多图正则多核非负矩阵分解高光谱图像解混

多图正则多核非负矩阵分解高光谱图像解混 Multi-graph regularized multi-kernel nonnegative matrix factorization for hyperspectral image unmixing 1 corresp first-author E-mail： zyhalj1975@163.com E-mail： zyhalj1975@163.com 刘 敬（1975-），女，安徽宿州人，博士，副教授，硕士生导师，2004年，2009年于西安电子科技大学分别获得硕士、博士学位，主要从事高光谱遥感影像光谱特征提取和解混方面的研究。E-mail：zyhalj1975@163.com http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994766&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994775&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994771&type= 22.01333237 32.00399780 刘 敬 （1975-），女，安徽宿州人，博士，副教授，硕士生导师，2004年，2009年于西安电子科技大学分别获得硕士、博士学位，主要从事高光谱遥感影像光谱特征提取和解混方面的研究。E-mail： zyhalj1975@163.com zyhalj1975@163.com 1 李康欣（1996-），女，陕西武功人，硕士研究生，2018年于西安邮电大学获得学士学位，目前主要从事高光谱遥感影像解混和光谱特征提取方面的研究。E-mail：Luminous_lkx@163.com http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994783&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994791&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994787&type= 22.01332855 32.00399780 李康欣 （1996-），女，陕西武功人，硕士研究生，2018年于西安邮电大学获得学士学位，目前主要从事高光谱遥感影像解混和光谱特征提取方面的研究。E-mail： Luminous_lkx@163.com Luminous_lkx@163.com 1 2 西安邮电大学 电子工程学院，陕西 西安 710121 西安邮电大学 电子工程学院，陕西 西安 710121 School of Electronic Engineering， Xi'an University of Posts and Telecommunications， Xi'an 710121， China School of Electronic Engineering， Xi'an University of Posts and Telecommunications， Xi'an 710121， China 西安电子科技大学 电子工程学院，陕西 西安 710071 西安电子科技大学 电子工程学院，陕西 西安 710071 School of Electronic Engineering， Xidian University， Xi'an 710071， China School of Electronic Engineering， Xidian University， Xi'an 710071， China 针对高光谱遥感图像的非线性解混问题，提出一种多图正则多核非负矩阵分解（MGMKNMF）算法，构造了多核空间中的多图正则项，并基于此构造了包含多核空间的多图正则项、多核权重正则项和多图权重正则项的MGMKNMF目标函数。MGMKNMF可在学习端元与丰度的过程中更新多核权重和多图权重，在合适的多核空间精确构造输入数据的图，解决了图权重和核权重的参数选择的问题。相比核非负矩阵分解（KNMF）的单一核，多核可确定更合适的核空间；相比图正则非负矩阵分解（GNMF）的单一图，多图更准确可靠。2个实测数据集和2个模拟数据集上的实验结果表明MGMKNMF算法是有效的。与GNMF、不含纯像元的核非负矩阵分解、核稀疏非负矩阵分解、基于核的字典剪枝非线性光谱解混、多图正则核非负矩阵分解算法相比，所提MGMKNMF算法在Cuprite和Jasper Ridge真实地物数据集上平均光谱角距离（SAD）值最优，分别为0.092 1和0.097 0；在HAPKE和广义双线性模型模拟数据集上平均SAD最优，分别是0.137 5和0.145 6，均方根误差值表现也最好，分别为0.050 6和0.057 0。 To solve the nonlinear unmixing problem of hyperspectral remote sensing images， a multi-graph regularized multi-kernel nonnegative matrix factorization （MGMKNMF） method is proposed， and the multi-graph regularization term in multi-kernel space is constructed. Moreover， the MGMKNMF objective function， which includes multi-graph in multi-kernel regularization， multi-kernel weights regularization， and multi-graph weights regularization terms， is constructed. MGMKNMF can update the multi-kernel and multi-graph weights during the process of learning endmembers and abundance， and precisely construct the graph of the input data in the appropriate multi-kernel space， thereby solving the problem of selecting the graph and kernel weights. Compared with the single kernel function used in kernel nonnegative matrix factorization （KNMF）， multiple kernel functions can determine a more appropriate kernel space. Further， compared with the single graph in graph regularized nonnegative matrix factorization （GNMF）， multiple graphs describe the relationship