Some counterexamples in multidimensional scaling |
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Authors: | John S Lew |
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Affiliation: | Mathematical Sciences Department, IBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598 USA |
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Abstract: | Given a set X with elements x, y,… which has a partial order < on the pairs of the Cartesian product X2, one may seek a distance function ? on such pairs (x, y) which satisfies ?(x1, y1) < ?(x2, y2) precisely when (x1, y1) < (x2, y2), and even demand a metric space (X, ?) with some such compatible ? which has an isometric imbedding into a finite-dimensional Euclidean space or a separable Hilbert space. We exhibit here systems (X, <) which cannot meet the latter demand. The space of real m-tuples (ξ1,…,ξm) with either the “city-block” norm Σi ∥ξi∥ or the “dominance” norm maxi, ∥ξi∥ cannot possibly become a subset of any finite-dimensional Euclidean space. The set of real sequences (ξ1, ξ2,…) with finitely many nonzero elements and the supremum norm supi, ∥ξi∥ cannot even become a subset of any separable Hilbert space. |
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