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Dense Associative Memory Is Robust to Adversarial Inputs
Open accessAbstract
Deep neural networks (DNNs) trained in a supervised way suffer from two known problems. First, the minima of the objective function used in learning correspond to data points (also known as rubbish examples or fooling images) that lack semantic similarity with the training data. Second, a clean input can be changed by a small, and often imperceptible for human vision, perturbation so that the resulting deformed input is misclassified by the network. These findings emphasize the differences between the ways DNNs and humans classify patterns and raise a question of designing learning algorithms that more accurately mimic human perception compared to the existing methods. Our article examines these questions within the framework of dense associative memory (DAM) models. These models are defined by the energy function, with higher-order (higher than quadratic) interactions between the neurons. We show that in the limit when the power of the interaction vertex in the energy function is sufficiently large, these models have the following three properties. First, the minima of the objective function are free from rubbish images, so that each minimum is a semantically meaningful pattern. Second, artificial patterns poised precisely at the decision boundary look ambiguous to human subjects and share aspects of both classes that are separated by that decision boundary. Third, adversarial images constructed by models with small power of the interaction vertex, which are equivalent to DNN with rectified linear units, fail to transfer to and fool the models with higher-order interactions. This opens up the possibility of using higher-order models for detecting and stopping malicious adversarial attacks. The results we present suggest that DAMs with higher-order energy functions are more robust to adversarial and rubbish inputs than DNNs with rectified linear units.
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The energy function of modern Hopfield networks makes them robust against adversarial attacks.
“The energy function of modern Hopfield networks makes them robust against adversarial attacks (Krotov & Hopfield, 2018).”
From Hopfield Networks is All You Need (2020) · cites this paper as Krotov & Hopfield, 2018
The Abstract (p. 1) reports that adversarial images made for low-power models fail to fool dense associative memory models with sufficiently high-order energy functions. The claim is broader than that result: the Discussion (p. 11) says that changing the activation function alone is not enough and that other aspects of training are important. (LLM, conf=0.87)
Experimental — accuracy not yet measuredInteraction functions of the form F(x) = x^n lead to storage capacity of α_n d^{n-1}, where α_n depends on the allowed error probability.
“Interaction functions of the form F ( x ) = x n lead to storage capacity of α n d n -1 , where α n depends on the allowed error probability (Krotov & Hopfield, 2016; 2018; Demircigil et al., 2017) (see (Krotov & Hopfield, 2018) for the non-binary case).”
From Hopfield Networks is All You Need (2020) · cites this paper as Krotov & Hopfield, 2018
Section 2 (p. 3) describes power-dependent, rectified-polynomial energy functions, but does not give a storage-capacity result. Searches for the α_n d^{n−1} scaling and its dependence on allowed error probability found no passage addressing either point in our copy. (LLM, conf=0.74)
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Metadata
- Authors
- Krotov, D., & Hopfield, J.
- Affiliations
- Institute for Advanced Study (US)
- Princeton University (US)
- ORCID
- Krotov, Dmitry
- Venue
- Neural Computation
- Publisher
- MIT Press - Journals
- Published
- 2018-12
- Volume
- 30
- Issue
- 12
- Pages
- 3151-3167
- ISSN
- 0899-7667 (ISSN-L)
- 1530-888X
- OpenAlex
- W2584321260
- Cited by
- 106
- References
- 2
- Citations by year
- 2026: 6 · 2025: 13 · 2024: 13 · 2023: 8 · 2022: 9 · 2021: 16 · 2020: 14 · 2019: 11 · 2018: 12
- FWCI
- 7.42
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Neural Computation (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Adversarial Robustness in Machine Learning
- Physical Sciences › Computer Science › Artificial Intelligence › Neural Networks and Applications
- Physical Sciences › Computer Science › Computer Vision and Pattern Recognition › Advanced Neural Network Applications
- Keywords
- dense associative memory, adversarial robustness, adversarial examples, higher-order interactions, energy function, decision boundaries, rectified linear units, deep neural network, human perception
- MeSH terms
- Brain, Humans, Pattern Recognition, Visual, Neural Networks, Computer
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