Sparsity meets correlation in Gaussian sequence model
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917898914627584 |
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| author | Kotekal, Subhodh Gao, Chao |
| author_facet | Kotekal, Subhodh Gao, Chao |
| contents | We study estimation of an $s$-sparse signal in the $p$-dimensional Gaussian sequence model with equicorrelated observations and derive the minimax rate. A new phenomenon emerges from correlation, namely the rate scales with respect to $p-2s$ and exhibits a phase transition at $p-2s \asymp \sqrt{p}$. Correlation is shown to be a blessing provided it is sufficiently strong, and the critical correlation level exhibits a delicate dependence on the sparsity level. Due to correlation, the minimax rate is driven by two subproblems: estimation of a linear functional (the average of the signal) and estimation of the signal's $(p-1)$-dimensional projection onto the orthogonal subspace. The high-dimensional projection is estimated via sparse regression and the linear functional is cast as a robust location estimation problem. Existing robust estimators turn out to be suboptimal, and we show a kernel mode estimator with a widening bandwidth exploits the Gaussian character of the data to achieve the optimal estimation rate. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_09356 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sparsity meets correlation in Gaussian sequence model Kotekal, Subhodh Gao, Chao Statistics Theory Methodology We study estimation of an $s$-sparse signal in the $p$-dimensional Gaussian sequence model with equicorrelated observations and derive the minimax rate. A new phenomenon emerges from correlation, namely the rate scales with respect to $p-2s$ and exhibits a phase transition at $p-2s \asymp \sqrt{p}$. Correlation is shown to be a blessing provided it is sufficiently strong, and the critical correlation level exhibits a delicate dependence on the sparsity level. Due to correlation, the minimax rate is driven by two subproblems: estimation of a linear functional (the average of the signal) and estimation of the signal's $(p-1)$-dimensional projection onto the orthogonal subspace. The high-dimensional projection is estimated via sparse regression and the linear functional is cast as a robust location estimation problem. Existing robust estimators turn out to be suboptimal, and we show a kernel mode estimator with a widening bandwidth exploits the Gaussian character of the data to achieve the optimal estimation rate. |
| title | Sparsity meets correlation in Gaussian sequence model |
| topic | Statistics Theory Methodology |
| url | https://arxiv.org/abs/2312.09356 |