Transformed Low-Rank Parameterization Can Help Robust Generalization for Tensor Neural Networks

• URL: http://arxiv.org/abs/2303.00196v3
• Date: Wed, 20 Dec 2023 08:57:18 GMT
• Title: Transformed Low-Rank Parameterization Can Help Robust Generalization for Tensor Neural Networks
• Authors: Andong Wang, Chao Li, Mingyuan Bai, Zhong Jin, Guoxu Zhou, Qibin Zhao
• Abstract summary: tensor Singular Value Decomposition (t-SVD) has achieved extensive success in multi-channel data representation. It still remains unclear how t-SVD theoretically affects the learning behavior of t-NNs. This paper is the first to answer this question by deriving the upper bounds of the generalization error of both standard and adversarially trained t-NNs.
• Score: 32.87980654923361
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Achieving efficient and robust multi-channel data learning is a challenging task in data science. By exploiting low-rankness in the transformed domain, i.e., transformed low-rankness, tensor Singular Value Decomposition (t-SVD) has achieved extensive success in multi-channel data representation and has recently been extended to function representation such as Neural Networks with t-product layers (t-NNs). However, it still remains unclear how t-SVD theoretically affects the learning behavior of t-NNs. This paper is the first to answer this question by deriving the upper bounds of the generalization error of both standard and adversarially trained t-NNs. It reveals that the t-NNs compressed by exact transformed low-rank parameterization can achieve a sharper adversarial generalization bound. In practice, although t-NNs rarely have exactly transformed low-rank weights, our analysis further shows that by adversarial training with gradient flow (GF), the over-parameterized t-NNs with ReLU activations are trained with implicit regularization towards transformed low-rank parameterization under certain conditions. We also establish adversarial generalization bounds for t-NNs with approximately transformed low-rank weights. Our analysis indicates that the transformed low-rank parameterization can promisingly enhance robust generalization for t-NNs.

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