Fitting Unknown Number of Hyperplanes with Manifold Optimization

2026-05-27

AI Summary

This study addresses the challenging problem of fitting an unknown number of hyperplanes to data, which involves non-convexity, non-differentiability, and uncertainty in model order. To tackle these difficulties, the authors propose a two-stage unsupervised learning approach grounded in a unit-sphere manifold framework. In the first stage, they integrate Riemannian expectation-maximization with heavy-tailed kernel density estimation to robustly infer posterior probabilities. The second stage employs hard assignment annealing to obtain geometrically consistent local optima. Key innovations include a manifold optimization framework for handling non-convex constraints, a projection-based density estimation scheme for initialization, and the two-stage optimization strategy itself. Experimental results demonstrate that the proposed method significantly outperforms state-of-the-art baselines in both geometric accuracy and robustness.

Abstract

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 $\mathcal{S}^{\textbf{dim}-1}$. 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.

Problem

• hyperplane fitting
• unknown number of models
• non-convex optimization
• model order selection
• geometric consistency

Innovation

• Manifold Optimization
• Hyperplane Fitting
• Riemannian Expectation-Maximization
• Unknown Model Order
• Geometric Consistency