Jie Sun

Tengteng Yu, Xin-Wei Liu, Yu-Hong Dai et al.

Variable metric proximal gradient methods with different metric selections have been widely used in composite optimization. Combining the Barzilai-Borwein (BB) method with a diagonal selection strategy for the metric, the diagonal BB stepsize can keep low per-step computation cost as the scalar BB stepsize and better capture the local geometry of the problem. In this paper, we propose a variable metric mini-batch proximal stochastic recursive gradient algorithm VM-mSRGBB, which updates the metric using a new diagonal BB stepsize. The linear convergence of VM-mSRGBB is established for strongly convex, non-strongly convex and convex functions. Numerical experiments on standard data sets show that VM-mSRGBB is better than or comparable to some variance reduced stochastic gradient methods with best-tuned scalar stepsizes or BB stepsizes. Furthermore, the performance of VM-mSRGBB is superior to some advanced mini-batch proximal stochastic gradient methods.

Lu Tan, Ling Li, Wanquan Liu et al.

Euler's Elastica based unsupervised segmentation models have strong capability of completing the missing boundaries for existing objects in a clean image, but they are not working well for noisy images. This paper aims to establish a Euler's Elastica based approach that properly deals with random noises to improve the segmentation performance for noisy images. We solve the corresponding optimization problem via using the progressive hedging algorithm (PHA) with a step length suggested by the alternating direction method of multipliers (ADMM). Technically, all the simplified convex versions of the subproblems derived from the major framework of PHA can be obtained by using the curvature weighted approach and the convex relaxation method. Then an alternating optimization strategy is applied with the merits of using some powerful accelerating techniques including the fast Fourier transform (FFT) and generalized soft threshold formulas. Extensive experiments have been conducted on both synthetic and real images, which validated some significant gains of the proposed segmentation models and demonstrated the advantages of the developed algorithm.