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

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.

# A Spectral Phase Diagram for Binary Few-Shot Classification: Intrinsic Dimensionality, Geometric Saturation, and Representational Diagnosis
Source: [https://arxiv.org/abs/2606.24903](https://arxiv.org/abs/2606.24903)
Bibliographic Tools

## Bibliographic and Citation Tools

Bibliographic Explorer Toggle

Code, Data, Media

## Code, Data and Media Associated with this Article

Demos

## Demos

Related Papers

## Recommenders and Search Tools

IArxiv recommender toggle

About arXivLabs

## arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website\.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy\. arXiv is committed to these values and only works with partners that adhere to them\.

Have an idea for a project that will add value for arXiv's community?[**Learn more about arXivLabs**](https://info.arxiv.org/labs/index.html)\.