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

TL;DR

SparseGeoHOPCA introduces a geometric approach to sparse higher-order PCA that improves computational efficiency and interpretability without covariance estimation, suitable for high-dimensional tensor data.

Contribution

It presents a novel geometric framework for SHOPCA that reformulates the problem as structured binary linear optimization, eliminating covariance estimation and deflation.

Findings

Accurately recovers sparse supports in synthetic data

Maintains classification performance under 10× compression

Achieves high-quality image reconstruction on ImageNet

Abstract

We propose sparseGeoHOPCA, a novel framework for sparse higher-order principal component analysis (SHOPCA) that introduces a geometric perspective to high-dimensional tensor decomposition. By unfolding the input tensor along each mode and reformulating the resulting subproblems as structured binary linear optimization problems, our method transforms the original nonconvex sparse objective into a tractable geometric form. This eliminates the need for explicit covariance estimation and iterative deflation, enabling significant gains in both computational efficiency and interpretability, particularly in high-dimensional and unbalanced data scenarios. We theoretically establish the equivalence between the geometric subproblems and the original SHOPCA formulation, and derive worst-case approximation error bounds based on classical PCA residuals, providing data-dependent performance guarantees. The proposed algorithm achieves a total computational complexity of $O\left(\sum_{n=1}^{N} (k_n^3 + J_n k_n^2)\right)$, which scales linearly with tensor size. Extensive experiments demonstrate that sparseGeoHOPCA accurately recovers sparse supports in synthetic settings, preserves classification performance under 10$\times$ compression, and achieves high-quality image reconstruction on ImageNet, highlighting its robustness and versatility.