A Spectral Phase Diagram for Binary Few-Shot Classification: Intrinsic Dimensionality, Geometric Saturation, and Representational Diagnosis
This paper introduces a saturation index to determine when to stop collecting labeled data in binary few-shot classification. Computed from support features alone, it measures the effective rank of the within-class covariance relative to shot count. Empirical evaluation on 246 observations from 17 tasks shows strong correlation with marginal accuracy gain (median Spearman ρ=0.811) and identifies three phases: exploration, transition, saturation. As a stopping rule, it achieves AUC 0.752. A low saturation index with low accuracy indicates representational inadequacy.
[2606.24903] A Spectral Phase Diagram for Binary Few-Shot Classification: Intrinsic Dimensionality, Geometric Saturation, and Representational Diagnosis
[Submitted on 12 Jun 2026]
Title:A Spectral Phase Diagram for Binary Few-Shot Classification: Intrinsic Dimensionality, Geometric Saturation, and Representational Diagnosis
View a PDF of the paper titled A Spectral Phase Diagram for Binary Few-Shot Classification: Intrinsic Dimensionality, Geometric Saturation, and Representational Diagnosis, by Arnav Gupta
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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.
Comments: 85 pages, 5 figures, 32 tables
Subjects:
Machine Learning (cs.LG)
Cite as: arXiv:2606.24903 [cs.LG]
(or arXiv:2606.24903v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2606.24903
arXiv-issued DOI via DataCite
Submission history
From: Arnav Gupta [view email] [v1] Fri, 12 Jun 2026 16:46:24 UTC (115 KB)
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