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

• URL: http://arxiv.org/abs/2606.24903v1
• Date: Fri, 12 Jun 2026 16:46:24 GMT
• Title: A Spectral Phase Diagram for Binary Few-Shot Classification: Intrinsic Dimensionality, Geometric Saturation, and Representational Diagnosis
• Abstract summary: The saturation index $S(K) = operatornameerank(widehat_W(K)) / K$ measures the ratio of the effective rank of the pooled within-class sample covariance to the shot count.<n>The index is computable in $O(d3)$ time from support features alone, requiring no test labels or trained classifier.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Deciding when to stop collecting labeled examples is a fundamental but undertheorized problem in applied machine learning. The saturation index $S(K) = \operatorname{erank}(\widehatΣ_W^{(K)}) / K$ 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 $ρ= 0.811$). The pooled Spearman correlation is $ρ= 0.548$ ($p = 1.1 \times 10^{-20}$, $N = 246$). 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.

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