TY - JOUR

T1 - Nonnegative Blind Source Separation for Ill-Conditioned Mixtures via John Ellipsoid

AU - Lin, Chia Hsiang

AU - Bioucas-Dias, Jose M.

N1 - Funding Information:
Manuscript received May 14, 2019; revised January 8, 2020; accepted June 11, 2020. Date of publication July 1, 2020; date of current version May 3, 2021. This work was supported in part by the Fundação para a Ciência e a Tecnologia (FCT)-Portugal, Ministry of Education and Science (MEC) through national funds, in part by FEDER–PT2020 Partnership Agreement under Project UID/EEA/50008/2019, in part by the Young Scholar Fellowship Program (Einstein Program) of the Ministry of Science and Technology (MOST), Taiwan, under Grant MOST108-2636-E-006-012, and in part by the Higher Education Sprout Project of Ministry of Education (MOE) to the Headquarters of University Advancement at National Cheng Kung University (NCKU). (Corresponding author: Chia-Hsiang Lin.) Chia-Hsiang Lin is with the Department of Electrical Engineering, National Cheng Kung University, Tainan 70101, Taiwan (e-mail: chiahsiang.steven.lin@gmail.com).
Publisher Copyright:
© 2012 IEEE.

PY - 2021/5

Y1 - 2021/5

N2 - 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.

AB - 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.

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U2 - 10.1109/TNNLS.2020.3002618

DO - 10.1109/TNNLS.2020.3002618

M3 - Article

C2 - 32609616

AN - SCOPUS:85105599499

VL - 32

SP - 2209

EP - 2223

JO - IEEE Transactions on Neural Networks and Learning Systems

JF - IEEE Transactions on Neural Networks and Learning Systems

SN - 2162-237X

IS - 5

M1 - 9130919

ER -