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By A. I. Fetísov

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In preparation. B. Tenenbaum, V. C. Langford. A global geometric framework for nonlinear dimensionality reduction. Science, 290:2319–2323, 2000. [49] R. Tibshirani. Principal curves revisited. Statistics and Computing, 2:183–190, 1992. [50] M. Trosset. Applications of multidimensional scaling to molecular conformation. Computing Science and Statistics, (29):148–152, 1998. P. O. J. van den Herik. Dimensionality reduction: a comparative review. 2007. [52] Kilian Q. Weinberger and Lawrence K. Saul.

Recall that the order of vanishing at v = ∞ of a rational function pq ∈ C(t) (p, q ∈ C[t]) equals deg(q) − deg(p). The basic properties of these multiplicities are: • ordv (ρ) = (0, 0) except for a finite number of v ∈ P1 and • v∈P1 ordv (ρ) = (0, 0). We next define an auxiliary operation which produces a convex lattice polygon from a balanced family of vectors of the plane. Let B ⊂ Z2 be a family of vectors which are zero except for a finite number of them and such that b∈B b = (0, 0). We denote by P(B) ⊂ (R≥0 )2 the (unique) convex polygon obtained by: 1) rotating −90◦ the non-zero vectors of B, 2) concatenating them following their directions counterclockwise and 3) translating the resulting polygon to the first quadrant (R≥0 )2 in such a way that it “touches” the coordinate axes (Figure 5).

On Knowledge Discovery and Data Mining, 2003. [58] Hao Zhang, Oliver van Kaick, and Ramsay Dyer. Spectral mesh processing. Computer Graphics Forum (to appear), 2008. [59] A. M. C. Berendsen. Essential dynamics of proteins. Proteins: Structure, Function, and Genetics, 17(4):412–425, 1993. D. S. Kunz, F-Y. Li, A. J. Wales. From topographies to dynamics on multidimensional potential energy surfaces of atomic clusters. Science, 271(5251):963 – 966, 1996. [61] O. Becker and M. Karplus. The topology of multidimensional potential energy surfaces: Theory and application to peptide structure and kinetics.

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