Anchoring the Eigengap: Cross-Modal Spectral Stabilization for Sample-Efficient Representation Learning

• URL: http://arxiv.org/abs/2605.08764v1
• Date: Sat, 09 May 2026 07:49:43 GMT
• Title: Anchoring the Eigengap: Cross-Modal Spectral Stabilization for Sample-Efficient Representation Learning
• Abstract summary: We show that vision models degrade sharply in low-data regimes, particularly in medical imaging where labeled samples are scarce.<n>We develop a theory of finite-sample representation learning that quantifies the recoverable dimension K(N)<n>We show that multimodal training maintains more stable modes and improves class separation, even when unimodal models achieve comparable few-shot accuracy.
• Score: 0.8326214193432122
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Deep vision models degrade sharply in low-data regimes, particularly in medical imaging where labeled samples are scarce. We show this arises not merely from overfitting but from a geometric failure: finite-sample noise corrupts the embedding covariance, collapsing the eigengap and limiting the number of recoverable signal-bearing modes. We develop a spectral theory of finite-sample representation learning that quantifies the recoverable dimension K(N), the number of eigenmodes that can be stably estimated from N samples. Using perturbation theory and concentration bounds, we show that only modes with eigenvalues above the noise floor $\|\hatΣ - Σ\|_{\mathrm{op}} \sim \sqrt{D/N}$ are reliable, yielding a truncated Mahalanobis energy that governs classification performance. Under a power-law spectral model, this energy can be approximated by a truncated Riemann zeta function, linking eigenvalue decay to data efficiency and AUC. Within this framework, multimodal learning acts as spectral stabilization: vision-language models impose low-rank constraints that suppress noise-dominated directions and preserve the eigengap, increasing K(N) under data scarcity. Across MNIST and multi-disease neuroimaging, we show that multimodal training maintains more stable modes and improves class separation, even when unimodal models achieve comparable few-shot accuracy. These results identify spectral collapse as a fundamental bottleneck in low-data learning. We use truncated Mahalanobis energy and K(N) to diagnose encoder quality, and introduce zeta-based spectral filtering as a principled approach to improve data efficiency.

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