# Low-Rank Principal Eigenmatrix Analysis

## TL;DR

This paper introduces low-rank principal eigenmatrix analysis, a novel approach that leverages low-rank structures in eigenvectors for high-dimensional data, offering an alternative to sparse PCA with efficient algorithms.

## Contribution

It proposes a new low-rank eigenmatrix analysis method, including a rank-truncated power algorithm, with theoretical guarantees and practical efficiency.

## Key findings

- The method performs competitively on synthetic datasets.
- The proposed algorithm is computationally efficient.
- It effectively captures low-rank structures in eigenvectors.

## Abstract

Sparse PCA is a widely used technique for high-dimensional data analysis. In this paper, we propose a new method called low-rank principal eigenmatrix analysis. Different from sparse PCA, the dominant eigenvectors are allowed to be dense but are assumed to have a low-rank structure when matricized appropriately. Such a structure arises naturally in several practical cases: Indeed the top eigenvector of a circulant matrix, when matricized appropriately is a rank-1 matrix. We propose a matricized rank-truncated power method that could be efficiently implemented and establish its computational and statistical properties. Extensive experiments on several synthetic data sets demonstrate the competitive empirical performance of our method.