A unified framework for manifold landmarking

TL;DR

This paper introduces a unified active manifold landmarking framework that combines geometric and algebraic methods to select representative samples, improving semi-supervised manifold learning performance.

Contribution

It proposes a novel landmarking approach using Gershgorin circle theorem to optimize landmark selection by minimizing an error bound, unifying geometric and algebraic criteria.

Findings

Outperforms existing landmarking methods in experiments

Demonstrates robustness, scalability, and efficiency through simulations

Improves regression and classification results with the proposed method

Abstract

The success of semi-supervised manifold learning is highly dependent on the quality of the labeled samples. Active manifold learning aims to select and label representative landmarks on a manifold from a given set of samples to improve semi-supervised manifold learning. In this paper, we propose a novel active manifold learning method based on a unified framework of manifold landmarking. In particular, our method combines geometric manifold landmarking methods with algebraic ones. We achieve this by using the Gershgorin circle theorem to construct an upper bound on the learning error that depends on the landmarks and the manifold's alignment matrix in a way that captures both the geometric and algebraic criteria. We then attempt to select landmarks so as to minimize this bound by iteratively deleting the Gershgorin circles corresponding to the selected landmarks. We also analyze the complexity, scalability, and robustness of our method through simulations, and demonstrate its superiority compared to existing methods. Experiments in regression and classification further verify that our method performs better than its competitors.