# Hyperplane Clustering Via Dual Principal Component Pursuit

arXiv: https://arxiv.org/abs/1706.01604 · HTML: https://mlsanity.com/paper/1706.01604/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Jun 2017.

## Built on by (most cited first, 6 of 6)

- [Image Clustering via the Principle of Rate Reduction in the Age of Pretrained Models](https://mlsanity.com/paper/2306.05272/) (2023)
- [Unlabeled Principal Component Analysis and Matrix Completion](https://mlsanity.com/paper/2101.09446/) (2021)
- [Imbalanced Mixed Linear Regression](https://mlsanity.com/paper/2301.12559/) (2023)
- [Boosting RANSAC via Dual Principal Component Pursuit](https://mlsanity.com/paper/2110.02918/) (2021)
- [Provable Clustering of a Union of Linear Manifolds Using Optimal Directions](https://mlsanity.com/paper/2201.02745/) (2022)
- [Robust Fuzzy local k-plane clustering with mixture distance of hinge loss and L1 norm](https://mlsanity.com/paper/2604.22405/) (2026)

## Abstract (arXiv)

We extend the theoretical analysis of a recently proposed single subspace learning algorithm, called Dual Principal Component Pursuit (DPCP), to the case where the data are drawn from of a union of hyperplanes. To gain insight into the properties of the ℓ1 non-convex problem associated with DPCP, we develop a geometric analysis of a closely related continuous optimization problem. Then transferring this analysis to the discrete problem, our results state that as long as the hyperplanes are sufficiently separated, the dominant hyperplane is sufficiently dominant and the points are uniformly distributed inside the associated hyperplanes, then the non-convex DPCP problem has a unique global solution, equal to the normal vector of the dominant hyperplane. This suggests the correctness of a sequential hyperplane learning algorithm based on DPCP. A thorough experimental evaluation reveals that hyperplane learning schemes based on DPCP dramatically improve over the state-of-the-art methods for the case of synthetic data, while are competitive to the state-of-the-art in the case of 3D plane clustering for Kinect data.