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

arXiv: https://arxiv.org/abs/2506.08670 · HTML: https://mlsanity.com/paper/2506.08670/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 26 references that are in the map.
- First posted on arXiv 10 Jun 2025.

## Builds on (most cited first, 10 of 10)

- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [TensorLy: Tensor Learning in Python](https://mlsanity.com/paper/1610.09555/) (2016)
- [Tensor Methods and Recommender Systems](https://mlsanity.com/paper/1603.06038/) (2016)
- [Linked Component Analysis from Matrices to High Order Tensors: Applications to Biomedical Data](https://mlsanity.com/paper/1508.07416/) (2015)
- [The Complexity of Sparse Tensor PCA](https://mlsanity.com/paper/2106.06308/) (2021)
- [Sparse PCA: a Geometric Approach](https://mlsanity.com/paper/2210.06590/) (2022)
- [A Tensor SVD-based Classification Algorithm Applied to fMRI Data](https://mlsanity.com/paper/2111.00587/) (2021)
- [On the Worst-Case Approximability of Sparse PCA](https://mlsanity.com/paper/1507.05950/) (2015)
- [Efficient Sparse PCA via Block-Diagonalization](https://mlsanity.com/paper/2410.14092/) (2024)
- [Image classification using local tensor singular value decompositions](https://mlsanity.com/paper/1706.09693/) (2017)

## Abstract (arXiv)

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 Oleft(sumn=1N (kn3 + Jn kn2)right), which scales linearly with tensor size. Extensive experiments demonstrate that sparseGeoHOPCA accurately recovers sparse supports in synthetic settings, preserves classification performance under 10× compression, and achieves high-quality image reconstruction on ImageNet, highlighting its robustness and versatility.