Frobenius norm of a rank-one update

Read the equation

The matrix norm is the product of the two factor lengths multiplied by the absolute scale. The final manuscript writes s without absolute bars because the pinned coefficient is positive. If either factor is zero, the effective operator and its norm are zero regardless of the other factor.

Terms & definitions

How it works

Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.

Assumptions & limits

Source recovery establishes that the expression appears in this record. Its mathematical validity, implementation behavior, and interpretation must be assessed separately. See the linked source and research notes for the checks available for this version.

Original context

⟨ΔWu,ΔWv⟩F=susvbuTbvauTav.

The norm is:

∥ΔWt∥F=st∥bt∥2∥at∥2.

Consequently, the matrix distance follows from:

∥ΔWv−ΔWu∥F2=∥ΔWv∥F2+∥ΔWu∥F2−2⟨ΔWu,ΔWv⟩F.

Normalizing the inner product gives a cosine similarity when both norms are nonzero. If an initial adapter update is exactly zero, its direction is undefined; assigning a cosine of zero would manufacture an apparent transition. The prepared implementation reports an undefined value rather than inventing a direction.

\Vert \Delta {W}_{t}{\Vert }_{F}=\left|{s}_{t}\right|\Vert {b}_{t}{\Vert }_{2}\Vert {a}_{t}{\Vert }_{2}.

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