  This paper presents a novel non-rigid point set registration method that is
inspired by unsupervised clustering analysis. Unlike previous approaches that
treat the source and target point sets as separate entities, we develop a
holistic framework where they are formulated as clustering centroids and
clustering members, separately. We then adopt Tikhonov regularization with an
$\ell_1$-induced Laplacian kernel instead of the commonly used Gaussian kernel
to ensure smooth and more robust displacement fields. Our formulation delivers
closed-form solutions, theoretical guarantees, independence from dimensions,
and the ability to handle large deformations. Subsequently, we introduce a
clustering-improved Nystr\"om method to effectively reduce the computational
complexity and storage of the Gram matrix to linear, while providing a rigorous
bound for the low-rank approximation. Our method achieves high accuracy results
across various scenarios and surpasses competitors by a significant margin,
particularly on shapes with substantial deformations. Additionally, we
demonstrate the versatility of our method in challenging tasks such as shape
transfer and medical registration.
