TL;DR
This paper proposes a novel active manifold learning method based on a unified framework of manifold landmarking that combines geometric manifold landmarking methods with algebraic ones and demonstrates its superiority compared to existing methods.
A novel supervised feature selection method named PRISM is proposed, which leverages product manifold to theoretically disentangle class-specific features from the shared structure and provides a theoretical guarantee by deriving an asymptotic convergence proof under the product manifold setting, guaranteeing the relia...
Manifold-Based Clustering (MBC) is proposed, which returns an explicit bracket interval to quantify the underlying data uncertainty, and suggests that ambiguity in cluster number is often intrinsic, and should be quantified rather than resolved.
This work develops a bootstrap algorithm on Riemannian manifolds that is both computationally efficient and accurate for hypothesis testing and confidence region construction, and establishes high-order asymptotics under an appropriate coordinate representation induced by a second-order retraction.
This work introduces a Nested Inductive Bias framework that utilizes a two-stage diffeomorphic composition to formally pull back non-Euclidean target geometries onto the SPD manifold, and proposes the Rational Conformal Metric (RCM), designed to establish state-of-the-art geometric robustness against outliers by boundi...
A unifying NEPv framework is established to efficiently deal with optimization on the Stiefel manifold with the $(2,1)$-norm regularization and numerical experiments are presented to illustrate the use of the NEPv approach and to gain insights as to what a proper regularizing parameter should have in real-world applica...
SHOPCA is computationally tractable, parameter-efficient, and applicable to domains requiring fully unsupervised, geometry-aware dimensionality reduction, and surpasses UMAP on small-sample settings, where iterative neighborhood-based manifold estimation can degrade.
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