Alternative Measures of Fit for the Schönemann-Carroll Matrix Fitting Algorithm

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Abstract
In connection with a least-squares solution for fitting one matrix, A, to another, B, under optimal choice of a rigid motion and a dilation, Schönemann and Carroll suggested two measures of fit: a raw measure, e, and a refined similarity measure, es, which is symmetric. Both measures share the weakness of depending upon the norm of the target matrix, B, e.g., e(A, kB) ≠ e(A, B) for k ≠ 1. Therefore, both measures are useless for answering questions of the type: “Does A fit B better than A fits C?”. In this note two new measures of fit are suggested which do not depend upon the norms of A and B, which are (0, 1)-bounded, and which, therefore, provide meaningful answers for comparative analyses.

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