Quadratic Matrix Factorization with Applications to Manifold Learning
This work addresses manifold learning for data analysis, offering a novel method to capture curved structures, but it appears incremental as it builds on existing matrix factorization techniques.
The paper tackled the problem of learning curved manifolds in data by proposing a quadratic matrix factorization (QMF) framework, which outperformed competitors on synthetic and real datasets like MNIST and cryogenic electron microscopy.
Matrix factorization is a popular framework for modeling low-rank data matrices. Motivated by manifold learning problems, this paper proposes a quadratic matrix factorization (QMF) framework to learn the curved manifold on which the dataset lies. Unlike local linear methods such as the local principal component analysis, QMF can better exploit the curved structure of the underlying manifold. Algorithmically, we propose an alternating minimization algorithm to optimize QMF and establish its theoretical convergence properties. Moreover, to avoid possible over-fitting, we then propose a regularized QMF algorithm and discuss how to tune its regularization parameter. Finally, we elaborate how to apply the regularized QMF to manifold learning problems. Experiments on a synthetic manifold learning dataset and two real datasets, including the MNIST handwritten dataset and a cryogenic electron microscopy dataset, demonstrate the superiority of the proposed method over its competitors.