62033 · doi:10.1080/03610927808827707
[15] K. V. Mardia, J. T. Kent, and J. M. Bibby, Multivariate analysis, Academic Press, London, 1979. · Zbl 0432.62029
[16] F. Mémoli, “Gromov-Wasserstein distances and the metric approach to object matching”, Found. Comput. Math. 11:4 (2011), 417-487. · Zbl 1244.68078 · doi:10.1007/s10208-011-9093-5
[17] F. Mémoli, “The Gromov-Wasserstein distance: a brief overview”, Axioms 3:3 (2014), 335-341. · Zbl 1311.68174
[18] J. von Neumann and I. J. Schoenberg, “Fourier integrals and metric geometry”, Trans. Amer. Math. Soc. 50 (1941), 226-251. · Zbl 0028.41002 · doi:10.1090/S0002-9947-1941-0004644-8
[19] E. P\kekalska, P. Paclík, and R. P. W. Duin, “A generalized kernel approach to dissimilarity-based classification”, J. Mach. Learn. Res. 2:2 (2002), 175-211. · Zbl 1037.68127
[20] J. W. Sammon, “A nonlinear mapping for data structure analysis”, IEEE Trans. Computers C-18:5 (1969), 401-409.
[21] R. Sibson, “Studies in the robustness of multidimensional scaling: Procrustes statistics”, J. Roy. Statist. Soc. Ser. B 40:2 (1978), 234-238. · Zbl 0389.62086 · doi:10.1111/j.2517-6161.1978.tb01669.x
[22] R. Sibson, “Studies in the robustness of multidimensional scaling: perturbational analysis of classical scaling”, J. Roy. Statist. Soc. Ser. B 41:2 (1979), 217-229. · Zbl 0413.62046 · doi:10.1111/j.2517-6161.1979.tb01076.x
[23] R. Sibson, A. Bowyer, and C. Osmond, “Studies in the robustness of multidimensional scaling: Euclidean models and simulation studies”, J. Stat. Comp. Sim. 13:3-4 (1981), 273-296. · Zbl 0475.62049 · doi:10.1080/00949658108810502
[24] M. Trosset, “Computing distances between convex sets and subsets of the positive semidefinite matrices”, technical report, Rice University, 1997, https://hdl.handle.net/1911/101889.
[25] M. W. Trosset, “A new formulation of the nonmetric strain problem in multidimensional scaling”, J. Classification 15:1 (1998), 15-35. · Zbl 0916.92031 · doi:10.1007/s003579900018
[26] “Functional principal component analysis”, Wikipedia entry, 2018, https://tinyurl.com/FPCAwiki.
[27] W. · Zbl 0010.37802 · doi:10.2307/2372019
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