Oja's Algorithm for Streaming Sparse PCA
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913729101168640 |
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| author | Kumar, Syamantak Sarkar, Purnamrita |
| author_facet | Kumar, Syamantak Sarkar, Purnamrita |
| contents | Oja's algorithm for Streaming Principal Component Analysis (PCA) for $n$ data-points in a $d$ dimensional space achieves the same sin-squared error $O(r_{\mathsf{eff}}/n)$ as the offline algorithm in $O(d)$ space and $O(nd)$ time and a single pass through the datapoints. Here $r_{\mathsf{eff}}$ is the effective rank (ratio of the trace and the principal eigenvalue of the population covariance matrix $Σ$). Under this computational budget, we consider the problem of sparse PCA, where the principal eigenvector of $Σ$ is $s$-sparse, and $r_{\mathsf{eff}}$ can be large. In this setting, to our knowledge, \textit{there are no known single-pass algorithms} that achieve the minimax error bound in $O(d)$ space and $O(nd)$ time without either requiring strong initialization conditions or assuming further structure (e.g., spiked) of the covariance matrix. We show that a simple single-pass procedure that thresholds the output of Oja's algorithm (the Oja vector) can achieve the minimax error bound under some regularity conditions in $O(d)$ space and $O(nd)$ time. We present a nontrivial and novel analysis of the entries of the unnormalized Oja vector, which involves the projection of a product of independent random matrices on a random initial vector. This is completely different from previous analyses of Oja's algorithm and matrix products, which have been done when the $r_{\mathsf{eff}}$ is bounded. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_07240 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Oja's Algorithm for Streaming Sparse PCA Kumar, Syamantak Sarkar, Purnamrita Statistics Theory Machine Learning Oja's algorithm for Streaming Principal Component Analysis (PCA) for $n$ data-points in a $d$ dimensional space achieves the same sin-squared error $O(r_{\mathsf{eff}}/n)$ as the offline algorithm in $O(d)$ space and $O(nd)$ time and a single pass through the datapoints. Here $r_{\mathsf{eff}}$ is the effective rank (ratio of the trace and the principal eigenvalue of the population covariance matrix $Σ$). Under this computational budget, we consider the problem of sparse PCA, where the principal eigenvector of $Σ$ is $s$-sparse, and $r_{\mathsf{eff}}$ can be large. In this setting, to our knowledge, \textit{there are no known single-pass algorithms} that achieve the minimax error bound in $O(d)$ space and $O(nd)$ time without either requiring strong initialization conditions or assuming further structure (e.g., spiked) of the covariance matrix. We show that a simple single-pass procedure that thresholds the output of Oja's algorithm (the Oja vector) can achieve the minimax error bound under some regularity conditions in $O(d)$ space and $O(nd)$ time. We present a nontrivial and novel analysis of the entries of the unnormalized Oja vector, which involves the projection of a product of independent random matrices on a random initial vector. This is completely different from previous analyses of Oja's algorithm and matrix products, which have been done when the $r_{\mathsf{eff}}$ is bounded. |
| title | Oja's Algorithm for Streaming Sparse PCA |
| topic | Statistics Theory Machine Learning |
| url | https://arxiv.org/abs/2402.07240 |