An Iterative Geometric Approach to Optimizing Separating Hyperplanes

We study the fundamental classification problem of computing a separating hyperplane for a binary-labeled dataset of size $n$ with normalized $d$-dimensional features. Letting $Φ\in \mathbb{R}^{n \times d}$ denote the feature matrix and $γ$ the margin of the maximum-margin separating hyperplane, we present a randomized algorithm that solves this problem in $\tilde{O}(γ^{-2/3}\, \operatorname{nnz}(Φ) + γ^{-2(ω+1)/3})$-sequential running time (work), $\tilde{O}(γ^{-2/3})$-parallel (computational) depth, and accesses $Φ$ only through $\tilde{O}(γ^{-2/3})$-matrix-vector queries (matvecs). We also present a second, faster randomized algorithm with a $\tilde{O}(γ^{-2/3}\, \operatorname{nnz}(Φ) + γ^{-2})$-sequential running time that uses $\tilde{O}(γ^{-2/3})$-matvecs to $Φ$, but achieves only $\tilde{O}(γ^{-4/3})$-parallel depth. Both algorithms match the near-optimal deterministic matvec complexity recently established by Kornowski and Shamir [2025], Karmarkar et al. [2026] and achieve improved sequential runtime and parallel depth, albeit at the expense of using randomness.

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.

This paper proposes a novel algorithm for semisupervised learning. This algorithm learns graph cuts that maximize the margin with respect to the labels induced by the harmonic function solution. We motivate the approach, compare it to existing work, and prove a bound on its generalization error. The quality of our solutions is evaluated on a synthetic problem and three UCI ML repository datasets. In most cases, we outperform manifold regularization of support vector machines, which is a state-of-the-art approach to semi-supervised max-margin learning.