between samples in different ways， which is more accurate and reliable than the single graph. The experimental results with two real measured datasets and two simulated datasets show that the presented MGMKNMF algorithm is effective. Compared with GNMF， non-pure pixels kernel nonnegative matrix factorization kernel sparse non-negative matrix factorization， kernel-based nonlinear spectral unmixing with dictionary pruning methods， and multi-graph regularized kernel nonnegative matrix factorization method， the average spectral angle distance （SAD） values of the proposed MGMKNMF are the best， that is， 0.092 1 and 0.097 0 on the real Cuprite dataset and Jasper ridge dataset， respectively. The average SAD values of MGMKNMF on the Hapke and generalized bilinear model simulated datasets are 0.137 5 and 0.145 6， respectively. Finally， the root mean square error values are 0.050 6 and 0.057 0， respectively. 解混 多图正则多核非负矩阵分解 正则化 高光谱图像 unmixing multi-graph regularized multi-kernel nonnegative matrix factorization （MGMKNMF） regularization hyperspectral image 1 引 言 1 高光谱图像 ［ 1 1 ］ 有高的光谱分辨率，包含丰富的图像及光谱信息，而由于高光谱传感器的低空间分辨率和不同纯物质波谱的混合，导致混合像元 ［ 2 2 ］ 的产生，极大地影响了高光谱遥感图像的应用。为改善高光谱图像分解精度，高光谱解混 ［ 3 3 - 5 5 3-5 ］ 已成为热点，可用线性或非线性方式将混合像元分解，同时提取端元与丰度。端元是混合像元分解出的纯物质光谱，而丰度 ［ 6 6 ］ 则是每个像元中端元的贡献。 早期高光谱解混算法主要采用线性混合模型（Linear Mixture Model， LMM），如基于几何的顶点成分分析法（Vertex Component Analysis， VCA） ［ 7 7 ］ 、最小封闭体积的单纯形法（Minimum Volume Enclosing Simplex， MVES） ［ 8 8 ］ ，基于统计的贝叶斯方法、独立成分分析（Independent Component Analysis， ICA） ［ 9 9 ］ 和非负矩阵分解（Nonnegative Matrix Factorization， NMF） ［ 10 10 ］ 算法等。在NMF算法中，先找到一组非负基，然后将原始数据映射到这组基上，且数据在每个基上的表达非负。NMF十分适合应用于高光谱解混，得到端元和丰度矩阵。为保持数据空间固有的流形结构，Yang等人在NMF中加入了图正则算法，称为图正则非负矩阵分解 ［ 11 11 ］ ，该算法利用一种图相似性描述样本之间的关系，充分考虑了局部不变性。由于定义样本之间的关系及其权重矩阵的方法很多，单图的选择至关重要。为解决图选择的问题，Shu等人提出多图正则非负矩阵分解（Multi-graph Regularized Nonnegative Matrix Factorization， MGNMF） ［ 12 12 ］ ，采用一组已知的多个图拉普拉斯矩阵，通过学习得到的加权参数组合，去逼近原始数据。多图可更准确地刻画样本的相似性，进而更好地表达原始数据的结构。虽然多图解决了图正则非负矩阵分解（Graph Regularized Nonnegative Matrix Factorization， GNMF）算法中图选择困难的问题，但其属于线性解混算法 ［ 13 13 - 14 14 13-14 ］ ，难以适应真实场景中复杂的非线性光谱混合结构，所以，这一问题也促进了非线性解混算法研究。 NMF是一种典型的线性解混算法，不适合提取数据的非线性混合结构 ［ 15 15 ］ ，而核方法可以解决这个问题。核方法 ［ 16 16 ］ 是将非线性关系转变为线性关系的一种过程方法，将低维非线性混合结构的数据映射到高维核空间，在核空间中数据混合结构呈现线性，因此在核空间可实现高光谱图像的非线性解混。Yan等人提出的包含纯像元的核非负矩阵分解（Pure-pixels Kernel Nonnegative Matrix Factorization， pKNMF）与不含纯像元的核非负矩阵分解（Non-pure-pixel Kernel Nonnegtive Matrix Factorization， npKNMF），是将KNMF ［ 17 17 ］ 算法直接应用于高光谱数据，得到比NMF好的分类效果。但KNMF的性能很大程度上取决于核函数的选择。多核非负矩阵分解（Multi-kernel Nonnegtive Matrix Factorization， MKNMF）算法采用多个核函数的组合，并为每个核函数设置适当的权重参数。相比KNMF算法，多核NMF可自适应地选择核函数并加权，有更好的映射能力。Yao等人将MKNMF引入图正则NMF算法中，在高光谱数据集中得到了更好的验证 ［ 18 18 ］ 。 在许多复杂自然场景中存在大量的非线性混合现象，如沙地和矿物混合区的密集混合现象，以及植被和建筑物覆盖区的多级混合现象。基于线性混合模型的线性解混算法不适合于非线性混合情况，所以，本文提出了一种非线性解混算法——多图正则多核非负矩阵分解（Multi-graph Regularized Multi-kernel Nonnegative Matrix Factorization，MGMKNMF），先使用多核函数构造适合于高光谱数据的核空间，然后在多核空间的基础上为目标函数添加多图正则项。本文提出的算法有以下两个优点：（1）与KNMF算法相比，MGMKNMF算法采用核函数权重将多个不同参数的高斯核函数联合起来，并在学习过程中不断更新核函数权重，避免了单核的唯一性，使构造的核空间更合适，也解决了多个高斯核函数权重选择困难的问题。（2）与GNMF和MGNMF算法相比，MGMKNMF算法是非线性方法，在多核空间构造多图，图权重将多个图拉普拉斯矩阵线性组合，并与丰度矩阵最终构成多核空间的多图正则项，且在学习过程中不断更新图权重。