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The Frobenius inner product separates into two vector dot products, while the norm separates into the product of the two vector lengths and |s|. These identities retain the geometry of the effective matrix while avoiding its explicit construction. They are exact in real arithmetic; numerical rounding still matters near cancellation.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| U | Rank-one update |
| a | Input factor |
| b | Output factor |
| s | Scale coefficient |
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
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Original context
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The factorization is not unique. For nonzero ct, replacing bt by ctbt and at by at/ct preserves Ut. In particular, simultaneous sign reversal changes both factor directions but not the effective operator. Consequently, we calculate factor metrics and corresponding effective-update metrics. This checks for a purely representational explanation of a factor change without assuming that the source study’s finding is such an artifact. 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
Exact original notation
⟨{U}_{u},{U}_{v}{⟩}_{F}={s}^{2}\left({b}_{u}^{T}{b}_{v}\right)\left({a}_{u}^{T}{a}_{v}\right), \Vert {U}_{t}{\Vert }_{F}=\left|s\right|\Vert {b}_{t}{\Vert }_{2}\Vert {a}_{t}{\Vert }_{2}.Related builds
Sources & provenance
⟨{U}_{u},{U}_{v}{⟩}_{F}={s}^{2}\left({b}_{u}^{T}{b}_{v}\right)\left({a}_{u}^{T}{a}_{v}\right), \Vert {U}_{t}{\Vert }_{F}=\left|s\right|\Vert {b}_{t}{\Vert }_{2}\Vert {a}_{t}{\Vert }_{2}.
⟨{U}_{u},{U}_{v}{⟩}_{F}={s}^{2}\left({b}_{u}^{T}{b}_{v}\right)\left({a}_{u}^{T}{a}_{v}\right), \Vert {U}_{t}{\Vert }_{F}=\left|s\right|\Vert {b}_{t}{\Vert }_{2}\Vert {a}_{t}{\Vert }_{2}.