document: Universal multi-dimensional scaling In this paper , we propose a unified algorithmic framework for solving many known variants of MDS . Our algorithm is a simple iterative scheme with guaranteed convergence , and is modular ; by changing the internals of a single subroutine in the algorithm , we can switch cost functions and target spaces easily . In addition to the formal guarantees of convergence , our algorithms are accurate ; in most cases , they converge to better quality solutions than existing methods in comparable time . Moreover , they have a small memory footprint and scale effectively for large data sets . We expect that this framework will be useful for a number of MDS variants that have not yet been studied . Our framework extends to embedding high-dimensional points lying on a sphere to points on a lower dimensional sphere , preserving geodesic distances . As a complement to this result , we also extend the Johnson-Lindenstrauss Lemma to this spherical setting , by showing that projecting to a random O ( ( 1 \/ µ2 ) log n ) - dimensional sphere causes only an eps-distortion in the geodesic distances .
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