The ℓ 1 oblique procrustes problem

Trendafilov, Nickolay T. and Watson, G. A. (2004). The ℓ 1 oblique procrustes problem. Statistics and Computing, 14(1) pp. 39–51.

DOI: https://doi.org/10.1023/B:STCO.0000009415.14785.2a

Abstract

In this paper, we reconsider the well-known oblique Procrustes problem where the usual least-squares objective function is replaced by a more robust discrepancy measure, based on the ℓ1 norm or smooth approximations of it. We propose two approaches to the solution of this problem. One approach is based on convex analysis and uses the structure of the problem to permit a solution to the ℓ1 norm problem. An alternative approach is to smooth the problem by working with smooth approximations to the ℓ 1 norm, and this leads to a solution process based on the solution of ordinary differential equations on manifolds. The general weighted Procrustes problem (both orthogonal and oblique) can also be solved by the latter approach. Numerical examples to illustrate the algorithms which have been developed are reported and analyzed.

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