# Fitting Unknown Number of Hyperplanes with Manifold Optimization

arXiv: https://arxiv.org/abs/2605.28501 · HTML: https://mlsanity.com/paper/2605.28501/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 39 references that are in the map.
- First posted on arXiv 27 May 2026.

## Builds on (most cited first, 16 of 16)

- [PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space](https://mlsanity.com/paper/1706.02413/) (2017)
- [PointCNN: Convolution On X-Transformed Points](https://mlsanity.com/paper/1801.07791/) (2018)
- [The Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017)
- [Point Transformer V3: Simpler, Faster, Stronger](https://mlsanity.com/paper/2312.10035/) (2023)
- [Towards K-means-friendly Spaces: Simultaneous Deep Learning and Clustering](https://mlsanity.com/paper/1610.04794/) (2016)
- [Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation](https://mlsanity.com/paper/1603.03236/) (2016)
- [Deep Clustering: A Comprehensive Survey](https://mlsanity.com/paper/2210.04142/) (2022)
- [A Brief Introduction to Manifold Optimization](https://mlsanity.com/paper/1906.05450/) (2019)
- [Towards Explaining Distribution Shifts](https://mlsanity.com/paper/2210.10275/) (2022)
- [First-Order Methods for Nonsmooth Nonconvex Functional Constrained Optimization with or without Slater Points](https://mlsanity.com/paper/2212.00927/) (2022)
- [SupeRANSAC: One RANSAC to Rule Them All](https://mlsanity.com/paper/2506.04803/) (2025)
- [Geometry of Deep Generative Models for Disentangled Representations](https://mlsanity.com/paper/1902.06964/) (2019)
- [PARSAC: Accelerating Robust Multi-Model Fitting with Parallel Sample Consensus](https://mlsanity.com/paper/2401.14919/) (2024)
- [Dying Clusters Is All You Need -- Deep Clustering With an Unknown Number of Clusters](https://mlsanity.com/paper/2410.09491/) (2024)
- [Breaking the Reclustering Barrier in Centroid-based Deep Clustering](https://mlsanity.com/paper/2411.02275/) (2024)
- [On the multisource hyperplanes location problem to fitting set of points](https://mlsanity.com/paper/2002.01692/) (2020)

## Abstract (arXiv)

Fitting an unknown number of hyperplanes to data is a fundamental yet challenging problem in machine learning, characterized by its non-convexity, non-differentiability, and unknown model order. Existing approaches often struggle with local optima or lack geometric consistency. To address these limitations, we propose a novel framework based on Manifold Optimization. We reformulate the problem as an unsupervised learning task on the unit sphere manifold Sdim-1. This formulation effectively handles the non-convex constraints and linearizes the distance measurement, rendering the gradient descent tractable. We propose a Two-Stage Manifold Optimization algorithm. In Phase I, we employ a Riemannian Expectation-Maximization process with a heavy-tailed kernel to robustly estimate posterior probabilities, effectively resolving the ambiguities of point distribution between intersecting hyperplanes. In Phase II, upon convergence of the soft estimates, the probabilistic weights degenerate into hard matching, generating a precise local optimum that strictly satisfies the geometric definition. Furthermore, we introduce a projected density estimation strategy for initialization to facilitate global convergence by significantly reducing the feature description space and search complexity. Extensive experiments demonstrate that our method outperforms state-of-the-art baselines in both geometric accuracy and robustness.