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

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 P∈ℝm× d trained from only a few examples per class. We study whether this matrix is over-parameterized by factorizing it as P=BA, which cuts the trainable prompt parameters from md to r(m+d), and to rd once the token-side factor 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 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 show why fixing B is far less restrictive than fixing A, and a smoothness-only guarantee certifies that optimizing A over a fixed B converges. In the CLIP prompt setting, the embedding-side coefficients carry the adaptation while the token basis can simply be fixed.