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首页> 外文期刊>Neural Networks and Learning Systems, IEEE Transactions on >Nonnegative Blind Source Separation for Ill-Conditioned Mixtures via John Ellipsoid
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Nonnegative Blind Source Separation for Ill-Conditioned Mixtures via John Ellipsoid

机译：通过John EllipsoId的不良盲源分离。

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摘要

Nonnegative blind source separation (nBSS) is often a challenging inverse problem, namely, when the mixing system is ill-conditioned. In this work, we focus on an important nBSS instance, known as hyperspectral unmixing (HU) in remote sensing. HU is a matrix factorization problem aimed at factoring the so-called endmember matrix, holding the material hyperspectral signatures, and the abundance matrix, holding the material fractions at each image pixel. The hyperspectral signatures are usually highly correlated, leading to a fast decay of the singular values (and, hence, high condition number) of the endmember matrix, so HU often introduces an ill-conditioned nBSS scenario. We introduce a new theoretical framework to attack such tough scenarios via the John ellipsoid (JE) in functional analysis. The idea is to identify the maximum volume ellipsoid inscribed in the data convex hull, followed by affinely mapping such ellipsoid into a Euclidean ball. By applying the same affine mapping to the data mixtures, we prove that the endmember matrix associated with the mapped data has condition number 1, the lowest possible, and that these (preconditioned) endmembers form a regular simplex. Exploiting this regular structure, we design a novel nBSS criterion with a provable identifiability guarantee and devise an algorithm to realize the criterion. Moreover, for the first time, the optimization problem for computing JE is exactly solved for a large-scale instance; our solver employs a split augmented Lagrangian shrinkage algorithm with all proximal operators solved by closed-form solutions. The competitiveness of the proposed method is illustrated by numerical simulations and real data experiments.
机译：非负盲源分离（NBSS）经常是一个具有挑战性的逆问题，即，当混合系统是病态。在这项工作中，我们专注于一个重要的NBSS例如，称为遥感高光谱混合像元分解（HU）。 HU是一个矩阵因式分解问题旨在因式分解所谓端元矩阵，保持该材料的高光谱签名，以及丰度矩阵，保持该材料级分在每个图像像素。高光谱签名通常高度相关，导致端元矩阵的奇异值（以及，因此，高条件数）的快速衰减，因此常常HU介绍了一种病态NBSS场景。我们推出了新的理论框架，通过在功能分析的约翰椭球（JE）攻击这样的艰难情景。这样做是为了识别数据中的凸包内接椭圆体的最大体积，随后仿射映射这种椭球成欧几里德球。通过应用相同的仿射映射到数据混合物，我们证明了与映射的数据相关联的端元矩阵具有条件号1，最低可能的，并且这些（预处理）端元形成规则单纯形。利用这个规则的结构，我们设计一个可证明的可识别性担保一种新型NBSS标准和设计一个算法来实现的标准。此外，在第一时间，计算JE优化问题正好解决了大规模的实例;我们的求解器采用分体式增广拉格朗日收缩算法与封闭形式的解决方案，解决了所有运营商近。所提出的方法的竞争力是通过数值模拟和实际实验数据示出。

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