Linear Independence of Generalized Neurons and Related Functions

• URL: http://arxiv.org/abs/2410.03693v1
• Date: Sun, 22 Sep 2024 21:09:15 GMT
• Title: Linear Independence of Generalized Neurons and Related Functions
• Authors: Leyang Zhang,
• Abstract summary: Linear independence of neurons plays a significant role in theoretical analysis of neural networks. We study the problem for neurons with arbitrary layers and widths, giving a simple but complete characterization for generic analytic activation functions.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The linear independence of neurons plays a significant role in theoretical analysis of neural networks. Specifically, given neurons $H_1, ..., H_n: \bR^N \times \bR^d \to \bR$, we are interested in the following question: when are $\{H_1(\theta_1, \cdot), ..., H_n(\theta_n, \cdot)\}$ are linearly independent as the parameters $\theta_1, ..., \theta_n$ of these functions vary over $\bR^N$. Previous works give a complete characterization of two-layer neurons without bias, for generic smooth activation functions. In this paper, we study the problem for neurons with arbitrary layers and widths, giving a simple but complete characterization for generic analytic activation functions.

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