{
  "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_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28029",
  "author_total": 2,
  "authors": [
    {
      "name": "Rahul Krishnan"
    },
    {
      "name": "Volker Schulz"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/12kkmrqwr447z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28029",
  "fulltext_url": null,
  "id": "12kkmrqwr447z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28029",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CL"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2605|Tensor decomposition and applications"
    ]
  ],
  "openalex": "W7214155527",
  "pdf_url": "https://arxiv.org/pdf/2609.28029",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/12kkmrqwr447z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.28029",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28029"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Tensor Decomposition of Transformer Key-Value Caches: Spectral Structure and Format Comparison",
  "title_text": "Tensor Decomposition of Transformer Key-Value Caches: Spectral Structure and Format Comparison",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/12kkmrqwr447z",
  "venue": "arXiv",
  "versions": []
}