Low-Rank Prompt Learning for Vision-Language Models with Fixed-Token Bases

A new arXiv paper (2609.09462v1) reports that factorizing the dense CoOp prompt matrix P ∈ R^(m×d) as P = BA cuts trainable prompt parameters from md to r(m+d), and to rd once the token-side factor B is fixed, matching or improving dense CoOp across seven few-shot benchmarks and two CLIP backbones. The authors find the token-side factor B need not be learned at all: fixing B to a Gaussian, orthogonal, SVD-derived, or even random basis and training only the embedding-side factor A stays on par with the fully trainable factorization, and a source-trained B offers no advantage over a random one. A prompt-factor asymmetry and a local update-space dimension gap explain why fixing B is far less restrictive than fixing A, with a smoothness-only guarantee certifying convergence when optimizing A over a fixed B.

arXiv:2609.09462v1 Announce Type: new Abstract: Prompt learning adapts CLIP to downstream recognition by replacing hand-written templates with learned continuous context vectors, which in Context Optimization CoOp form a dense prompt matrix $\mathbf{P}\in\mathbb{R}^{m\times d}$ trained from only a few examples per class. We study whether this matrix is over-parameterized by factorizing it as $\mathbf{P}=\mathbf{B}\mathbf{A}$, which cuts the trainable prompt parameters from $md$ to $r m+d $, and to $rd$ once the token-side factor $\mathbf{B}$ is fixed. Across seven few-shot benchmarks and two CLIP backbones, low-rank prompts match or improve dense CoOp at far fewer parameters, with the clearest gains on low-shot base-to-new generalization. We then find that the token-side factor need not be learned at all: fixing $\mathbf{B}$ to a Gaussian, orthogonal, SVD-derived, or even random basis and training only the embedding-side factor $\mathbf{A}$ stays on par with the fully trainable factorization, and a source-trained $\mathbf{B}$ offers no advantage over a random one. A prompt-factor asymmetry and a local update-space dimension gap show why fixing $\mathbf{B}$ is far less restrictive than fixing $\mathbf{A}$, and a smoothness-only guarantee certifies that optimizing $\mathbf{A}$ over a fixed $\mathbf{B}$ converges. In the CLIP prompt setting, the embedding-side coefficients carry the adaptation while the token basis can simply be fixed.