{
  "abstract": "Recovering piecewise-smooth functions with jump discontinuities from noisy, indirect, and incomplete data is a central task in computational inverse problems. Bayesian neural networks (BNNs) with heavy-tailed weight priors have emerged as effective edge-preserving priors for such problems. However, posterior inference for these models remains computationally challenging, and existing approaches settle for point estimates obtained by directly optimizing heavy-tailed log-densities. We propose GSBL-BNNs, which model the neural network parameters hierarchically using sparsity-promoting priors from generalized sparse Bayesian learning (GSBL). The hierarchical formulation recovers several popular marginal priors, including Laplace and Student's $t$ (including Cauchy). A key modeling ingredient, novel among edge-preserving BNNs and native to the hierarchical formulation, couples each neuron's bias to its weights through a shared hyperparameter, so that the induced jump discontinuities are free to occur anywhere in the domain while the conditionally Gaussian structure underlying efficient inference is preserved. Furthermore, we develop a block-coordinate descent (BCD) algorithm for maximum a posteriori (MAP) estimation. This BCD approach alternates between standard network training with iteratively reweighted $\\ell^2$-regularization and exact, closed-form hyperparameter updates for gamma and inverse gamma hyperpriors. Building on the BCD algorithm, we perform approximate Bayesian inference and uncertainty quantification (UQ) using deep ensembles. Computational experiments---including signal denoising, deblurring, and inverting a two-dimensional hyperbolic conservation law--- show that GSBL-BNNs deliver accurate, edge-preserving reconstructions with built-in UQ. Compared with Gaussian- and Cauchy-BNNs, they yield more accurate predictive means and avoid overconfident uncertainty estimates.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17023",
  "author_total": 2,
  "authors": [
    {
      "name": "Jan Habscheid"
    },
    {
      "name": "Jan Glaubitz"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ftabm5sphbh2b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17023",
  "fulltext_url": null,
  "id": "ftabm5sphbh2b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17023",
  "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",
      "math.NA"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1702|Gaussian Processes and Bayesian Inference"
    ]
  ],
  "openalex": "W7221803600",
  "pdf_url": "https://arxiv.org/pdf/2610.17023",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ftabm5sphbh2b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.17023",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17023"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Edge-preserving Bayesian neural networks via hierarchical sparsity-promoting priors",
  "title_text": "Edge-preserving Bayesian neural networks via hierarchical sparsity-promoting priors",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/ftabm5sphbh2b",
  "venue": "arXiv",
  "versions": []
}