Overparametrized linear dimensionality reductions: From projection pursuit to two-layer neural networks
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866908347393900544 |
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| author | Montanari, Andrea Zhou, Kangjie |
| author_facet | Montanari, Andrea Zhou, Kangjie |
| contents | Given a cloud of $n$ data points in $\mathbb{R}^d$, consider all projections onto $m$-dimensional subspaces of $\mathbb{R}^d$ and, for each such projection, the empirical distribution of the projected points. What does this collection of probability distributions look like when $n,d$ grow large?
We consider this question under the null model in which the points are i.i.d. standard Gaussian vectors, focusing on the asymptotic regime in which $n,d\to\infty$, with $n/d\toα\in (0,\infty)$, while $m$ is fixed. Denoting by $\mathscr{F}_{m, α}$ the set of probability distributions in $\mathbb{R}^m$ that arise as low-dimensional projections in this limit, we establish new inner and outer bounds on $\mathscr{F}_{m, α}$. In particular, we characterize the Wasserstein radius of $\mathscr{F}_{m,α}$ up to constant multiplicative factors, and determine it exactly for $m=1$. We also prove sharp bounds in terms of Kullback-Leibler divergence and Rényi information dimension.
The previous question has application to unsupervised learning methods, such as projection pursuit and independent component analysis. We introduce a version of the same problem that is relevant for supervised learning, and prove a sharp Wasserstein radius bound. As an application, we establish an upper bound on the interpolation threshold of two-layers neural networks with $m$ hidden neurons. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_06526 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Overparametrized linear dimensionality reductions: From projection pursuit to two-layer neural networks Montanari, Andrea Zhou, Kangjie Machine Learning Given a cloud of $n$ data points in $\mathbb{R}^d$, consider all projections onto $m$-dimensional subspaces of $\mathbb{R}^d$ and, for each such projection, the empirical distribution of the projected points. What does this collection of probability distributions look like when $n,d$ grow large? We consider this question under the null model in which the points are i.i.d. standard Gaussian vectors, focusing on the asymptotic regime in which $n,d\to\infty$, with $n/d\toα\in (0,\infty)$, while $m$ is fixed. Denoting by $\mathscr{F}_{m, α}$ the set of probability distributions in $\mathbb{R}^m$ that arise as low-dimensional projections in this limit, we establish new inner and outer bounds on $\mathscr{F}_{m, α}$. In particular, we characterize the Wasserstein radius of $\mathscr{F}_{m,α}$ up to constant multiplicative factors, and determine it exactly for $m=1$. We also prove sharp bounds in terms of Kullback-Leibler divergence and Rényi information dimension. The previous question has application to unsupervised learning methods, such as projection pursuit and independent component analysis. We introduce a version of the same problem that is relevant for supervised learning, and prove a sharp Wasserstein radius bound. As an application, we establish an upper bound on the interpolation threshold of two-layers neural networks with $m$ hidden neurons. |
| title | Overparametrized linear dimensionality reductions: From projection pursuit to two-layer neural networks |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2206.06526 |