Exploring Sparsity and Smoothness of Arbitrary $\ell_p$ Norms in Adversarial Attacks
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914310809190400 |
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| author | Duhme, Christof Eilers, Florian Jiang, Xiaoyi |
| author_facet | Duhme, Christof Eilers, Florian Jiang, Xiaoyi |
| contents | Adversarial attacks against deep neural networks are commonly constructed under $\ell_p$ norm constraints, most often using $p=1$, $p=2$ or $p=\infty$, and potentially regularized for specific demands such as sparsity or smoothness. These choices are typically made without a systematic investigation of how the norm parameter \( p \) influences the structural and perceptual properties of adversarial perturbations. In this work, we study how the choice of \( p \) affects sparsity and smoothness of adversarial attacks generated under \( \ell_p \) norm constraints for values of $p \in [1,2]$. To enable a quantitative analysis, we adopt two established sparsity measures from the literature and introduce three smoothness measures. In particular, we propose a general framework for deriving smoothness measures based on smoothing operations and additionally introduce a smoothness measure based on first-order Taylor approximations. Using these measures, we conduct a comprehensive empirical evaluation across multiple real-world image datasets and a diverse set of model architectures, including both convolutional and transformer-based networks. We show that the choice of $\ell_1$ or $\ell_2$ is suboptimal in most cases and the optimal $p$ value is dependent on the specific task. In our experiments, using $\ell_p$ norms with $p\in [1.3, 1.5]$ yields the best trade-off between sparse and smooth attacks. These findings highlight the importance of principled norm selection when designing and evaluating adversarial attacks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_06578 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Exploring Sparsity and Smoothness of Arbitrary $\ell_p$ Norms in Adversarial Attacks Duhme, Christof Eilers, Florian Jiang, Xiaoyi Machine Learning Artificial Intelligence Adversarial attacks against deep neural networks are commonly constructed under $\ell_p$ norm constraints, most often using $p=1$, $p=2$ or $p=\infty$, and potentially regularized for specific demands such as sparsity or smoothness. These choices are typically made without a systematic investigation of how the norm parameter \( p \) influences the structural and perceptual properties of adversarial perturbations. In this work, we study how the choice of \( p \) affects sparsity and smoothness of adversarial attacks generated under \( \ell_p \) norm constraints for values of $p \in [1,2]$. To enable a quantitative analysis, we adopt two established sparsity measures from the literature and introduce three smoothness measures. In particular, we propose a general framework for deriving smoothness measures based on smoothing operations and additionally introduce a smoothness measure based on first-order Taylor approximations. Using these measures, we conduct a comprehensive empirical evaluation across multiple real-world image datasets and a diverse set of model architectures, including both convolutional and transformer-based networks. We show that the choice of $\ell_1$ or $\ell_2$ is suboptimal in most cases and the optimal $p$ value is dependent on the specific task. In our experiments, using $\ell_p$ norms with $p\in [1.3, 1.5]$ yields the best trade-off between sparse and smooth attacks. These findings highlight the importance of principled norm selection when designing and evaluating adversarial attacks. |
| title | Exploring Sparsity and Smoothness of Arbitrary $\ell_p$ Norms in Adversarial Attacks |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2602.06578 |