Linear Independence of Generalized Neurons and Related Functions
This work addresses a theoretical problem in neural network analysis, offering incremental insights by extending prior results from two-layer neurons to more complex architectures.
The paper tackles the problem of characterizing linear independence in neurons with arbitrary layers and widths, providing a complete characterization for generic analytic activation functions.
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.