Signal detection from spiked noise via asymmetrization
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915412275363840 |
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| author | Bao, Zhigang Cheong, Kha Man Lee, Jaehun Li, Yuji |
| author_facet | Bao, Zhigang Cheong, Kha Man Lee, Jaehun Li, Yuji |
| contents | The signal plus noise model $H=S+Y$ is a fundamental model in signal detection when a low rank signal $S$ is polluted by noise $Y$. In the high-dimensional setting, one often uses the leading singular values and corresponding singular vectors of $H$ to conduct the statistical inference of the signal $S$. Especially, when $Y$ consists of iid random entries, the singular values of $S$ can be estimated from those of $H$ as long as the signal $S$ is strong enough. However, when the $Y$ entries are heteroscedastic or heavy-tailed, this standard approach may fail. Especially in this work, we consider a situation that can easily arise with heteroscedastic or heavy-tailed noise but is particularly difficult to address using the singular value approach, namely, when the noise $Y$ itself may create spiked singular values. It has been a recurring question how to distinguish the signal $S$ from the spikes in $Y$, as this seems impossible by examining the leading singular values of $H$. Inspired by the work \cite{CCF21}, we turn to study the eigenvalues of an asymmetrized model when two samples $H_1=S+Y_1$ and $H_2=S+Y_2$ are available. We show that by looking into the leading eigenvalues (in magnitude) of the asymmetrized model $H_1H_2^*$, one can easily detect $S$. We will primarily discuss the heteroscedastic case and then discuss the extension to the heavy-tailed case. As a byproduct, we also derive the fundamental result regarding the outlier of non-Hermitian random matrix in \cite{Tao} under the minimal 2nd moment condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_19450 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Signal detection from spiked noise via asymmetrization Bao, Zhigang Cheong, Kha Man Lee, Jaehun Li, Yuji Statistics Theory The signal plus noise model $H=S+Y$ is a fundamental model in signal detection when a low rank signal $S$ is polluted by noise $Y$. In the high-dimensional setting, one often uses the leading singular values and corresponding singular vectors of $H$ to conduct the statistical inference of the signal $S$. Especially, when $Y$ consists of iid random entries, the singular values of $S$ can be estimated from those of $H$ as long as the signal $S$ is strong enough. However, when the $Y$ entries are heteroscedastic or heavy-tailed, this standard approach may fail. Especially in this work, we consider a situation that can easily arise with heteroscedastic or heavy-tailed noise but is particularly difficult to address using the singular value approach, namely, when the noise $Y$ itself may create spiked singular values. It has been a recurring question how to distinguish the signal $S$ from the spikes in $Y$, as this seems impossible by examining the leading singular values of $H$. Inspired by the work \cite{CCF21}, we turn to study the eigenvalues of an asymmetrized model when two samples $H_1=S+Y_1$ and $H_2=S+Y_2$ are available. We show that by looking into the leading eigenvalues (in magnitude) of the asymmetrized model $H_1H_2^*$, one can easily detect $S$. We will primarily discuss the heteroscedastic case and then discuss the extension to the heavy-tailed case. As a byproduct, we also derive the fundamental result regarding the outlier of non-Hermitian random matrix in \cite{Tao} under the minimal 2nd moment condition. |
| title | Signal detection from spiked noise via asymmetrization |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2504.19450 |