# Adversarial Training Without Input Gradients via Low-Rank Householder Expansions

arXiv: https://arxiv.org/abs/2608.26963 · HTML: https://mlsanity.com/paper/2608.26963/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 30 of its 34 references that are in the map.
- First posted on arXiv 27 Aug 2026.

## Builds on (most cited first, 30 of 30)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Reliable evaluation of adversarial robustness with an ensemble of diverse parameter-free attacks](https://mlsanity.com/paper/2003.01690/) (2020)
- [Natural Adversarial Examples](https://mlsanity.com/paper/1907.07174/) (2019)
- [Obfuscated Gradients Give a False Sense of Security: Circumventing Defenses to Adversarial Examples](https://mlsanity.com/paper/1802.00420/) (2018)
- [Robustness May Be at Odds with Accuracy](https://mlsanity.com/paper/1805.12152/) (2018)
- [Fast is better than free: Revisiting adversarial training](https://mlsanity.com/paper/2001.03994/) (2020)
- [Square Attack: a query-efficient black-box adversarial attack via random search](https://mlsanity.com/paper/1912.00049/) (2019)
- [Adversarial Training for Free!](https://mlsanity.com/paper/1904.12843/) (2019)
- [On Adaptive Attacks to Adversarial Example Defenses](https://mlsanity.com/paper/2002.08347/) (2020)
- [On Evaluating Adversarial Robustness](https://mlsanity.com/paper/1902.06705/) (2019)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [Minimally distorted Adversarial Examples with a Fast Adaptive Boundary Attack](https://mlsanity.com/paper/1907.02044/) (2019)
- [Parseval Networks: Improving Robustness to Adversarial Examples](https://mlsanity.com/paper/1704.08847/) (2017)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Improving the Adversarial Robustness and Interpretability of Deep Neural Networks by Regularizing their Input Gradients](https://mlsanity.com/paper/1711.09404/) (2017)
- [Understanding and Improving Fast Adversarial Training](https://mlsanity.com/paper/2007.02617/) (2020)
- [You Only Propagate Once: Accelerating Adversarial Training via Maximal Principle](https://mlsanity.com/paper/1905.00877/) (2019)
- [Robustness via curvature regularization, and vice versa](https://mlsanity.com/paper/1811.09716/) (2018)
- [The Space of Transferable Adversarial Examples](https://mlsanity.com/paper/1704.03453/) (2017)
- [Adversarial Robustness through Local Linearization](https://mlsanity.com/paper/1907.02610/) (2019)
- [Robust Learning with Jacobian Regularization](https://mlsanity.com/paper/1908.02729/) (2019)
- [Orthogonalizing Convolutional Layers with the Cayley Transform](https://mlsanity.com/paper/2104.07167/) (2021)
- [Preventing Gradient Attenuation in Lipschitz Constrained Convolutional Networks](https://mlsanity.com/paper/1911.00937/) (2019)
- [Skew Orthogonal Convolutions](https://mlsanity.com/paper/2105.11417/) (2021)
- [Improved deterministic l2 robustness on CIFAR-10 and CIFAR-100](https://mlsanity.com/paper/2108.04062/) (2021)
- [Subspace Adversarial Training](https://mlsanity.com/paper/2111.12229/) (2021)

## Abstract (arXiv)

This work concerns adversarial training against the small-norm adversarial examples that arise from the inherent input instability of a trained deep neural network. Examples in this class are small as measured in the relative ℓ2-norm, and therefore lie in the neighborhood of the input on which the model acts approximately linearly, the regime in which the perturbation remains imperceptible. We first show that such examples can be computed directly from the trained network parameters, without input gradient iterations, by means of a linearization called the low-rank Householder expansion (LRHE). The expansion describes the composed affine map rather than any individual layer, and the directions it identifies are read from the activation pattern already available in the forward pass. We then propose a simple adversarial training scheme built on this construction. No differentiation with respect to the input is performed at any point: training requires only additional forward evaluations, with weight parameters updated by the standard backward pass, and the inner maximization of the usual min-max formulation is eliminated entirely. That such a regularizer exists is our main finding: the methods that dispense with the inner search all obtain their local geometry by differentiating with respect to the input, and we show this is not necessary. The regularizer costs the equivalent of 2.8 PGD steps per epoch, an 8.7× reduction relative to 40-step adversarial training on MNIST and below the cost of 3-step training. The resulting models match three-step PGD adversarial training for relative ℓ2 budgets varepsilon ≤ 0.02 and 40-step training for varepsilon ≤ 0.012, falling away beyond, consistent with the locality of the expansion.