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Robust One-Class Classification with Signed Distance Function using 1-Lipschitz Neural Networks

Louis Bethune, Paul Novello, Thibaut Boissin, Guillaume Coiffier, Mathieu Serrurier, Quentin Vincenot, Andres Troya-Galvis

TL;DR

OCSDF introduces learning the Signed Distance Function $\mathcal{S}$ to the boundary $\partial\mathcal{X}$ of the data support for One Class Classification by training a $1$-Lipschitz neural network with the Hinge Kantorovich-Rubinstein (HKR) loss against a carefully constructed complementary distribution $Q$. Through an Adapted Newton-Raphson sampling scheme and alternating minimization, the method yields a robust normality score whose behavior is underpinned by the Eikonal condition $\|\nabla_x \mathcal{S}(x)\|=1$, enabling certifiable $l_2$-robustness and a certified AUROC computable at the same cost as standard AUROC. Empirically, OCSDF is competitive on tabular and image OCC benchmarks, while offering provable robustness advantages and a natural link to implicit surface parametrization and generative visualization. The approach also opens avenues for shape reconstruction from point clouds and deeper integration of OCC with geometric representations, with code available at the authors' repository. Overall, OCSDF addresses core OCC challenges—negative data scarcity and adversarial vulnerability—by grounding the decision boundary in a geometrically meaningful SDF learned via Lipschitz-constrained networks.

Abstract

We propose a new method, dubbed One Class Signed Distance Function (OCSDF), to perform One Class Classification (OCC) by provably learning the Signed Distance Function (SDF) to the boundary of the support of any distribution. The distance to the support can be interpreted as a normality score, and its approximation using 1-Lipschitz neural networks provides robustness bounds against $l2$ adversarial attacks, an under-explored weakness of deep learning-based OCC algorithms. As a result, OCSDF comes with a new metric, certified AUROC, that can be computed at the same cost as any classical AUROC. We show that OCSDF is competitive against concurrent methods on tabular and image data while being way more robust to adversarial attacks, illustrating its theoretical properties. Finally, as exploratory research perspectives, we theoretically and empirically show how OCSDF connects OCC with image generation and implicit neural surface parametrization. Our code is available at https://github.com/Algue-Rythme/OneClassMetricLearning

Robust One-Class Classification with Signed Distance Function using 1-Lipschitz Neural Networks

TL;DR

OCSDF introduces learning the Signed Distance Function to the boundary of the data support for One Class Classification by training a -Lipschitz neural network with the Hinge Kantorovich-Rubinstein (HKR) loss against a carefully constructed complementary distribution . Through an Adapted Newton-Raphson sampling scheme and alternating minimization, the method yields a robust normality score whose behavior is underpinned by the Eikonal condition , enabling certifiable -robustness and a certified AUROC computable at the same cost as standard AUROC. Empirically, OCSDF is competitive on tabular and image OCC benchmarks, while offering provable robustness advantages and a natural link to implicit surface parametrization and generative visualization. The approach also opens avenues for shape reconstruction from point clouds and deeper integration of OCC with geometric representations, with code available at the authors' repository. Overall, OCSDF addresses core OCC challenges—negative data scarcity and adversarial vulnerability—by grounding the decision boundary in a geometrically meaningful SDF learned via Lipschitz-constrained networks.

Abstract

We propose a new method, dubbed One Class Signed Distance Function (OCSDF), to perform One Class Classification (OCC) by provably learning the Signed Distance Function (SDF) to the boundary of the support of any distribution. The distance to the support can be interpreted as a normality score, and its approximation using 1-Lipschitz neural networks provides robustness bounds against adversarial attacks, an under-explored weakness of deep learning-based OCC algorithms. As a result, OCSDF comes with a new metric, certified AUROC, that can be computed at the same cost as any classical AUROC. We show that OCSDF is competitive against concurrent methods on tabular and image data while being way more robust to adversarial attacks, illustrating its theoretical properties. Finally, as exploratory research perspectives, we theoretically and empirically show how OCSDF connects OCC with image generation and implicit neural surface parametrization. Our code is available at https://github.com/Algue-Rythme/OneClassMetricLearning
Paper Structure (50 sections, 4 theorems, 23 equations, 11 figures, 13 tables, 3 algorithms)

This paper contains 50 sections, 4 theorems, 23 equations, 11 figures, 13 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. One Class Classification (OCC)
  4. SDF for neural implicit surfaces
  5. 1-Lipschitz neural nets
  6. Robustness and certification
  7. Tackling the lack of OOD data
  8. Method
  9. SDF learning formulated as binary classification
  10. Finding the complementary distribution by targeting the boundary
  11. Alternating minimization for SDF learning
  12. Properties
  13. Certificates against adversarial attacks
  14. Rank normal and anomalous examples
  15. Experiments
  16. ...and 35 more sections

Key Result

Theorem 1

SDF Learning with HKR loss. Let $\mathcal{L}^{\text{hkr}}_{m,\lambda}(yf(x))=\lambda\max{(0, m-yf(x))}-yf(x)$ be the Hinge Kantorovich Rubinstein loss, with margin $m=\epsilon$, regularization $\lambda>0$, prediction $f(x)$ and label $y\in\{-1,1\}$. Let $Q$ be a probability distribution on $B$. Let Let $f^{*}$ be the minimizer of population risk, whose existence is guaranteed with Arzelà-Ascoli t

Figures (11)

  • Figure 1: Summary of One Class Signed Distance Function (OCSDF). We start with an uniform negative sampling, then we fit a 1-Lipschitz classifier $f_{\theta}$ using the Hinge Kantorovich-Rubinstein loss. We apply the Adapted Newton Raphson algorithm \ref{['alg:newtonraphson']} to attract the points towards the boundary of the domain $\partial \mathcal{X}$ thanks to the smoothness of $f_{\theta}$, which in addition allows providing robustness certificates.
  • Figure 2: Contour plots of our method with 1-Lipschitz (LIP) network and $\mathcal{L}^{\text{hkr}}_{m,\lambda}$ (HKR) loss on toy examples of Scikit-learn.
  • Figure 3: Empirical Mean AUROC on all classes against adversarial attacks of various radii in One Class setting, using default parameters of FoolBox rauber2017foolboxnative.
  • Figure 4: Examples from algorithm \ref{['alg:newtonraphson']} with $T=64$ and $\eta=1$.
  • Figure 5: Visualization of the SDF (3rd column) from sparse point clouds of size $2048$ (2nd column) sampled from ground truth meshes (1st column) with Trimesh library, against the SSSR algorithm boltcheva2017surface that attempts to reproduce the meshes solely from a point cloud (4th column). The SDF exhibits better extrapolation properties and provides smooth surfaces.
  • ...and 6 more figures

Theorems & Definitions (12)

  • Definition 1: $\stackrel{B,\epsilon}{\sim}$ Complementary Distribution (informal)
  • Theorem 1
  • Definition 2: 1-Lipschitz neural network (informal)
  • Remark
  • Remark
  • Proposition 1: certifiable AUROC
  • Definition 3: $\stackrel{B,\epsilon}{\sim}$ Complementary Distribution
  • Theorem 1
  • proof
  • proof
  • ...and 2 more