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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
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Original context
Read the surrounding source passage
This includes simultaneous reversal of both signs. I compare both the factors and their product to distinguish an operator change from an equivalent factorization. That comparison does not presume an error in the original study. The required matrix quantities reduce to vector operations: ⟨Uu,Uv⟩F=s2(buTbv)(auTav), ∥Ut∥F=s∥bt∥2∥at∥2. (4.3) Here the subscript F denotes the Frobenius inner product or norm, and s is the fixed positive coefficient. It follows that
Exact original notation
⟨{U}_{u},{U}_{v}{⟩}_{F}={s}^{2}({b}_{u}^{T}{b}_{v})({a}_{u}^{T}{a}_{v}),Related builds
Sources & provenance
⟨{U}_{u},{U}_{v}{⟩}_{F}={s}^{2}({b}_{u}^{T}{b}_{v})({a}_{u}^{T}{a}_{v}),
⟨{U}_{u},{U}_{v}{⟩}_{F}={s}^{2}({b}_{u}^{T}{b}_{v})({a}_{u}^{T}{a}_{v}),
⟨{U}_{u},{U}_{v}{⟩}_{F}={s}^{2}({b}_{u}^{T}{b}_{v})({a}_{u}^{T}{a}_{v}),
⟨{U}_{u},{U}_{v}{⟩}_{F}={s}^{2}({b}_{u}^{T}{b}_{v})({a}_{u}^{T}{a}_{v}),