Effective rank-one LoRA update

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.

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Original context

6.1 Comparing the effective update

LoRA represents an update through factors. For a rank-one case, write:

ΔWt=stbtatT.

Here st is the effective scaling coefficient from the adapter configuration, at is the input-side direction, and bt the output-side direction. Standard and rank-stabilized scaling differ in general; at rank one their rank denominators coincide. Configuration values, per-layer patterns, orientation conventions, and optional methods such as DoRA must be checked rather than assumed [1,4].

Comparing bt alone is useful within a fixed parameterization, but the factorization is not unique. For any nonzero scalar c:

cbtat/cT=btatT.

\Delta {W}_{t}={s}_{t}{b}_{t}{a}_{t}^{T}.

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