sparseGeoHOPCA: A Geometric Solution to Sparse Higher-Order PCA Without Covariance Estimation

Renjie Xu, Chong Wu, Zhuoheng Ran, Maolin Che, Yimin WEI, Hong Yan

Keywords: Sparse Higher-Order Principal Component Analysis, Tensor Optimization

Abstract:

This paper proposes sparseGeoHOPCA, a geometric framework for sparse higher-order principal component analysis (SHOPCA).
The method unfolds the input tensor along each mode and reformulates the resulting subproblems as binary linear programs, transforming the nonconvex sparse objective into a tractable geometric form.
This eliminates covariance estimation and iterative deflation, leading to improved efficiency and interpretability in high-dimensional and unbalanced settings.
Theoretical equivalence with the original SHOPCA formulation is established, and error bounds linked to PCA residuals are derived, providing data-dependent guarantees.
The algorithm has total complexity $O(\sum_{n=1}^{N} (k_n^3 + J_n k_n^2))$ per iteration, scaling linearly with tensor size.
Extensive experiments demonstrate accurate sparse support recovery, stable classification under 10× compression, high-quality ImageNet reconstruction, and semantic reduction, highlighting robustness and versatility.

Primary Area: optimization

Submission Number: 2695