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Estimation of Ridge Using Nonlinear Transformation on Density Function
Jun 09, 2023
Ridges play a vital role in accurately approximating the underlying structure of manifolds. In this paper, we explore the ridge's variation by applying a concave nonlinear transformation to the density function. Through the derivation of the Hessian matrix, we observe that nonlinear transformations yield a rank-one modification of the Hessian matrix. Leveraging the variational properties of eigenvalue problems, we establish a partial order inclusion relationship among the corresponding ridges. We intuitively discover that the transformation can lead to improved estimation of the tangent space via rank-one modification of the Hessian matrix. To validate our theories, we conduct extensive numerical experiments on synthetic and real-world datasets that demonstrate the superiority of the ridges obtained from our transformed approach in approximating the underlying truth manifold compared to other manifold fitting algorithms.
Bounded Projection Matrix Approximation with Applications to Community Detection
May 21, 2023
Community detection is an important problem in unsupervised learning. This paper proposes to solve a projection matrix approximation problem with an additional entrywise bounded constraint. Algorithmically, we introduce a new differentiable convex penalty and derive an alternating direction method of multipliers (ADMM) algorithm. Theoretically, we establish the convergence properties of the proposed algorithm. Numerical experiments demonstrate the superiority of our algorithm over its competitors, such as the semi-definite relaxation method and spectral clustering.
Quadratic Matrix Factorization with Applications to Manifold Learning
Jan 30, 2023
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.