Read the equation
The matrix inner product factorizes into an inner product of the output factors and an inner product of the input factors, multiplied by their scale coefficients. With one fixed s, that coefficient is s². This identity permits exact operator comparisons using small vector calculations without allocating the dense matrix.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| U_u | Update at checkpoint u |
| U_v | Update at checkpoint v |
| a | Input factors |
| b | Output factors |
| 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
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
Read the surrounding source passage
6.2 Low-memory exact formulas The Frobenius inner product between two rank-one updates can be computed without constructing the dense matrices: ⟨ΔWu,ΔWv⟩F=susvbuTbvauTav. The norm is: ∥ΔWt∥F=st∥bt∥2∥at∥2. Consequently, the matrix distance follows from:
Exact original notation
⟨\Delta {W}_{u},\Delta {W}_{v}{⟩}_{F}={s}_{u}{s}_{v}\left({b}_{u}^{T}{b}_{v}\right)\left({a}_{u}^{T}{a}_{v}\right).Related builds
Sources & provenance
⟨\Delta {W}_{u},\Delta {W}_{v}{⟩}_{F}={s}_{u}{s}_{v}\left({b}_{u}^{T}{b}_{v}\right)\left({a}_{u}^{T}{a}_{v}\right).
⟨\Delta {W}_{u},\Delta {W}_{v}{⟩}_{F}={s}_{u}{s}_{v}\left({b}_{u}^{T}{b}_{v}\right)\left({a}_{u}^{T}{a}_{v}\right).