On the Spectral Flattening of Quantized Embeddings

TL;DR

This paper investigates how quantization affects the spectral properties of language model embeddings, revealing that spectral flattening caused by quantization leads to instability and collapse in low-precision training.

Contribution

It formalizes the relationship between spectral decay, quantization noise, and stability in LLMs, providing theoretical bounds and empirical validation.

Findings

Quantization introduces a noise floor that truncates spectral tails.

Spectral flattening correlates with representational collapse.

Spectral fidelity is essential for stable low-bit training.

Abstract

Training Large Language Models (LLMs) at ultra-low precision is critically impeded by instability rooted in the conflict between discrete quantization constraints and the intrinsic heavy-tailed spectral nature of linguistic data. By formalizing the connection between Zipfian statistics and random matrix theory, we prove that the power-law decay in the singular value spectra of embeddings is a fundamental requisite for semantic encoding. We derive theoretical bounds showing that uniform quantization introduces a noise floor that disproportionately truncates this spectral tail, which induces spectral flattening and a strictly provable increase in the stable rank of representations. Empirical validation across diverse architectures including GPT-2 and TinyLlama corroborates that this geometric degradation precipitates representational collapse. This work not only quantifies the spectral sensitivity of LLMs but also establishes spectral fidelity as a necessary condition for stable low-bit optimization.