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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.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| U_u | Earlier update |
| U_v | Later update |
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Original context
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The Frobenius inner product is available without materializing a dense matrix: ⟨Uu,Uv⟩F=s2buTbvauTav, ∥Ut∥F=s∥bt∥2∥at∥2. The squared distance is ∥Uv−Uu∥F2=∥Uv∥F2+∥Uu∥F2−2⟨Uu,Uv⟩F. When both updates are nonzero, their cosine is the normalized inner product. At fixed positive scale it is also the product of the two factor cosines. Matrix-level metrics are invariant to the reciprocal rescalings above, unlike raw factor norms. 4.4 Local path bend and amplitude eligibility The local metric used in the released analysis compares displacements from a center checkpoint to earlier and later checkpoints [3]. For Zt=bt or Zt=Ut, define cZt;k=⟨Zt−k−Zt,Zt+k−Zt⟩∥Zt−k−Zt∥ ∥Zt+k−Zt∥.
Exact original notation
\Vert {U}_{v}-{U}_{u}{\Vert }_{F}^{2}=\Vert {U}_{v}{\Vert }_{F}^{2}+\Vert {U}_{u}{\Vert }_{F}^{2}-2⟨{U}_{u},{U}_{v}{⟩}_{F}Related builds
Sources & provenance
\Vert {U}_{v}-{U}_{u}{\Vert }_{F}^{2}=\Vert {U}_{v}{\Vert }_{F}^{2}+\Vert {U}_{u}{\Vert }_{F}^{2}-2⟨{U}_{u},{U}_{v}{⟩}_{F}
\Vert {U}_{v}-{U}_{u}{\Vert }_{F}^{2}=\Vert {U}_{v}{\Vert }_{F}^{2}+\Vert {U}_{u}{\Vert }_{F}^{2}-2⟨{U}_{u},{U}_{v}{⟩}_{F}