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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 |
How it works
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Original context
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∥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. 6.3 Executed versus prepared work The package includes adapter_geometry.py, which reads local .safetensors and configuration files and calculates both factor-level and effective-update quantities. Its algebra was checked against dense synthetic matrices and a synthetic simultaneous sign change. Those tests passed. The program was not run on the released model tensors because the binary acquisition path failed.
Exact original notation
\Vert \Delta {W}_{v}-\Delta {W}_{u}{\Vert }_{F}^{2}=\Vert \Delta {W}_{v}{\Vert }_{F}^{2}+\Vert \Delta {W}_{u}{\Vert }_{F}^{2}-2⟨\Delta {W}_{u},\Delta {W}_{v}{⟩}_{F}.Related builds
Sources & provenance
\Vert \Delta {W}_{v}-\Delta {W}_{u}{\Vert }_{F}^{2}=\Vert \Delta {W}_{v}{\Vert }_{F}^{2}+\Vert \Delta {W}_{u}{\Vert }_{F}^{2}-2⟨\Delta {W}_{u},\Delta {W}_{v}{⟩}_{F}.
\Vert \Delta {W}_{v}-\Delta {W}_{u}{\Vert }_{F}^{2}=\Vert \Delta {W}_{v}{\Vert }_{F}^{2}+\Vert \Delta {W}_{u}{\Vert }_{F}^{2}-2⟨\Delta {W}_{u},\Delta {W}_{v}{⟩}_{F}.