Uniform error bound for PCA matrix denoising
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866929476840980480 |
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| author | Tong, Xin T. Wang, Wanjie Wang, Yuguan |
| author_facet | Tong, Xin T. Wang, Wanjie Wang, Yuguan |
| contents | Principal component analysis (PCA) is a simple and popular tool for processing high-dimensional data. We investigate its effectiveness for matrix denoising.
We consider the clean data are generated from a low-dimensional subspace, but masked by independent high-dimensional sub-Gaussian noises with standard deviation $σ$. Under the low-rank assumption on the clean data with a mild spectral gap assumption, we prove that the distance between each pair of PCA-denoised data point and the clean data point is uniformly bounded by $O(σ\log n)$. To illustrate the spectral gap assumption, we show it can be satisfied when the clean data are independently generated with a non-degenerate covariance matrix. We then provide a general lower bound for the error of the denoised data matrix, which indicates PCA denoising gives a uniform error bound that is rate-optimal. Furthermore, we examine how the error bound impacts downstream applications such as clustering and manifold learning. Numerical results validate our theoretical findings and reveal the importance of the uniform error. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_12690 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Uniform error bound for PCA matrix denoising Tong, Xin T. Wang, Wanjie Wang, Yuguan Statistics Theory Methodology 62H25(primary), 62H30, 62R30 Principal component analysis (PCA) is a simple and popular tool for processing high-dimensional data. We investigate its effectiveness for matrix denoising. We consider the clean data are generated from a low-dimensional subspace, but masked by independent high-dimensional sub-Gaussian noises with standard deviation $σ$. Under the low-rank assumption on the clean data with a mild spectral gap assumption, we prove that the distance between each pair of PCA-denoised data point and the clean data point is uniformly bounded by $O(σ\log n)$. To illustrate the spectral gap assumption, we show it can be satisfied when the clean data are independently generated with a non-degenerate covariance matrix. We then provide a general lower bound for the error of the denoised data matrix, which indicates PCA denoising gives a uniform error bound that is rate-optimal. Furthermore, we examine how the error bound impacts downstream applications such as clustering and manifold learning. Numerical results validate our theoretical findings and reveal the importance of the uniform error. |
| title | Uniform error bound for PCA matrix denoising |
| topic | Statistics Theory Methodology 62H25(primary), 62H30, 62R30 |
| url | https://arxiv.org/abs/2306.12690 |