研究神经元的线性无关性，揭示深层网络中参数变化对函数独立性的影响。
Linear Independence of Generalized Neurons and Related Functions
- 针对任意层数与宽度的神经元，给出线性无关性的完整判定条件。
- 对通用解析激活函数，证明了参数变化时函数组线性无关的充要条件。
- 理论成果适用于深度网络分析，适合从事神经网络数学建模的研究者。
神经元的线性无关性在神经网络的理论分析中具有重要意义。给定神经元 $H_1, ..., H_n: R^N imes R^d o R$，我们关注的问题是：当参数 $θ_1, ..., θ_n$ 在 $R^N$ 范围内变化时，函数组 $\\{H_1(θ_1, \ cdot), ..., H_n(θ_n, \ cdot)\\\$ 是否线性无关。已有工作对无偏置的两层神经元、通用光滑激活函数给出了完整刻画。本文将该问题拓展至任意层数与宽度的神经元，对通用解析激活函数给出了简洁而完整的线性无关性判定条件。
原文摘要 · Abstract (English)
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 $\{H_1(θ_1, \cdot), ..., H_n(θ_n, \cdot)\}$ are linearly independent as the parameters $θ_1, ..., θ_n$ 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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