{
  "abstract": "We present a novel algorithm for calculating the preimage of an affine space through a product ${J}={J}_{T-1}\\cdots{J}_0$ of matrices ${J}_t$ of special form: finding the largest input space $\\mathbf{X}$ such that $\\mathbf{x}\\in\\mathbf{X}$ implies ${J}\\mathbf{x}\\in \\mathbf{Y}$, where $\\mathbf{Y}$ is a given output affine space. These special matrices arise in AD, where the Jacobians $J$ describing the linearized computation have precisely this structure: the product of a series of linearized primitive numeric operations. This allows us to use the new algorithm to formulate Null-A mode preimage AD, which finds the affine preimage through the Jacobian or Jacobian transpose of a numeric computation. This is a generalization of the inverse AD problem of solving ${J}\\acute{\\mathbf{x}}^{\\ast}=\\acute{\\mathbf{y}}^{\\ast}$ or ${J}^{T}\\grave{\\mathbf{y}}^{\\ast}=\\grave{\\mathbf{x}}^{\\ast}$. The key is to represent affine spaces in a fashion which lends itself to efficient preimage calculation, in a compositional and quasi-local fashion, through a succession of matrices ${J}_t$. Unlike previous methods, Null-A preimage mode AD allows the ${J}_t$ matrices to be non-square, corresponding to a computer program whose number of active variables swells and shrinks during the computation. When ${J}$ is square and the initial affine space is a single point, this finds the conventional inverse. But in the more general case, having the entire affine space provides freedom which can be leveraged in a problem-specific manner. We apply the method to small problems on-CPU where the $J_t$ are linearized scalar unary or binary numeric functions; and to larger problems on-GPU where the $J_t$ are linearized aggregate array operations like convolution and attention.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38497",
  "author_total": 2,
  "authors": [
    {
      "name": "Barak A. Pearlmutter"
    },
    {
      "name": "Jeffrey Mark Siskind"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/399f5xtpfxa40.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38497",
  "fulltext_url": null,
  "id": "399f5xtpfxa40",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38497",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "cs.PL"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "1703|Numerical Methods and Algorithms"
    ]
  ],
  "openalex": "W7215039741",
  "pdf_url": "https://arxiv.org/pdf/2609.38497",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/399f5xtpfxa40.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.38497",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38497"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "The Mode of Null-A: Compositional Computation of a Generalized Inverse",
  "title_text": "The Mode of Null-A: Compositional Computation of a Generalized Inverse",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/399f5xtpfxa40",
  "venue": "arXiv",
  "versions": []
}