Linear Independence of Generalized Neurons and Related Functions
<|reference_start|>Linear Independence of Generalized Neurons and Related 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 $\{H_1(\theta_1, \cdot), ..., H_n(\theta_n, \cdot)\}$ 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.<|reference_end|>
@article{zhang2024linear,
    title={Linear Independence of Generalized Neurons and Related Functions},
    author={Leyang Zhang},
    journal={arXiv preprint arXiv:2410.03693},
    year={2024},
    archivePrefix={arXiv},
    eprint={2410.03693},
    primaryClass={cs.LG}
}