Efficient Estimation of the Central Mean Subspace via Smoothed Gradient Outer Products
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866909313889468416 |
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| author | Yuan, Gan Xu, Mingyue Kpotufe, Samory Hsu, Daniel |
| author_facet | Yuan, Gan Xu, Mingyue Kpotufe, Samory Hsu, Daniel |
| contents | We consider the problem of sufficient dimension reduction (SDR) for multi-index models. The estimators of the central mean subspace in prior works either have slow (non-parametric) convergence rates, or rely on stringent distributional conditions (e.g., the covariate distribution $P_{\mathbf{X}}$ being elliptical symmetric). In this paper, we show that a fast parametric convergence rate of form $C_d \cdot n^{-1/2}$ is achievable via estimating the \emph{expected smoothed gradient outer product}, for a general class of distribution $P_{\mathbf{X}}$ admitting Gaussian or heavier distributions. When the link function is a polynomial with a degree of at most $r$ and $P_{\mathbf{X}}$ is the standard Gaussian, we show that the prefactor depends on the ambient dimension $d$ as $C_d \propto d^r$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_15469 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Efficient Estimation of the Central Mean Subspace via Smoothed Gradient Outer Products Yuan, Gan Xu, Mingyue Kpotufe, Samory Hsu, Daniel Machine Learning Methodology 62B05, 62G08 We consider the problem of sufficient dimension reduction (SDR) for multi-index models. The estimators of the central mean subspace in prior works either have slow (non-parametric) convergence rates, or rely on stringent distributional conditions (e.g., the covariate distribution $P_{\mathbf{X}}$ being elliptical symmetric). In this paper, we show that a fast parametric convergence rate of form $C_d \cdot n^{-1/2}$ is achievable via estimating the \emph{expected smoothed gradient outer product}, for a general class of distribution $P_{\mathbf{X}}$ admitting Gaussian or heavier distributions. When the link function is a polynomial with a degree of at most $r$ and $P_{\mathbf{X}}$ is the standard Gaussian, we show that the prefactor depends on the ambient dimension $d$ as $C_d \propto d^r$. |
| title | Efficient Estimation of the Central Mean Subspace via Smoothed Gradient Outer Products |
| topic | Machine Learning Methodology 62B05, 62G08 |
| url | https://arxiv.org/abs/2312.15469 |