Read the equation
Expanding the two outer products shows that the operator difference equals a changed output factor times the later input factor plus the earlier output factor times a changed input factor. This rearrangement makes small factor differences explicit and can avoid cancellation in a norm subtraction. It is algebraically exact for the shared scale s.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| U_j | Later effective update |
| U_i | Earlier effective update |
| a | Input-side factors |
| b | Output-side factors |
| s | Shared scale |
How it works
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Original context
Read the surrounding source passage
4.5 Numerical stability and PCA The main implementation forms small Gram matrices from the checkpoint factors. Subtraction of nearly equal Gram entries can lose precision when displacements are small. We cross-check every available effective corner using the rank-two difference identity Uj−Ui=sbj−biajT+biaj−aiT. Inner products of these two-term expressions are sums of vector dot products and avoid subtracting large, nearly equal matrix norms. All 462 available corner comparisons are checked. The maximum absolute difference from the Gram implementation is reported rather than assumed to vanish. For principal-components analysis, the 158 regularly sampled bt vectors are stacked and centered across checkpoints. Eigenanalysis of the centered Gram matrix yields component scores and explained-variance fractions. Component signs are arbitrary; a deterministic sign convention makes the delivered scores reproducible. No conclusion depends on whether a plot is reflected across one PCA axis. 4.6 Statistical scope
Exact original notation
{U}_{j}-{U}_{i}=s\left[\left({b}_{j}-{b}_{i}\right){a}_{j}^{T}+{b}_{i}{\left({a}_{j}-{a}_{i}\right)}^{T}\right].Related builds
Sources & provenance
{U}_{j}-{U}_{i}=s\left[\left({b}_{j}-{b}_{i}\right){a}_{j}^{T}+{b}_{i}{\left({a}_{j}-{a}_{i}\right)}^{T}\right].
{U}_{j}-{U}_{i}=s\left[\left({b}_{j}-{b}_{i}\right){a}_{j}^{T}+{b}_{i}{\left({a}_{j}-{a}_{i}\right)}^{T}\right].