{
  "abstract": "Adversarial perturbations can alter neural-network predictions, motivating closer analysis of the geometry of learned feature representations. For a given local displacement in feature space, small-norm feature vectors can undergo larger first-order directional changes than larger-norm vectors. This study examines this geometric effect in a two-dimensional (2D) bottleneck and proposes a composite objective that regulates feature radius and class-aligned direction. A norm-dependent auxiliary eXtra-node contributes a direct outward feature-gradient component for nonzero features near the origin, whereas a prototype-based cosine similarity (PCS) term aligns features with normalized classifier-weight directions. An exact comparison with semantic cross-entropy further shows that auxiliary softmax normalization introduces an additional semantic-normalization coupling term with generally mixed radial and tangential components. Experiments on a 10-class subset of ImageNet with six matched CNN and Transformer backbones show that the proposed objective increases both the mean feature radius and the mean similarity to the ground-truth class prototype. Under the FGSM, PGD, FDA, UAP, and DeepFool settings used in the experiments, post-attack accuracy is higher in 26 of 30 backbone–procedure comparisons, while mean clean accuracy changes from 98.69% to 98.15%. These results show that the eXtra-node–PCS objective reshapes the 2D representation in the intended radial and angular directions and support further investigation in standard high-dimensional feature spaces and under adaptive attack evaluation.",
  "abstract_shown": true,
  "affiliations": [
    "Department of Computer Engineering, Korea National University of Transportation, Chungju-si 27469, Republic of Korea"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 2,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Eun-Sung Lee"
    },
    {
      "affs": [
        0
      ],
      "name": "Hyun-Cheol Park",
      "orcid": "0000-0001-8280-4026"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b7nrsfh0s3b8a.bib",
  "cited_by": 0,
  "doi": "10.3390/math14203746",
  "fulltext_url": null,
  "id": "b7nrsfh0s3b8a",
  "issns": [
    "2227-7390"
  ],
  "landing_url": "https://www.mdpi.com/2227-7390/14/20/3746",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "1702|Adversarial Robustness in Machine Learning"
    ]
  ],
  "openalex": "W7222058246",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": "MDPI AG",
  "ris": "https://research.draconitic.com/paper/b7nrsfh0s3b8a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:29Z",
      "native_id": "10.3390/math14203746",
      "source": "crossref",
      "url": "https://www.mdpi.com/2227-7390/14/20/3746"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "issn:2227-7390",
      "subfield": "general"
    }
  ],
  "title": "Radial–Angular Feature Shaping with an eXtra-Node and Prototype-Based Cosine Similarity Under Adversarial Perturbations",
  "title_text": "Radial–Angular Feature Shaping with an eXtra-Node and Prototype-Based Cosine Similarity Under Adversarial Perturbations",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T12:18:29Z",
  "url": "https://research.draconitic.com/paper/b7nrsfh0s3b8a",
  "venue": "Mathematics",
  "versions": []
}