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

arXiv: https://arxiv.org/abs/2605.08764 · HTML: https://mlsanity.com/paper/2605.08764/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 21 references that are in the map.
- First posted on arXiv 9 May 2026.

## Builds on (most cited first, 7 of 7)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [Sigmoid Loss for Language Image Pre-Training](https://mlsanity.com/paper/2303.15343/) (2023)
- [Prevalence of Neural Collapse during the terminal phase of deep learning training](https://mlsanity.com/paper/2008.08186/) (2020)
- [Med3DVLM: An Efficient Vision-Language Model for 3D Medical Image Analysis](https://mlsanity.com/paper/2503.20047/) (2025)
- [A Geometric Modeling of Occam's Razor in Deep Learning](https://mlsanity.com/paper/1905.11027/) (2019)
- [How Much Data is Enough? The Zeta Law of Discoverability in Biomedical Data, featuring the enigmatic Riemann zeta function](https://mlsanity.com/paper/2604.17581/) (2026)

## Abstract (arXiv)

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 |hatSigma - Sigma|op ~ √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.