Sparse spectral estimation with missing and corrupted measurements

Elsener, Andreas; Geer, Sara van de

doi:10.48550/arxiv.1811.10443

Cited by 2 publications

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“…The inadequacy of the complete-case approach in many applications has motivated numerous methodological developments in the field of missing data over the past 60 years or so, including imputation (Ford, 1983;Rubin, 2004), factored likelihood (Anderson, 1957) and maximum likelihood approaches (Dempster et al, 1977); see, for example, Little & Rubin (2019) for an introduction to the area. Recent years have also witnessed increasing emphasis on understanding the performance of methods for dealing with missing data in a variety of high-dimensional problems, including sparse regression (Belloni et al, 2017;Loh & Wainwright, 2012), classification , sparse principal component analysis (Elsener & van de Geer, 2018) and covariance and precision matrix estimation (Loh & Tan, 2018;Lounici, 2014).…”

Section: Introductionmentioning

confidence: 99%

High-Dimensional Principal Component Analysis with Heterogeneous Missingness

Zhu

Wang

Samworth

2022

Journal of the Royal Statistical Society Series B: Statistical Methodology

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We study the problem of high‐dimensional Principal Component Analysis (PCA) with missing observations. In a simple, homogeneous observation model, we show that an existing observed‐proportion weighted (OPW) estimator of the leading principal components can (nearly) attain the minimax optimal rate of convergence, which exhibits an interesting phase transition. However, deeper investigation reveals that, particularly in more realistic settings where the observation probabilities are heterogeneous, the empirical performance of the OPW estimator can be unsatisfactory; moreover, in the noiseless case, it fails to provide exact recovery of the principal components. Our main contribution, then, is to introduce a new method, which we call primePCA, that is designed to cope with situations where observations may be missing in a heterogeneous manner. Starting from the OPW estimator, primePCA iteratively projects the observed entries of the data matrix onto the column space of our current estimate to impute the missing entries, and then updates our estimate by computing the leading right singular space of the imputed data matrix. We prove that the error of primePCA converges to zero at a geometric rate in the noiseless case, and when the signal strength is not too small. An important feature of our theoretical guarantees is that they depend on average, as opposed to worst‐case, properties of the missingness mechanism. Our numerical studies on both simulated and real data reveal that primePCA exhibits very encouraging performance across a wide range of scenarios, including settings where the data are not Missing Completely At Random.

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Section: Introductionmentioning

confidence: 99%

High-Dimensional Principal Component Analysis with Heterogeneous Missingness

Zhu

Wang

Samworth

2022

Journal of the Royal Statistical Society Series B: Statistical Methodology

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“…The inadequacy of the complete-case approach in many applications has motivated numerous methodological developments in the field of missing data over the past 60 years or so, including imputation (Ford, 1983;Rubin, 2004), factored likelihood (Anderson, 1957) and Expectation-Maximisation approaches (Dempster, Laird and Rubin, 1977); see, e.g., Little and Rubin (2014) for an introduction to the area. Recent years have also witnessed increasing emphasis on understanding the performance of methods for dealing with missing data in a variety of high-dimensional problems, including sparse regression (Loh and Wainwright, 2012;Belloni, Rosenbaum and Tsybakov, 2017), classification (Cai and Zhang, 2018b), sparse principal component analysis (Elsener and van de Geer, 2018) and covariance and precision matrix estimation (Lounici, 2014;Loh and Tan, 2018).…”

Section: Introductionmentioning

confidence: 99%

High-dimensional principal component analysis with heterogeneous missingness

Zhu¹,

Wang²,

Samworth³

2019

Preprint

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We study the problem of high-dimensional Principal Component Analysis (PCA) with missing observations. In simple, homogeneous missingness settings with a noise level of constant order, we show that an existing inverse-probability weighted (IPW) estimator of the leading principal components can (nearly) attain the minimax optimal rate of convergence. However, deeper investigation reveals both that, particularly in more realistic settings where the missingness mechanism is heterogeneous, the empirical performance of the IPW estimator can be unsatisfactory, and moreover that, in the noiseless case, it fails to provide exact recovery of the principal components. Our main contribution, then, is to introduce a new method for high-dimensional PCA, called primePCA, that is designed to cope with situations where observations may be missing in a heterogeneous manner. Starting from the IPW estimator, primePCA iteratively projects the observed entries of the data matrix onto the column space of our current estimate to impute the missing entries, and then updates our estimate by computing the leading right singular space of the imputed data matrix. It turns out that the interaction between the heterogeneity of missingness and the low-dimensional structure is crucial in determining the feasibility of the problem. We therefore introduce an incoherence condition on the principal components and prove that in the noiseless case, the error of primePCA converges to zero at a geometric rate when the signal strength is not too small. An important feature of our theoretical guarantees is that they depend on average, as opposed to worst-case, properties of the missingness mechanism. Our numerical studies on both simulated and real data reveal that primePCA exhibits very encouraging performance across a wide range of scenarios.

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Sparse spectral estimation with missing and corrupted measurements

Cited by 2 publications

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High-Dimensional Principal Component Analysis with Heterogeneous Missingness

High-Dimensional Principal Component Analysis with Heterogeneous Missingness

High-dimensional principal component analysis with heterogeneous missingness

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