学术报告： Spherical designs and non-convex minimization for recovery of sparse signals on the sphere
报告人：Chair Professor Xiaojun Chen， Dean of Department of Applied Mathematics, The Hong Kong Polytechnic University
摘要：This talk considers the use of spherical designs and non-convex minimization for recovery of sparse signals on the unit sphere S^2. The available information consists of low order, potentially noisy, Fourier coefficients for S^2. 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 S^2, 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 S^2 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 model. Numerical performance on nodes using spherical t-designs and t_ε-designs (extremal fundamental systems) are compared with tensor product nodes. We also compare the basis pursuit denoise problem with q=1 and 0<q<1.