Alternative Measures of Fit for the Schönemann-Carroll Matrix Fitting Algorithm
Generate an AI Snapshot to get a quick, structured summary of this paper.
A concise AI-generated summary of the paper will appear here once you click Generate AI Snapshot.
TL;DR
Two new measures of fit are suggested which do not depend upon the norms of A andB, which are (0, 1)-bounded, and which, therefore, provide meaningful answers for comparative analyses.
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, e s , 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.