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Yes, I adopted those strategies earlier. But in my cases, it is not satisfied when two spectrum curves intersect. Because, the eigenvectors computed are not numerical stable when their eigenvalues are close.
As you said, a permutation with a scale transform will minimise the Frobenius norm. Thank you anyway. I tested the Frobenius norm by a general proposed optimisation program. The result is also what you proved. I think Frobenius norm is not what I need in application. I may consider induced norms.
Sorry, I failed to express my idea. My focus is about local structure, I misuse the concept of quasi-isometry. However I did not found a way to describe which two manifold are "almost" isometric. For example, the bounding surfaces of a non-rigid object may be "near" isometric of different poses. Their spectrum data seems be continuously change with respect to pose change. The deformation of a metric space will have corresponding change of their spectrum data. For two metric, considering their map $f$. inf{e: |d(x,y)-d(f(x),f(y)|< e} estimate compute min w.r.t $f$
