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% \title{Beyond Sin-Squared Error: Linear-Time Entrywise Uncertainty Quantification for Streaming PCA}

% \author{Syamantak Kumar\thanks{University of Texas at Austin, \texttt{syamantak@utexas.edu}} 
% \and Shourya Pandey \thanks{University of Texas at Austin, \texttt{shouryap@utexas.edu}}
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\title{Beyond Sin-Squared Error: Linear Time Entrywise Uncertainty Quantification for Streaming PCA}
\date{}

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\author[1]{Syamantak Kumar}
\author[1]{Shourya Pandey}
\author[2]{Purnamrita Sarkar}
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    Department of Computer Science\\
    University of Texas at Austin
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    Department of Statistics and Data Sciences\\
    University of Texas at Austin
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\begin{abstract}
    We propose a novel statistical inference framework for streaming principal component analysis (PCA) using Oja’s algorithm, enabling the construction of confidence intervals for individual entries of the estimated eigenvector. Most existing works on streaming PCA focus on providing sharp sin-squared error guarantees. Recently, there has been some interest in uncertainty quantification for the sin-squared error. However, uncertainty quantification or sharp error guarantees for \textit{entries of the estimated eigenvector} in the streaming setting remains largely unexplored. We derive a sharp Bernstein-type concentration bound for elements of the estimated vector matching the optimal error rate up to logarithmic factors. We also establish a Central Limit Theorem for a suitably centered and scaled subset of the entries.  To efficiently estimate the coordinate-wise variance, we introduce a provably consistent subsampling algorithm that leverages the median-of-means approach, empirically achieving similar accuracy to multiplier bootstrap methods while being significantly more computationally efficient. Numerical experiments demonstrate its effectiveness in providing reliable uncertainty estimates with a fraction of the computational cost of existing methods.
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\section*{Acknowledgments}
We gratefully acknowledge NSF grants 2217069, 2019844, and DMS 2109155. We are thankful to Soumendu Sundar Mukherjee and Arun Kuchibhotla for helpful discussions.

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