带正交和非负约束的大规模矩阵恢复模型、算法及应用

Hyperspectral images have a high spectral resolution and can provide richer information on the Earth's surface. Their applications have covered all aspects of earth science. Because the mixed pixels widely exist in hyperspectral images, which have become the bottleneck problem that restricts the high-precision unmixing of hyperspectral images. Hyperspectral unmixing is an important mean of interpreting mixed pixels. Matrix recovery with orthogonal and nonnegative constraints is the core scientific problem. In big-data scenario, matrices to be completed may be large-scale and may not fit in RAM. Matrix recovery becomes challenging in this large-scale setting. In order to solve practical problems and adhere to the national big data strategy, this project aims at studying models and algorithms of large-scale matrix recovery with orthogonal and nonnegative constraints. New models and fast algorithms will be designed. This project will also try to solve linear hyperspectral unmixing. Based on matrix recovery, augmented Lagrangian algorithm and hierarchical alternating least squares algorithm will be designed for matrix recovery under orthogonal and nonnegative constraints. When the matrix is large-scale, random projection method is used to reduce dimensionality, two randomized algorithms will be proposed accordingly. We give a new model for linear hyperspectral unmixing, which can be solved by the proximal gradient method. Adaptive moments method is then used to improve the proximal gradient. When the number of given pixels is large, we use proximal stochastic variance reduction gradient algorithm to reduce computational time and complexity.

高光谱遥感影像具有很高的光谱分辨率，能够提供丰富的地球表面信息，广泛应用于地球科学的各个方面。但是混合像元是制约高光谱遥感影像高精度解译的瓶颈问题。高光谱解混技术是解译混合像元的重要手段，带正交和非负约束的矩阵恢复是背后的核心科学问题。在大数据时代，矩阵规模很大，甚至超出主储存器的空间，导致矩阵恢复面临巨大的挑战。为了解决实际问题并响应国家大数据战略，本项目对带正交和非负约束的大规模矩阵恢复的模型和算法进行研究，并求解线性高光谱解混问题。基于矩阵分解，为带正交和非负约束大规模矩阵恢复设计增广拉格朗日算法和分层最小二乘算法；当矩阵的规模很大时，用随机方法降维，提出随机投影算法；基于矩阵恢复，为线性高光谱解混设计新模型，设计近端梯度算法和自适应动量算法，给出收敛性分析；当高光谱图像的规模很大时，引入方差缩减的随机方法降维，提出随机梯度算法。

带正交和非负约束的矩阵恢复是许多实际问题的核心科学问题。在大数据时代，矩阵规模很大，甚至超出主储存器的空间，导致矩阵恢复面临巨大的挑战。本项目旨在开展带正交和非负约束的大规模矩阵恢复的模型、算法及应用研究，获得如下三个方面的结果：(1) 基于矩阵分解，为带正交和非负约束大规模矩阵恢复建立了增广拉格朗日算法和分层最小二乘算法；(2) 当矩阵的规模很大时，用随机方法降维，提出随机投影算法；（3）基于矩阵恢复，我们为线性高光谱解混设计出新模型，设计了近端梯度算法和自适应动量算法，给出了收敛性分析；当高光谱图像的规模很大时，引入方差缩减的随机方法降维，提出了随机梯度算法。这些结果不仅能为求解带正交和非负约束的大规模矩阵恢复问题提供新模型和新算法，而且也可为最优化、信息科学、数据科学、计算机科学技术的交叉融合提供新元素，具有重要的科学意义和实用价值。