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

How much data is enough to make a scientific discovery? As biomedical datasets scale to millions of samples and AI models grow in capacity, progress increasingly depends on predicting when additional data will substantially improve performance. In practice, model development often relies on empirical scaling curves measured across architectures, modalities, and dataset sizes, with limited theoretical guidance on when performance should improve, saturate, or exhibit cross-over behavior. We propose a scaling-law framework for cross-modal discoverability based on spectral structure of data covariance operators, task-aligned signal projections, and learned representations. Many performance metrics, including AUC, can be expressed in terms of cumulative signal-to-noise energy accumulated across identifiable spectral modes of an encoder and cross-modal operator. Under mild assumptions, this accumulation follows a zeta-like scaling law governed by power-law decay of covariance spectra and aligned signal energy, leading naturally to the appearance of the Riemann zeta function. Representation learning methods such as sparse models, low-rank embeddings, and multimodal contrastive objectives improve sample efficiency by concentrating useful signal into earlier stable modes, effectively steepening spectral decay and shifting scaling curves. The framework predicts cross-over regimes in which simpler models perform best at small sample sizes, while higher-capacity or multimodal encoders outperform them once sufficient data stabilizes additional degrees of freedom. Applications include multimodal disease classification, imaging genetics, functional MRI, and topological data analysis. The resulting zeta law provides a principled way to anticipate when scaling data, improving representations, or adding modalities is most likely to accelerate discovery.

Real-world perception and decision making are inherently multimodal, integrating complementary signals across modalities. However, training multimodal models faces two main obstacles. First, collecting large-scale, well-aligned paired multimodal datasets is often impractical, making end-to-end multimodal training difficult. Second, existing multimodal representations frequently entangle information shared across modalities with modality-specific information, hindering interpretability and control. We introduce MultiLoReFT, an efficient and scalable low-rank representation fine-tuning framework for multimodal learning with pretrained unimodal models. MultiLoReFT extends low-rank adaptation to the multimodal setting and learns interpretable projection subspaces that decouple shared and modality-specific information. Across simulated and real-world benchmarks, it produces representations that support multimodal prediction while explicitly revealing how shared and modality-specific information is distributed across modalities.

Vision-language models map visual features into a shared embedding space through learned projection layers, yet it remains unclear how these transformations alter the structure of visual information. This study examines changes in representation through spatial-frequency accessibility, measured by the linear recoverability of band-limited Fourier energy from model representations. To isolate effects beyond dimensionality reduction, we introduce Residual Spectral Loss (RSL), which evaluates changes relative to a dimension-matched random projection baseline. To reduce confounding effects from optimization, the analysis uses pretrained models with all parameters frozen. The experimental results show consistent frequency-dependent changes in accessibility across CLIP and DINOv2 on ImageNet and MS-COCO datasets. Spectral accessibility follows a non-monotonic trajectory across depth, peaking at intermediate layers before decreasing toward the output representation. The final transformation differs across architectures: CLIP's learned projection is spectrally neutral, with changes explained by compression, whereas DINOv2's [CLS] pooling induces a structured loss across the spectrum. These findings identify intermediate layers and pooling mechanisms as primary drivers of spectral transformation in modern vision encoders.