Behaviorally Effective LoRA Writes Are Sparse and Structured

arXiv:2609.01374v1 Announce Type: new Abstract: Low-rank adaptation fixes the rank of the update, but it does not identify which parts of a trained write actually carry behavior. We study that question directly and show that behaviorally effective LoRA writes are sparse, structured, and far more concentrated than the raw low-rank parameterization suggests. We use Learned-Basis LoRA, a learned-basis continuation recipe, to expose that structure. The recipe warms up an unconstrained adapter, converts its learned write columns into a module-wise orthonormal basis, freezes that basis, and continues training inside the constrained parameterization. Across 14 exact switches from unconstrained to constrained form, held-out accuracy is unchanged at the conversion step and reconstructed write matrices differ by at most 0.25% relative Frobenius error. Same-state continuation then shows that the same trained checkpoint develops differently under different write subspaces, establishing write geometry as a causal state variable. A no-retraining projection test shows that useful write signal stays inside the learned write space and largely disappears from random or frozen-activation PCA controls. The concentration pattern is strong at both local and global scales. Across GSM8K, MathQA, and AQuA, per-module top-k continuation reaches its optimum at k in {2, 4} in all twelve seed-level cases we test. A stricter global ranking test shows that learned top-16 and top-32 subsets outperform matched random subsets, especially on GSM8K/Qwen and MathQA/Qwen. Single-direction ablations further reveal a sparse set of late q_proj, o_proj, and down_proj components with outsized behavioral impact.

Abstract:Low-rank adaptation fixes the rank of the update, but it does not identify which parts of a trained write actually carry behavior. We study that question directly and show that behaviorally effective LoRA writes are sparse, structured, and far more concentrated than the raw low-rank parameterization suggests. We use Learned-Basis LoRA, a learned-basis continuation recipe, to expose that structure. The recipe warms up an unconstrained adapter, converts its learned write columns into a module-wise orthonormal basis, freezes that basis, and continues training inside the constrained parameterization. Across 14 exact switches from unconstrained to constrained form, held-out accuracy is unchanged at the conversion step and reconstructed write matrices differ by at most 0.25% relative Frobenius error. Same-state continuation then shows that the same trained checkpoint develops differently under different write subspaces, establishing write geometry as a causal state variable. A no-retraining projection test shows that useful write signal stays inside the learned write space and largely disappears from random or frozen-activation PCA controls. The concentration pattern is strong at both local and global scales. Across GSM8K, MathQA, and AQuA, per-module top-k continuation reaches its optimum at k in {2, 4} in all twelve seed-level cases we test. A stricter global ranking test shows that learned top-16 and top-32 subsets outperform matched random subsets, especially on GSM8K/Qwen and MathQA/Qwen. Single-direction ablations further reveal a sparse set of late q_proj, o_proj, and down_proj components with outsized behavioral impact.

Computation and Language (cs.CL)

arXiv:2609.01374 [cs.CL]

(or arXiv:2609.01374v1 [cs.CL] for this version)

https://doi.org/10.48550/arXiv.2609.01374 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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Behaviorally Effective LoRA Writes Are Sparse and Structured

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