无需协方差估计，用几何方法解决高阶主成分分析的稀疏化问题。
sparseGeoHOPCA: A Geometric Solution to Sparse Higher-Order PCA Without Covariance Estimation
- 将张量展开后转化为结构化二元线性优化问题，避免迭代降维。
- 计算复杂度线性于张量规模，10倍压缩下仍保持分类性能。
- 适合高维稀疏数据，兼具高效与可解释性，适用于图像重建等任务。
我们提出 sparseGeoHOPCA，一种基于几何视角的稀疏高阶主成分分析框架。通过沿各模式展开输入张量，并将子问题重构成结构化二元线性优化问题，该方法将原始非凸稀疏目标转化为可处理的几何形式，无需显式协方差估计与迭代去偏。这在高维和不平衡数据场景中显著提升计算效率与可解释性。我们理论证明了几何子问题与原SHOPCA公式的等价性，并基于经典PCA残差推导出最坏情况近似误差界，提供数据依赖的性能保证。算法总复杂度为 $O\left(\sum_{n=1}^{N} (k_n^3 + J_n k_n^2)\right)$，随张量大小线性增长。大量实验表明，sparseGeoHOPCA在合成数据中准确恢复稀疏支持，在10×压缩下保持分类性能，并在ImageNet上实现高质量图像重建，展现其鲁棒性与通用性。
原文摘要 · Abstract (English)
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.
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