Sharp phase transitions in high-dimensional changepoint detection
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909552764518400 |
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| author | Xiang, Daniel Gao, Chao |
| author_facet | Xiang, Daniel Gao, Chao |
| contents | We study a hypothesis testing problem in the context of high-dimensional changepoint detection. Given a matrix $X \in \R^{p \times n}$ with independent Gaussian entries, the goal is to determine whether or not a sparse, non-null fraction of rows in $X$ exhibits a shift in mean at a common index between $1$ and $n$. We focus on three aspects of this problem: the sparsity of non-null rows, the presence of a single, common changepoint in the non-null rows, and the signal strength associated with the changepoint. Within an asymptotic regime relating the data dimensions $n$ and $p$ to the signal sparsity and strength, the information-theoretic limits of this testing problem are characterized by a formula that determines whether or not there exists a testing procedure whose sum of Type I and II errors tends to zero as $n,p \to \infty$. The formula, called the \emph{detection boundary}, partitions the parameter space into a two regions: one where it is possible to detect the presence of a single aligned changepoint (detectable region), and another where no test is able to consistently distinguish the mean matrix from one with constant rows (undetectable region). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_11704 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharp phase transitions in high-dimensional changepoint detection Xiang, Daniel Gao, Chao Statistics Theory We study a hypothesis testing problem in the context of high-dimensional changepoint detection. Given a matrix $X \in \R^{p \times n}$ with independent Gaussian entries, the goal is to determine whether or not a sparse, non-null fraction of rows in $X$ exhibits a shift in mean at a common index between $1$ and $n$. We focus on three aspects of this problem: the sparsity of non-null rows, the presence of a single, common changepoint in the non-null rows, and the signal strength associated with the changepoint. Within an asymptotic regime relating the data dimensions $n$ and $p$ to the signal sparsity and strength, the information-theoretic limits of this testing problem are characterized by a formula that determines whether or not there exists a testing procedure whose sum of Type I and II errors tends to zero as $n,p \to \infty$. The formula, called the \emph{detection boundary}, partitions the parameter space into a two regions: one where it is possible to detect the presence of a single aligned changepoint (detectable region), and another where no test is able to consistently distinguish the mean matrix from one with constant rows (undetectable region). |
| title | Sharp phase transitions in high-dimensional changepoint detection |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2403.11704 |