{
  "abstract": "We give an algorithm for PAC learning intersections of $k$ degree-2 polynomial threshold functions over $R^d$ under a Gaussian distribution. Our algorithm achieves classification error $ε$ using $d^{O(k)}2^{O_k(\\log^3(1/ε))}$ time and samples, improving over the $d^{O(k^2/ε^2)}$ running time of the polynomial regression approach. To our knowledge, our result is the first to obtain a dimension exponent independent of the accuracy parameter, even for $k=2$. We complement our algorithmic result with an SQ lower bound showing that the complexity dependence of $d^{Ω(k)}$ is unavoidable. Lifting the input to polynomial features turns the target into an intersection of $k$ halfspaces. This suggests using the first two labeled moments for subspace recovery. In the lifted space, however, these moments can reveal nothing. Our starting point is a Gaussian comparison theorem showing that, for sufficiently regular defining quadratics, the labeled moments needed for subspace recovery are approximately preserved. To handle arbitrary defining quadratics, we establish a new structural result showing that every direction with significant resampling influence can be detected through degree-$O(k)$ moments. These results yield an initial $d^{O(k)}2^{\\mathrm{poly}_k(1/ε)}$-time learner using adaptive restrictions. To improve the exponential dependence on $1/ε$, we develop a two-stage dimension-reduction approach that avoids the costly adaptive partitioning. After identifying a subspace containing all directions with large influence, we use generalized regularity and Gaussian comparison to recover the remaining relevant quadratic features through a new derivative-based moment operator. The recovered features support an intersection close to the target. Combining them with robust hinge-loss minimization gives our quasipolynomial-time learner.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05114",
  "author_total": 5,
  "authors": [
    {
      "name": "Ilias Diakonikolas"
    },
    {
      "name": "Daniel M. Kane"
    },
    {
      "name": "Mingchen Ma"
    },
    {
      "name": "Lisheng Ren"
    },
    {
      "name": "Trung Tran"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ekhswsh3wrt38.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05114",
  "fulltext_url": null,
  "id": "ekhswsh3wrt38",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05114",
  "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.DS"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "arxiv",
      "stat.TH"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05114",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ekhswsh3wrt38.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.05114",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05114"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Beyond Multi-Index Models: Learning Intersections of Degree-2 PTFs",
  "title_text": "Beyond Multi-Index Models: Learning Intersections of Degree-2 PTFs",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:10Z",
  "url": "https://research.draconitic.com/paper/ekhswsh3wrt38",
  "venue": "arXiv",
  "versions": []
}