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

4.1. The effective update, not only its factors

Let at and bt be column vectors containing the flattened input and output factors. The effective learned contribution is

Ut=s btatT,  s=64.

(4.1)

The adapted layer applies (W0+Ut)h to input h, where W0 is its frozen base weight. A change in one factor need not change that operator. For any nonzero scalar λt,

(λtbt)(at/λt)T=btatT.

{U}_{t}=s {b}_{t}{a}_{t}^{T},  s=64.

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