A Spectral Phase Diagram for Binary Few-Shot Classification: Intrinsic Dimensionality, Geometric Saturation, and Representational Diagnosis

Researchers introduced the saturation index S(K) to determine when to stop collecting labeled examples in binary few-shot classification, proving it falls below a threshold when the covariance estimator is well-concentrated. Across 246 observations from 17 tasks, the index showed a median Spearman correlation of 0.811 with marginal accuracy gain and achieved AUC of 0.752 as a stopping rule, enabling annotation decisions without test labels or trained classifiers.

arXiv:2606.24903v1 Announce Type: new Abstract: Deciding when to stop collecting labeled examples is a fundamental but undertheorized problem in applied machine learning. The saturation index $S K = \operatorname{erank} \widehat{\Sigma} W^{ K } / K$ measures the ratio of the effective rank of the pooled within-class sample covariance to the shot count; we prove it falls below a threshold precisely when the covariance estimator is well-concentrated around the population covariance and the linear discriminant has stabilized. The index is computable in $O d^3 $ time from support features alone, requiring no test labels or trained classifier. Evaluated across $N = 246$ doubling-pair observations from seventeen binary tasks and six datasets, sixteen of seventeen tasks have a positive within-task Spearman correlation between $S K $ and marginal accuracy gain median $\rho = 0.811$ . The pooled Spearman correlation is $\rho = 0.548$ $p = 1.1 \times 10^{-20}$, $N = 246$ . A three-phase diagram exploration, transition, saturation with mean marginal gains of $3.48\%$, $2.40\%$, and $0.82\%$ is supported by all pairwise significance tests $p \leq 0.008$ . As a binary stopping rule, the index achieves AUC $= 0.752$, providing meaningful probabilistic guidance for annotation decisions. Asymptotic effective rank and peak accuracy show no significant monotone relationship across tasks Spearman $r s = 0.380$, $p = 0.133$, $N = 17$ . A small saturation index paired with low accuracy diagnoses representational inadequacy. All results are for binary classification with a fixed linear classifier; extensions to $N$-way settings and pretrained backbone representations are discussed as future work.