Adversarially robust neural network decision boundaries via tropical geometry.

Area of Science:

• Artificial Intelligence
• Machine Learning
• Deep Learning

Background:

• Adversarial attacks pose a significant threat to the reliability of neural networks.
• Existing defenses like adversarial training are computationally expensive.
• Piece-wise linear neural networks possess inherent tropical properties.

Purpose of the Study:

• To introduce a novel, computationally efficient tropical convolutional neural network architecture.
• To enhance the robustness of neural networks against adversarial attacks.
• To investigate the geometric properties of decision boundaries in tropical neural networks.

Main Methods:

• Embedding data into the tropical projective torus using a tropical embedding layer.
• Integrating the tropical embedding layer into existing neural network architectures.
• Analyzing the geometry of decision boundaries and the number of linear regions.

Main Results:

• The proposed tropical convolutional neural network architecture demonstrates state-of-the-art adversarial robustness.
• The tropical embedding layer increases the number of linear regions in decision boundaries.
• The method requires significantly less computational time compared to adversarial training.

Conclusions:

• The tropical embedding layer offers a simple, efficient, and effective defense against adversarial attacks.
• This approach provides a new avenue for building more secure and robust deep learning models.
• The tropical convolutional neural network architecture presents a promising alternative for practical adversarial defense.