报告摘要：

This talk considers the use of spherical designs and non-convex optimization for recovery of sparse signals on the unit sphere. The available information consists of low order, potentially noisy, Fourier coefficients for the unit sphere. As Fourier coefficients are integrals of the product of a function and spherical harmonics, a good cubature rule is essential for the recovery.   A spherical t-design is a set of points on the unit sphere, which are nodes of an equal weight cubature rule integrating exactly all spherical polynomials of degree ≤ t. We will show that a spherical t-design provides a sharp error bound for the approximation signals. Moreover, the resulting coefficient matrix has orthonormal rows. In general the L_1 minimization model for recovery of sparse signals on the unit sphere using spherical harmonics has infinitely many minimizers, which means that most existing sufficient conditions for sparse recovery do not hold.  To induce the sparsity, we replace the L_1-norm by the L_q-norm (0<q<1) in the basis pursuit denoise model.  Recovery properties and optimality conditions are discussed. Moreover, we show that  the penalty method with a starting point obtained from the re-weighted L_1 method is promising  to solve the L_q basis pursuit denoise optimization model.  Numerical performance on nodes using spherical t-designs is compared tensor product nodes. We also compare the basis pursuit denoise problem with the L_1-norm and the L_q-norm (0<q<1).

    主讲人简介：

  陈小君教授分别于西安交通大学(1987年)和日本冈山理科大学(1991年)获得理学博士学位。目前陈小君教授是香港理工大学数学系的讲座教授，应用数学系主任。她的主要研究领域包括随机优化、随机变分不等式问题及其应用，非光滑稀疏最优化算法，大规模非光滑方程组的快速算法、大数据优化等。她主持的项目包括香港RGC项目10项，日本JSPS项目5项，澳大利亚ARC项目3项等。陈小君教授在优化领域顶级期刊Mathematical Programming，SIAM Journal on Optimization，SIAM Journal on Numerical Analysis等发表学术论文八十余篇，也是多个国际知名期刊的编委和编辑。她还曾在数学优化领域的顶级会议国际数学规划大会(2012年)上作大会特邀报告。陈小君教授从2015年起担任国际数学规划大会的学术委员会成员，也是迄今为止中国唯一担当此任的科学家。

此讲座为前沿课题讲座，欢迎全校师生踊跃参加。