A unified probabilistic framework for robust manifold learning and embedding

This paper focuses on learning a smooth skeleton structure from noisy data—an emerging topic in the fields of computer vision and computational biology. Many dimensionality reduction methods have been proposed, but none are specially designed for this purpose. To achieve this goal, we propose a unified probabilistic framework that directly models the posterior distribution of data points in an embedding space so as to suppress data noise and reveal the smooth skeleton structure. Within the proposed framework, a sparse positive similarity matrix is obtained by solving a box-constrained convex optimization problem, in which the sparsity of the matrix represents the learned neighborhood graph and the positive weights stand for the new similarity. Embedded data points are then obtained by applying the maximum a posteriori estimation to the posterior distribution expressed by the learned similarity matrix. The embedding process naturally provides a probabilistic interpretation of Laplacian eigenmap and maximum variance unfolding. Extensive experiments on various datasets demonstrate that our proposed method obtains the embedded points that accurately uncover inherent smooth skeleton structures in terms of data visualization, and the method yields superior clustering performance compared to various baselines.