Read the equation
The outer product bₜaₜᵀ forms a rank-at-most-one matrix, and the scalar s sets its contribution to the model weight update. The pinned study uses s=64. Comparing this effective matrix describes the implemented linear operator more directly than comparing one factor alone, because different factor pairs can represent the same product.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| U_t | Effective update; ΔW in another version |
| b_t | Output-side column factor |
| a_t | Input-side column factor |
| s | Adapter scaling coefficient |
How it works
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Assumptions & limits
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Original context
Read the surrounding source passage
4.3 Effective weight update and factorization invariance For a rank-one adapter, let at∈R13824 and bt∈R5120 be the flattened input and output factors. The learned contribution to the layer is Ut=s btatT, s=64, and the adapted layer acts as W0+Uth. This is the usual low-rank update structure [4]. Rank-stabilized and ordinary rank denominators coincide at rank one; inspecting the configuration is nevertheless necessary [9]. 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:
Exact original notation
{U}_{t}=s {b}_{t}{a}_{t}^{T}, s=64,Related builds
Sources & provenance
{U}_{t}=s {b}_{t}{a}_{t}^{T}, s=64,
{U}_{t}=s {b}_{t}{a}_{t}^{T}, s=64,