Paper · Apr 22, 2024

A mean curvature flow arising in adversarial training

Leon Bungert, Tim Laux, Kerrek Stinson

math.APWider flowsarXiv:2404.14402Journal de Mathématiques Pures et Appliquées, 192, 103625, 2024
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Abstract

We connect adversarial training for binary classification to a geometric evolution equation for the decision boundary. Relying on a perspective that recasts adversarial training as a regularization problem, we introduce a modified training scheme that constitutes a minimizing movements scheme for a nonlocal perimeter functional. We prove that the scheme is monotone and consistent as the adversarial budget vanishes and the perimeter localizes, and as a consequence we rigorously show that the scheme approximates a weighted mean curvature flow. This highlights that the efficacy of adversarial training may be due to locally minimizing the length of the decision boundary. In our analysis, we introduce a variety of tools for working with the subdifferential of a supremal-type nonlocal total variation and its regularity properties.

Mean curvature flowAnisotropic & nonlocal flowsWeak & singular flows

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Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Strong convergence of the thresholding scheme for the mean curvature flow of mean convex sets (Jakob Fuchs and Tim Laux, 2022)2Fuchs 2022 Diffuse-interface approximation and weak-strong uniqueness of anisotropic mean curvature flow (Tim Laux, Kerrek Stinson and Clemens Ullrich, 2022)1Laux 2022 Generic level sets in mean curvature flow are BV solutions (Anton Ullrich and Tim Laux, 2023)8Ullrich 2023 Minimizing movements for forced anisotropic curvature flow of droplets (Shokhrukh Yu. Kholmatov, 2024)5Kholmatov 2024 Convergence of a heterogeneous Allen-Cahn equation to weighted mean curvature flow (Likhit Ganedi, Alice Marveggio and Kerrek Stinson, 2024)6Ganedi 2024 Variational Nonlinear and Nonlocal Curvature Flows (Daniele De Gennaro, 2025)4De Gennaro 2025 A variational approach to the volume-preserving anisotropic mean curvature flow in 2D (Andrea Kubin, Domenico Angelo La Manna and Enrico Pasqualetto, 2025)7Kubin 2025 De Giorgi varifold solutions to Mean Curvature Flow: a minimizing movements approach (Tim Laux and Andrea Poiatti, 2026)3Laux 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Diffuse-interface approximation and weak-strong uniqueness of anisotropic mean curvature flow

    Tim Laux, Kerrek Stinson and Clemens UllrichWider flows PDF

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Same authors Tim Laux, Kerrek Stinson

  2. Strong convergence of the thresholding scheme for the mean curvature flow of mean convex sets

    Jakob Fuchs and Tim LauxWider flows PDF

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Same author Tim Laux · Shared terms “minimizing movements”

  3. De Giorgi varifold solutions to Mean Curvature Flow: a minimizing movements approach

    Tim Laux and Andrea PoiattiWider flows PDF

    Topics Weak & singular flows · Same author Tim Laux · Shared terms “minimizing movements”, “movement scheme”

  4. Variational Nonlinear and Nonlocal Curvature Flows

    Daniele De GennaroWider flows PDF

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Shared terms “movement scheme”, “minimizing movements”

  5. Minimizing movements for forced anisotropic curvature flow of droplets

    Shokhrukh Yu. KholmatovWider flows PDF

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Shared terms “minimizing movements”, “perimeter”

  6. Convergence of a heterogeneous Allen-Cahn equation to weighted mean curvature flow

    Likhit Ganedi, Alice Marveggio and Kerrek StinsonWider flows PDF

    Topics Weak & singular flows · Same author Kerrek Stinson · Shared terms “weighted mean”, “variation”

  7. A variational approach to the volume-preserving anisotropic mean curvature flow in 2D

    Andrea Kubin, Domenico Angelo La Manna and Enrico PasqualettoWider flows PDF

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Shared terms “movement scheme”, “minimizing movements”

  8. Generic level sets in mean curvature flow are BV solutions

    Anton Ullrich and Tim LauxWider flows PDF

    Topics Weak & singular flows · Same author Tim Laux · Shared terms “perimeter”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Anisotropic & nonlocal flows Wider flows When surface energy depends on direction, as in crystals, mean curvature flow becomes anisotropic, and the Wulff shape replaces the round sphere. Start with: Crystalline variational problems (Taylor, 1978) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  2. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.