Some mathematical appeoximation approaches in data science

来源：数学科学学院发布时间：2018-09-19 1080

浙江大学数学科学学院九十周年院庆系列活动之七十四

时间: 2018年9月20日下午2点30至6点

地点：浙大玉泉校区教十一417

报告人：沈益教授，浙江理工大学数学科学系

报告题目：On the Convergence of Alternating Minimization with Applications to Blind Inpainting

报告摘要：In this talk, we consider the problem of vectors (such as digital signals or images) recovery with missing data or corrupted with impluse noise. Under the assumption that the locations of the corrupted entries are unknown, both the vectors and impulse noise are estimated by solving blind inpainting models which includes smooth date fitting terms, a regularization on the image and

a non convex regularization term on the noise. Models are solved by a proximal alternating minimization algorithm. Systematic theoretical analysis of the proximal alternating minimization algorithm including stability, local minimizers and convergent rate are studied. Simulation results of proposed algorithm on both compressed sensing and blind inpainting problems verify the theoretical analysis.

报告人：夏羽，杭州师范大学理学院数学系

报告题目：Identifiability of Multichannel Blind Deconvolution and Nonconvex Regularization Algorithm
报告摘要：―In this talk, we consider the multichannel blind deconvolution problem, where we observe the output of channels hi 2 Rn(i = 1; :::;N) that all convolve with the same unknown
input signal x 2 Rn. We wish to estimate the input signal and blur kernels simultaneously. Existing theoretical results showed that the original inputs are identifiable under subspace assumptions.However, the subspaces discussed before were randomly or generically chosen. Here we propose deterministic subspace assumption which is widely used in practice, and give some
theoretical results. First of all, we derive tight sufficient condition for identifiability of signal and convolution kernels, which is only violated on a set of Lebesgue measure zero. Then we present a non-convex regularization algorithm by lifting method and approximate the rank-one constraint via the difference of nuclear norm and Frobenius norm. The global minimizer of the proposed non-convex algorithm is rank-one matrix under mild conditions
on parameters and noise level. The stability result is also shown under the assumption that the inputs lie in a compact set. Besides, the computation of our regularization model is carried out and any limit point of iterations converges to a stationary point of our model. Finally, we provide numerical experiments to show that our non-convex regularization model outperforms convex relaxation models, such as nuclear norm minimization and some non-convex methods, such as alternating minimization method and spectral method.

联系人：李松老师（songli@zju.edu.cn）