{
  "abstract": "Existing low-rank KV-cache indexes preserve model weights or key variance, not the attention scores that determine retrieval. We derive the expected score distortion of key-side rank-r compression, a two-sided covariance-weighted low-rank problem that uses both query and key statistics, and solve it exactly: the optimum is an optimal rank-r linear score map, computed in closed form from two d x d covariances per head, and its predicted per-head score-MSE reduction matches measurement (Pearson r = 0.997). The resulting training-free index, SAKI, beats key-PCA at every rank on LLaMA-3.1-8B, Qwen2.5-7B, Mistral-7B and Llama-3.2-3B, at every context length from 4K to 64K, and in every calibration-domain pair (0.748 -> 0.799 recall@64 at r = 32 on LLaMA-3.1-8B). Because RoPE rotates keys by their distance from the query, we make the closed form RoPE-aware by averaging the query covariance over rotations; this SAKI-RoPE, our main method, raises recall to 0.861, beats PCA on every head of all four models, and needs only 28-39% of PCA's rank to match it at r = 32 across the four models. Random-matrix theory of the covariance estimation noise shows that 512 calibration tokens suffice and that clipping the Marchenko-Pastur noise bulk helps SAKI when calibration is short. We make no perplexity claim: with the attention sink pinned, perplexity differences between indexes are small and depend on model and text, and we show that any low-rank index calibrated on a long prefix drops the attention-sink token, which collapses 32K perplexity unless the sink is pinned.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.03228",
  "author_total": 1,
  "authors": [
    {
      "name": "Lin Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/95k54ph59bnka.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.03228",
  "fulltext_url": null,
  "id": "95k54ph59bnka",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.03228",
  "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.LG"
    ],
    [
      "arxiv",
      "stat.AP"
    ],
    [
      "openalex",
      "1702|Topic Modeling"
    ]
  ],
  "openalex": "W7172489335",
  "pdf_url": "https://arxiv.org/pdf/2608.03228",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/95k54ph59bnka.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:49Z",
      "native_id": "2608.03228",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.03228"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.AP",
      "subfield": "statistics"
    }
  ],
  "title": "SAKI-RoPE: Score-Aware Low-Rank Key Indexing with RoPE-Aware Covariance and Random-Matrix Noise Analysis",
  "title_text": "SAKI-RoPE: Score-Aware Low-Rank Key Indexing with RoPE-Aware Covariance and Random-Matrix Noise Analysis",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:49Z",
  "url": "https://research.draconitic.com/paper/95k54ph59bnka",
  "venue": "arXiv",
  "versions": []
}