The Quadratic Optimization Bias Of Large Covariance Matrices
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917793941684224 |
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| author | Gurdogan, Hubeyb Shkolnik, Alex |
| author_facet | Gurdogan, Hubeyb Shkolnik, Alex |
| contents | We describe a puzzle involving the interactions between an optimization of a multivariate quadratic function and a "plug-in" estimator of a spiked covariance matrix. When the largest eigenvalues (i.e., the spikes) diverge with the dimension, the gap between the true and the out-of-sample optima typically also diverges. We show how to "fine-tune" the plug-in estimator in a precise way to avoid this outcome. Central to our description is a "quadratic optimization bias" function, the roots of which determine this fine-tuning property. We derive an estimator of this root from a finite number of observations of a high dimensional vector. This leads to a new covariance estimator designed specifically for applications involving quadratic optimization. Our theoretical results have further implications for improving low dimensional representations of data, and principal component analysis in particular. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_03053 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Quadratic Optimization Bias Of Large Covariance Matrices Gurdogan, Hubeyb Shkolnik, Alex Statistics Theory Applications We describe a puzzle involving the interactions between an optimization of a multivariate quadratic function and a "plug-in" estimator of a spiked covariance matrix. When the largest eigenvalues (i.e., the spikes) diverge with the dimension, the gap between the true and the out-of-sample optima typically also diverges. We show how to "fine-tune" the plug-in estimator in a precise way to avoid this outcome. Central to our description is a "quadratic optimization bias" function, the roots of which determine this fine-tuning property. We derive an estimator of this root from a finite number of observations of a high dimensional vector. This leads to a new covariance estimator designed specifically for applications involving quadratic optimization. Our theoretical results have further implications for improving low dimensional representations of data, and principal component analysis in particular. |
| title | The Quadratic Optimization Bias Of Large Covariance Matrices |
| topic | Statistics Theory Applications |
| url | https://arxiv.org/abs/2410.03053 |