How Much Does Random Rotation Help GPTQ? Sharp Worst-Case Rate Gaps in the High-Rate Regime

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Author

Runguo Li (single author).

Abstract

GPTQ is Babai’s nearest-plane algorithm on the lattice of the Cholesky factor of the layer’s input second moment Σ. In the high-rate, entropy-coded regime it needs ½ log₂(AM/GM) of its Cholesky pivots more bits than WaterSIC, which is within 0.255 bit of the rate-distortion limit for every Σ. Random rotations close most of this gap in practice, and Ordentlich and Polyanskiy asked for the worst-case gap between rotated GPTQ and WaterSIC.

We show that after a Haar rotation the k-th pivot converges almost surely, uniformly in k ≥ δn and over spectra of bounded condition number, to a classical formula from the theory of Lyapunov exponents, and use it to answer the question.

Key Findings

• Over spectra with condition number κ, the worst rotated gap is ½ log₂ κ − log₂ ln κ + ½ − 1/ln 2 + o(1) bits, attained only by two-point spectra — just ½ log₂ ln κ + O(1) bits below the worst case without rotation, so rotation cannot replace waterfilling in general.
• It nearly can on real layers: rotation cuts the cost of an η-fraction of outlier eigenvalues of size μ from about ημ to η²μ/2 to leading order, but cannot remove spectral spread on a logarithmic scale; for non-Gaussian weights it also adds a weight-side rate cost, characterized exactly under a standard high-rate assumption.
• On 96 Hessians of Qwen3 and Llama-3.1, the rotated gap has median 0.04 bit and maximum 0.36 bit, against 2–5 bits in the worst case for the same condition numbers, and the limit matches it to within 0.009 bit.

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Recommended citation: Runguo Li. (2026). "How Much Does Random Rotation Help GPTQ? Sharp Worst-Case Rate Gaps in the High-Rate Regime." Manuscript under review.
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