{"ID":5936928,"CreatedAt":"2026-07-07T03:14:33.014478982Z","UpdatedAt":"2026-07-09T17:51:18.37832961Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.05379","arxiv_id":"2607.05379","title":"Focused Width in Adversarial Fake Detection: A Separation","abstract":"We study the adversarial fake detection model introduced by Mendelson, Paouris and Vershynin. In this model, a genuine sample is $\\pmb{X}\\sim N(0,\\pmb{I}_n)$, while a fake sample is produced as $\\pmb{X}+r\\pmb{t}({\\pmb{X}})$, where the adversary first observes $\\pmb{X}$ and then chooses an admissible perturbation $\\pmb{t}({\\pmb{X}})$ from a prescribed set $\\mathscr{T}\\subset\\mathbb{R}^n$. The central quantity is the detectability radius $r(\\mathscr{T})$, which formalizes the transition scale at which fake samples become reliably distinguishable from genuine ones. Mendelson, Paouris and Vershynin introduced the focused width $\\widetilde{w}(\\mathscr{T})$ as a geometric parameter for this radius and conjectured that, for every origin-symmetric set $\\mathscr{T}$, it characterizes $r(\\mathscr{T})$ up to universal constants. In this note, we disprove this conjecture for a broad class of discrete sets. More precisely, we consider any origin-symmetric set $\\mathscr{T}_n$ lying between the hypercube and the odd integer grid: \\begin{equation*} \\{-1,1\\}^n\\subset\\mathscr{T}_n\\subset ( 2\\mathbb{Z}+1)^n. \\end{equation*} For every such $\\mathscr{T}_n$, we prove that $\\frac{\\widetilde{w}(\\mathscr{T}_n)}{r(\\mathscr{T}_n) }\\gtrsim \\sqrt{\\log n}$. Thus, in the Gaussian model, the focused width can overestimate the detectability radius by a $\\sqrt{\\log n}$ factor and therefore does not characterize it in general. We further show that this logarithmic scale is not intrinsic: in the corresponding non-Gaussian model with product Laplace data, the focused width benchmark can even exceed the detectability radius by at least a polynomial factor of order $n^{1/4}$.","short_abstract":"We study the adversarial fake detection model introduced by Mendelson, Paouris and Vershynin. In this model, a genuine sample is $\\pmb{X}\\sim N(0,\\pmb{I}_n)$, while a fake sample is produced as $\\pmb{X}+r\\pmb{t}({\\pmb{X}})$, where the adversary first observes $\\pmb{X}$ and then chooses an admissible perturbation $\\pmb{...","url_abs":"https://arxiv.org/abs/2607.05379","url_pdf":"https://arxiv.org/pdf/2607.05379v1","authors":"[\"Gao Huang\"]","published":"2026-07-06T17:55:41Z","proceeding":"math.ST","tasks":"[\"math.ST\"]","methods":"[]","has_code":false}