Squared distance between effective update matrices

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Expanding the squared Frobenius norm gives two squared norms minus twice the inner product. Combined with rank-one factorization identities, this calculates matrix distance without constructing the matrices. When the two updates are extremely close, subtraction of similar large numbers can lose precision and should be checked against a stable difference calculation.

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Original context

(4.3)

Here the subscript F denotes the Frobenius inner product or norm, and s is the fixed positive coefficient. It follows that

∥Uu−Uv∥F2=∥Uu∥F2+∥Uv∥F2−2⟨Uu,Uv⟩F.

(4.4)

When both effective norms are nonzero, their normalized inner product gives the effective cosine. At the fixed positive scale used here, it equals the product of the two factor cosines. These matrix metrics are invariant under Eq. (4.2), whereas individual factor norms are not.

4.2. Local bend and the role of the filter

\Vert {U}_{u}-{U}_{v}{\Vert }_{F}^{2}=\Vert {U}_{u}{\Vert }_{F}^{2}+\Vert {U}_{v}{\Vert }_{F}^{2}-2⟨{U}_{u},{U}_{v}{⟩}_{F}.

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