Nonconvex And Nonsmooth Optimization Problems In Image Processing And Machine Learning:Algorithms And Applications

In the research field of information science,solving a problem in an energy formulation consists of two steps:modeling the energy and solving it for a minimum.Unfortunately,one can not have both of these two steps in the sense that nonconvex and nonsmooth models have more ability on depicting problems,but at the same time,are more difficult to optimize.Thus proposing nonconvex and nonsmooth models for specific problems and designing general,effi-cient and converged algorithms have become hot spots in current research.This work deals with both algorithmic and practical aspects of nonconvex optimization problems in image processing and machine learning.The main works can be summarized as follows:(1)LADMP for Linear Equality-constrained Optimization Problem.We propose LADMP based on ADM for a general nonconvex and nonsmooth optimization.By intro-ducing an auxiliary variable and penalize its bringing constraint to the objective function,we prove that any limit point of our proposed algorithm is a KKT point of the primal prob-lem.In addition,our algorithm is a linearized method that avoids the difficulties of solving subproblems.Experiments on signal representation and image denoising have shown the effectiveness of our proposed algorithm.(2)Learn an Algorithm with Bregman Distance for Unconstrained Optimization Prob-lems.We develop an AMBM algorithm for solving unconstrained nonconvex optimization problems.AMBM flexibly solves each subproblem with a designated Bregman distance,and is proved to have a superior convergence result for general nonconvex and nonsmooth optimization.Moreover,we design a learning-based LAMBM,whose step sizes are adap-tively learnt from training data to force the objective value drop rapidly toward the mini-mum.(3)An Optimization Framework with Flexible Inexact Inner Iterations.For uncon-strained nonconvex and nonsmooth optimization problems,we propose an IPAD algorith-m which inexactly solves each subproblem.Under some mild conditions,any numerical algorithms can be incorporated into IPAD and a superior convergence result is guaranteed.Moreover,a hybrid form of IPAD(HIPAD)is proposed by taking the varieties of subprob-lems into consideration.Numerical evaluations on both synthetic data and real images demonstrated promising experimental results of the proposed algorithms.(4)A Nonlocal L0 Model for Saliency Detection and Extension.By observing the intrinsic sparsity of saliency map,we propose a graph-based nonlocal L0 model(NLL0)for image saliency detection.The nonlocal graph used in NLL0 is demonstrated to be much better than local graphs.We propose two types of guiding maps,one is from perception and the other is learnt from training image.Moreover,NLL0 with learning-based strategy can also be used for interactive segmentation task.Extensive experiments have verified the efficiency of our proposed nonconvex model.