{
  "abstract": "We consider a nonlinear model motivated by polychromatic computed tomography (CT). Here, $W$ distinct $d$-dimensional signals $x^*_1, \\ldots, x^*_W \\in \\mathbb{R}^d$ must be recovered from $n$ measurements $(a_i, y_i)_{i = 1}^n$ that obey the nonlinear model $\\mathbb{E}[y_i|a_i] = h(\\langle a_i, x^*_1 \\rangle, \\ldots, \\langle a_i, x^*_W \\rangle)$, where $h: \\mathbb{R}^W \\to \\mathbb{R}$ is a known nonlinearity that models a certain type of exponential attenuation law. Even when there is no noise in the measurements, the sample size $n$ (for any measurement ensemble $\\{a_i\\}_{i = 1}^n$) must exceed the number of unknowns $Wd$ to guarantee that the underlying signals are identifiable. We construct a measurement ensemble that ensures perfect signal recovery almost surely provided $n \\geq Wd + W - 1$, thereby isolating the injectivity threshold up to an additive factor $W - 1$. We also study computational complexity of signal recovery in this model with a general measurement ensemble $\\{a_i\\}$. We show that if $W \\geq 2$, then there is a measurement ensemble for which deciding whether there exist signals consistent with the measurements is NP-hard. In particular, this implies that polychromatic, multimaterial CT reconstruction is NP-hard in general. This finding stands in sharp contrast to the single-material setting, for which a polynomial-time algorithm can provably perform signal recovery for any measurement ensemble.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23040",
  "author_total": 2,
  "authors": [
    {
      "name": "Xuanzhou Chen"
    },
    {
      "name": "Ashwin Pananjady"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6kjnnd7ka2zn5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23040",
  "fulltext_url": null,
  "id": "6kjnnd7ka2zn5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23040",
  "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.IT"
    ],
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      "arxiv",
      "eess.IV"
    ],
    [
      "arxiv",
      "eess.SP"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "physics.med-ph"
    ],
    [
      "openalex",
      "2206|Sparse and Compressive Sensing Techniques"
    ]
  ],
  "openalex": "W7214053862",
  "pdf_url": "https://arxiv.org/pdf/2609.23040",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6kjnnd7ka2zn5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:19Z",
      "native_id": "2609.23040",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23040"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    },
    {
      "field": "physics",
      "origin": "arxiv:physics.med-ph",
      "subfield": "biological-medical"
    }
  ],
  "title": "Signal recovery in the polychromatic computed tomography model: Injectivity and complexity",
  "title_text": "Signal recovery in the polychromatic computed tomography model: Injectivity and complexity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/6kjnnd7ka2zn5",
  "venue": "arXiv",
  "versions": []
}