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On Degenerate Nonmetric Unfolding Solutions

Abstract

Loss functions proposed for nonmetric unfolding are discussed critically. Practical experience suggests that they do not work, mathematical reasons are sought why this is. Although the loss functions are constructed in such a way that they are ill-behaved at trivial solutions, it is shown that they are rather well-behaved along differentiable paths to trivial solutions. It is shown that in the neighborhood of trivial solutions we can find infinitely many nonmetric unfolding solutions which cannot be distinguished from stationary points, and which can differ very considerably in loss function value. The conclusion is that nonmetric unfolding, as currently formalized, is an inherently ill-posed problem and that a different approach is called for.

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