Wide stable neural networks: Sample regularity, functional convergence and Bayesian inverse problems
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915190313844736 |
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| author | Soto, Tomás |
| author_facet | Soto, Tomás |
| contents | We study the large-width asymptotics of random fully connected neural networks with weights drawn from $α$-stable distributions, a family of heavy-tailed distributions arising as the limiting distributions in the Gnedenko-Kolmogorov heavy-tailed central limit theorem. We show that in an arbitrary bounded Euclidean domain $\mathcal{U}$ with smooth boundary, the random field at the infinite-width limit, characterized in previous literature in terms of finite-dimensional distributions, has sample functions in the fractional Sobolev-Slobodeckij-type quasi-Banach function space $W^{s,p}(\mathcal{U})$ for integrability indices $p < α$ and suitable smoothness indices $s$ depending on the activation function of the neural network, and establish the functional convergence of the processes in the space of probability measures on $W^{s,p}(\mathcal{U})$. This convergence result is leveraged in the study of functional posteriors for edge-preserving Bayesian inverse problems with stable neural network priors. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_03909 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Wide stable neural networks: Sample regularity, functional convergence and Bayesian inverse problems Soto, Tomás Statistics Theory Probability Primary: 68T07, 62F15, 60G52, 60G17, Secondary: 46E35 We study the large-width asymptotics of random fully connected neural networks with weights drawn from $α$-stable distributions, a family of heavy-tailed distributions arising as the limiting distributions in the Gnedenko-Kolmogorov heavy-tailed central limit theorem. We show that in an arbitrary bounded Euclidean domain $\mathcal{U}$ with smooth boundary, the random field at the infinite-width limit, characterized in previous literature in terms of finite-dimensional distributions, has sample functions in the fractional Sobolev-Slobodeckij-type quasi-Banach function space $W^{s,p}(\mathcal{U})$ for integrability indices $p < α$ and suitable smoothness indices $s$ depending on the activation function of the neural network, and establish the functional convergence of the processes in the space of probability measures on $W^{s,p}(\mathcal{U})$. This convergence result is leveraged in the study of functional posteriors for edge-preserving Bayesian inverse problems with stable neural network priors. |
| title | Wide stable neural networks: Sample regularity, functional convergence and Bayesian inverse problems |
| topic | Statistics Theory Probability Primary: 68T07, 62F15, 60G52, 60G17, Secondary: 46E35 |
| url | https://arxiv.org/abs/2407.03909 |