Technical Report CS-2008-12

Title: On convex formulation of 1D scaling problem
Authors: Michael Zibulevsky
Abstract: The 1D-scaling problem is to locate a set of points on an axis using [noisy] matrix of pair-wise distances between them. Straight-forward least-squares formulation of this problem is non-convex, and there is a danger of being trapped in a local minimum. We propose a convex variational formulation, which provides reliable solution to the prob- lem. Noise statistics can be incorporated into the objective in quasi-optimal way. We also introduce a parametric family of objective functions, which gradually transforms from convex to "statistically optimal" non-convex. Using sequential optimizations with gradually updated "non-convexity" parameter, further improvement of solution can be achieved. A priori knowledge about the order of point locations can be incorporated as a set of linear constraints.
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