The Geometry of Saturation: Effective Rank Predicts When Labels Stop Helping in Few-Shot Classification

Abstract

Few-shot label acquisition lacks a label-free signal for when additional labels cease to improve accuracy: existing stopping criteria either require a held-out validation set (violating the few-shot premise) or rely on theoretically ungrounded heuristics, so we introduce the spectral saturation index $S(K)=\mathrm{erank}(\hat{\Sigma}_W^{(K)})/K$, the exponential spectral entropy of the pooled within-class covariance normalized by per-class support size $K$, which measures the exploration rate per label and falls below a fixed threshold $\tau=0.02$ once the explored spectral subspace saturates and marginal accuracy gains vanish; across 49 real tasks (binary, 5-way, 10-way) and three frozen backbones (PCA-50, CLIP ViT-B/32, DINOv2 ViT-S/14), $S(K)$ correlates strongly with the marginal gain on doubling the support set ($\rho_{\text{pool}}=0.6366$, $p=2.9\times10^{-57}$, cluster-bootstrap 95\% CI $[0.551,0.720]$), a fixed $\tau=0.02$ classifies stop/continue decisions with cluster-bootstrap $\mathrm{AUC}=0.787$ (95\% CI $[0.713,0.860]$) with high recall on meaningful gains ($\Delta A>1\%$), and a partial correlation controlling for $\log K$ yields $\rho_{\text{partial}}=0.324$ ($p=1.65\times10^{-13}$), confirming $S(K)$ carries spectral information beyond shared $K$-dependence; theory predicts this from first principles, since the population effective rank sets the saturation scale $K_{\text{sat}}\approx\mathrm{erank}(\Sigma_W)/\tau$, $\tau=0.02$ sits at the boundary between the first and second descent (Nakkiran et al., 2021), and $O(1/K)$ bias in the sample effective rank explains the small-$K$ hump in $S(K)$; for unregularized linear probes ($C=\infty$), practitioners should halt when $S(K)<0.02$ (PCA-50, hard stop) or monitor $S(K)$ dropping from $\sim0.3\to0.05$ (foundation models, diminishing-returns signal), with computation costing $\sim1$ ms at $d=50$.