A Lower Bound for the Number of Linear Regions of Ternary ReLU Regression Neural Networks
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| Format: | Preprint |
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2025
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| _version_ | 1866915956730626048 |
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| author | Nakahara, Yuta Kobayashi, Manabu Matsushima, Toshiyasu |
| author_facet | Nakahara, Yuta Kobayashi, Manabu Matsushima, Toshiyasu |
| contents | With the advancement of deep learning, reducing computational complexity and memory consumption has become a critical challenge, and ternary neural networks (NNs) that restrict parameters to $\{-1, 0, +1\}$ have attracted attention as a promising approach. While ternary NNs demonstrate excellent performance in practical applications such as image recognition and natural language processing, their theoretical understanding remains insufficient. In this paper, we theoretically analyze the expressivity of ternary NNs from the perspective of the number of linear regions. Specifically, we evaluate the number of linear regions of ternary regression NNs with Rectified Linear Unit (ReLU) for activation functions and prove that the number of linear regions increases polynomially with respect to network width and exponentially with respect to depth, similar to standard NNs. Moreover, we show that it suffices to first double the width, then either square the width or double the depth of ternary NNs with alternating ReLU and identity layers to achieve a lower bound on the maximum number of linear regions comparable to that of general ReLU regression NNs. When using ReLU in all the layers, a similar bound is obtained by further doubling the width. This provides a theoretical explanation, in some sense, for the practical success of ternary NNs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_16079 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Lower Bound for the Number of Linear Regions of Ternary ReLU Regression Neural Networks Nakahara, Yuta Kobayashi, Manabu Matsushima, Toshiyasu Machine Learning Artificial Intelligence With the advancement of deep learning, reducing computational complexity and memory consumption has become a critical challenge, and ternary neural networks (NNs) that restrict parameters to $\{-1, 0, +1\}$ have attracted attention as a promising approach. While ternary NNs demonstrate excellent performance in practical applications such as image recognition and natural language processing, their theoretical understanding remains insufficient. In this paper, we theoretically analyze the expressivity of ternary NNs from the perspective of the number of linear regions. Specifically, we evaluate the number of linear regions of ternary regression NNs with Rectified Linear Unit (ReLU) for activation functions and prove that the number of linear regions increases polynomially with respect to network width and exponentially with respect to depth, similar to standard NNs. Moreover, we show that it suffices to first double the width, then either square the width or double the depth of ternary NNs with alternating ReLU and identity layers to achieve a lower bound on the maximum number of linear regions comparable to that of general ReLU regression NNs. When using ReLU in all the layers, a similar bound is obtained by further doubling the width. This provides a theoretical explanation, in some sense, for the practical success of ternary NNs. |
| title | A Lower Bound for the Number of Linear Regions of Ternary ReLU Regression Neural Networks |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2507.16079 |