论文标题
小组稀疏优化用于球体上的随机字段
Group sparse optimization for inpainting of random fields on the sphere
论文作者
论文摘要
我们提出了一个稀疏优化模型，用于在单位球体上为平方累积的各向同性随机场注入，该模型由带有随机复杂系数的球形谐波表示该场。在提出的优化模型中，该变量是无限维复合体矢量，目标函数是由$ \ ell_2 $ norm和non-liptchitz $ \ ell_p（0 <p <1）$ norm的杂种定义的实值函数，可保留旋转不变性属性和随机复合物系数的旋转不可不良性和组结构。 我们表明，无限维度优化问题等效于一个有限约束的优化问题。此外，我们提出了一种平滑的惩罚算法，以通过不受约束的优化问题解决有限维问题。我们提供了由概率度量以概率度量在球体上的正方形综合空间中约束优化问题的缩放KKT点定义的成分随机场的近似误差。最后，我们对球体的带限随机场进行数值实验和地球地形数据的图像，以显示平滑惩罚算法的有希望的性能，以填充球体上随机场。
We propose a group sparse optimization model for inpainting of a square-integrable isotropic random field on the unit sphere, where the field is represented by spherical harmonics with random complex coefficients. In the proposed optimization model, the variable is an infinite-dimensional complex vector and the objective function is a real-valued function defined by a hybrid of the $\ell_2$ norm and non-Liptchitz $\ell_p (0<p<1)$ norm that preserves rotational invariance property and group structure of the random complex coefficients. We show that the infinite-dimensional optimization problem is equivalent to a convexly-constrained finite-dimensional optimization problem. Moreover, we propose a smoothing penalty algorithm to solve the finite-dimensional problem via unconstrained optimization problems. We provide an approximation error bound of the inpainted random field defined by a scaled KKT point of the constrained optimization problem in the square-integrable space on the sphere with probability measure. Finally, we conduct numerical experiments on band-limited random fields on the sphere and images from Earth topography data to show the promising performance of the smoothing penalty algorithm for inpainting of random fields on the sphere.
