Efficient block noise removal based on nonlinear manifolds
The problem of block noise removal is considered. It is assumed that the original image is on or close to a sub-space of admissible images in the form of a low dimensional nonlinear manifold. We propose to use a close variant of the total variation regularizer for measuring block noise. Based on this noise measure, we present an effective approach that reconstructs the original image in the presence of block noise. Our main computational task is the solution of a quadratic programming problem, for which we propose a very efficient interior point method. The effectiveness and efficiency of our approach is demonstrated by an example.
We study the problem of learning a kernel matrix from an apriori kernel and training data. An unconstrained convex optimization formulation is proposed, with an arbitrary convex smooth loss function on kernel entries and a LogDet divergence for regularization. Since the number of variables is of order O(n), standard Newton and quasi-Newton methods are too time-consuming. An operator form Hessian is used to develop an O(n) trust-region inexact Newton method, where the Newton direction is computed using several conjugate gradient steps on the Hessian operator equation. On the uspst dataset, our algorithm can handle 2 million optimization variables within one hour. Experiments are shown for both linear (Mahalanobis) metric learning and for kernel learning. The convergence rate, speed and performance of several loss functions and algorithms are discussed.