Tensor Decomposition of Transformer Key-Value Caches: Spectral Structure and Format Comparison

arXiv:2609.28029v1 Announce Type: cross Abstract: The key-value (KV) cache of autoregressive transformers can be viewed as a fourth-order tensor spanning attention heads, tokens, features, and grouped layers. We measure the singular-value spectra of all four mode unfoldings on Mistral-7B-v0.3 and LLaMA-2-13B and compare four standard tensor decompositions: Tucker, CP, tensor train, and t-SVD, at matched storage. The spectra partition the four axes into two classes. The token and feature modes carry low-rank structure, particularly for keys. The head and layer modes are nearly full-rank and resist compression at any practical error level. Among the four decompositions, Tucker achieves the lowest reconstruction error at every compression ratio from $2\times$ to $5\times$, because it can leave the full-rank modes untouched. Comparisons with two-dimensional unfolding baselines show that the preferred representation differs between keys and values: 2D methods achieve lower key error, while four-way Tucker achieves lower value error at matched storage. A mode-pinning theorem certifies the full-rank preservation from the measured spectra alone. Two further spectral properties affect the compressible modes without touching the full-rank ones: values reach a higher error floor than keys at every ratio, and post-RoPE keys lose $41\%$ - $64\%$ of their pre-RoPE compressibility on both models.

Abstract:The key-value (KV) cache of autoregressive transformers can be viewed as a fourth-order tensor spanning attention heads, tokens, features, and grouped layers. We measure the singular-value spectra of all four mode unfoldings on Mistral-7B-v0.3 and LLaMA-2-13B and compare four standard tensor decompositions: Tucker, CP, tensor train, and t-SVD, at matched storage. The spectra partition the four axes into two classes. The token and feature modes carry low-rank structure, particularly for keys. The head and layer modes are nearly full-rank and resist compression at any practical error level. Among the four decompositions, Tucker achieves the lowest reconstruction error at every compression ratio from $2\times$ to $5\times$, because it can leave the full-rank modes untouched. Comparisons with two-dimensional unfolding baselines show that the preferred representation differs between keys and values: 2D methods achieve lower key error, while four-way Tucker achieves lower value error at matched storage. A mode-pinning theorem certifies the full-rank preservation from the measured spectra alone. Two further spectral properties affect the compressible modes without touching the full-rank ones: values reach a higher error floor than keys at every ratio, and post-RoPE keys lose $41\%$ - $64\%$ of their pre-RoPE compressibility on both models.

18 pages, 3 figures, 8 tables. Submitted to SIAM Journal on Matrix Analysis and Applications (SIMAX)

Numerical Analysis (math.NA); Computation and Language (cs.CL); Machine Learning (cs.LG)

arXiv:2609.28029 [math.NA]

(or arXiv:2609.28029v1 [math.NA] for this version)

https://doi.org/10.48550/arXiv.2609.28029 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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Tensor Decomposition of Transformer Key-Value Caches: Spectral Structure and Format Comparison

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