# The Homogeneity Trap: Spectral Collapse in Doubly-Stochastic Deep Networks

arXiv: https://arxiv.org/abs/2601.02080 · HTML: https://mlsanity.com/paper/2601.02080/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 5 references that are in the map.
- First posted on arXiv 5 Jan 2026.

## Builds on (most cited first, 1 of 1)

- [Attention is Not All You Need: Pure Attention Loses Rank Doubly Exponentially with Depth](https://mlsanity.com/paper/2103.03404/) (2021)

## Built on by (most cited first, 1 of 1)

- [Spectral-Sphere-Constrained Hyper-Connections](https://mlsanity.com/paper/2603.20896/) (2026)

## Abstract (arXiv)

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 σ2 and filtering out high-frequency feature components. We derive a spectral bound linking σ2 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.