Linear Independence of Generalized Neurons and Related Functions
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910634307747840 |
|---|---|
| author | Zhang, Leyang |
| author_facet | Zhang, Leyang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_03693 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Linear Independence of Generalized Neurons and Related Functions Zhang, Leyang Machine Learning 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. |
| title | Linear Independence of Generalized Neurons and Related Functions |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2410.03693 |