[2605.08114] Statistical Inference and Quality Measures of KV Cache Quantisations Inspired by TurboQuant
[Submitted on 27 Apr 2026]
Title:Statistical Inference and Quality Measures of KV Cache Quantisations Inspired by TurboQuant
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Abstract:We analyse three KV cache quantization schemes under a fair bit budget: \textbf{KV} (scalar MSE baseline), \textbf{KQV} (WHT + MSE on $K$; WHT + MSE + QJL on $V$), and \textbf{QKQV} (WHT + MSE + QJL on both). Starting from the Beta distribution on the hypersphere, we trace how QJL on $K$ inflates inner product variance by $\pi/2$, which softmax amplifies nonlinearly via Jensen's inequality, and we present statistical inference and information metrics to highlight practical differences.
Three empirical findings emerge. (1)~At $n=4$ (the practically dominant budget), KQV wins on every measure -- KL divergence, geometric $K$ error, and 6D distance -- across all distributions and ranks tested. (2)~The K--V asymmetry is unconditional: QKQV is consistently worse than KQV in KL divergence at every budget and distribution. (3)~A budget-dependent crossover exists: QKQV achieves better geometric $K$ reconstruction at $n \in \{2,3,5\}$, KQV at $n \in \{4,6\}$, invariant to rank and tail weight -- an open rate-distortion problem.
$\mathrm{KL}(p_{\mathrm{ref}} \| p_{\mathrm{quant}})$, K-only by construction, bridges K direction error to routing corruption and output collapse. We present a sufficient condition when the Jensen mechanism amplifies superlinearly through the softmax. At $n \in \{2,3,5\}$, QKQV wins geometrically because this assumption does not bind. At $n=4$, elevated K error and KL divergence for QKQV strongly suggest the Jensen mechanism is the operative cause of the crossover, providing a new perspective and explanation.
Comments: 23 pages, 7 Figures, multiple tables, the process is highly assisted by AI
Subjects:
Machine Learning (cs.LG); Information Theory (cs.IT); Mathematical Software (cs.MS)
Cite as: arXiv:2605.08114 [cs.LG]
(or arXiv:2605.08114v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2605.08114
arXiv-issued DOI via DataCite
Submission history
From: Paolo D'Alberto [view email] [v1] Mon, 27 Apr 2026 23:11:12 UTC (1,119 KB)
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