The Homogeneity Trap: Spectral Collapse in Doubly-Stochastic Deep Networks
This addresses a fundamental trade-off between stability and expressivity in deep learning architectures using doubly-stochastic matrices, such as Optimal Transport layers, which is incremental but critical for improving model performance.
The paper identifies a spectral degradation phenomenon called the Homogeneity Trap in doubly-stochastic deep networks, showing that high-entropy constraints suppress feature components and restrict effective depth, with Layer Normalization failing to prevent collapse when Signal-to-Noise Ratio drops below a critical threshold.
Doubly-stochastic matrices (DSM) are increasingly utilized in structure-preserving deep architectures -- such as Optimal Transport layers and Sinkhorn-based attention -- to enforce numerical stability and probabilistic interpretability. In this work, we identify a critical spectral degradation phenomenon inherent to these constraints, termed the Homogeneity Trap. We demonstrate that the maximum-entropy bias, typical of Sinkhorn-based projections, drives the mixing operator towards the uniform barycenter, thereby suppressing the subdominant singular value σ_2 and filtering out high-frequency feature components. We derive a spectral bound linking σ_2 to the network's effective depth, showing that high-entropy constraints restrict feature transformation to a shallow effective receptive field. Furthermore, we formally demonstrate that Layer Normalization fails to mitigate this collapse in noise-dominated regimes; specifically, when spectral filtering degrades the Signal-to-Noise Ratio (SNR) below a critical threshold, geometric structure is irreversibly lost to noise-induced orthogonal collapse. Our findings highlight a fundamental trade-off between entropic stability and spectral expressivity in DSM-constrained networks.