相比原空间的单图和多图，多核空间中的多图可更精确地刻画原始数据的非线性流形结构，更适合对真实场景中复杂的非线性光谱混合结构进行非线性解混。本文采用两个真实高光谱数据集Jasper Ridge和Cuprite，并采用广义双线性模型（GBM） ［ 19 19 ］ 和Hapke ［ 20 20 ］ 非线性模型分别生成两个模拟数据集，将所提MGMKNMF算法与GNMF、npKNMF、核稀疏非负矩阵分解（Kernel Sparse Nonnegative Matrix Factorization， KSNMF）、基于核的字典剪枝非线性光谱解混（Kernel-based Nonlinear Spectral Unmixing with Dictionary Pruning， KDP）、多图正则核非负矩阵（Multi-graph Regularized Kernel Nonnegative Matrix Factorization， MGKNMF）算法比较，实验结果表明，MGMKNMF的光谱角距离（Spectral Angel Distance， SAD）和均方根误差（Root Mean Square Error， RMSE）相比其他算法均有较为显著的下降。 2 相关工作 1 2.1　非负矩阵分解NMF 2 NMF可用于高光谱遥感影像的无监督解混。给定一个数据矩阵 <math id="M1"><mi mathvariant="bold-italic">X</mi><mo>=</mo><mo stretchy="false">[</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>N</mi></mrow></msub><mo stretchy="false">]</mo><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mi>M</mi><mo>×</mo><mi>N</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994801&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994797&type= 37.42266846 4.65666676 ，其中 X 的每一列都是样本向量。NMF通过矩阵分解将原始高秩矩阵分解为两个低秩矩阵 <math id="M2"><mi mathvariant="bold-italic">A</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mi>M</mi><mo>×</mo><mi>K</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994812&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994806&type= 14.90133286 3.38666677 和 <math id="M3"><mi mathvariant="bold-italic">S</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mi>K</mi><mo>×</mo><mi>N</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994824&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994818&type= 14.13933277 3.38666677 的相乘，并加上非负的约束。 <math display="block" id="M4"><mi mathvariant="bold-italic">X</mi><mo>≈</mo><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994834&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994829&type= 12.61533356 2.79399991 . 非负矩阵分解的目标函数： <math display="block" id="M5"><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">A</mi><mo>,</mo><mi mathvariant="bold-italic">S</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">|</mo><mo stretchy="false">|</mo><mi mathvariant="bold-italic">X</mi><mo>-</mo><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">S</mi><mo stretchy="false">|</mo><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mtd></mtr><mtr><mtd><mi mathvariant="normal">s</mi><mo>.</mo><mi mathvariant="normal">t</mi><mo>.</mo><mi mathvariant="bold-italic">A</mi><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo><mtext> </mtext><mi mathvariant="bold-italic">S</mi><mo>≥</mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994845&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994839&type= 41.74066544 10.32933331 ， 其中： X 是原始数据矩阵，在高光谱数据中， A 和 S 分别代表端元矩阵与丰度矩阵。 2.2　图正则非负矩阵分解GNMF 2 NMF将非负矩阵 X 分解为基矩阵 A 和编码矩阵 S 的乘积 <math id="M6"><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo>≈</mo><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">S</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994852&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994848&type= 15.91733265 3.72533321 ，用于高光谱图像的无监督解混时，即将高光谱数据集 X 分解为端元矩阵 A 与丰度矩阵 S 的乘积。GNMF将图正则项添加到NMF的目标函数中，改善了未考虑数据流形结构所带来的问题。对有 N 个样本的数据集 X 构造一个 K 近邻图，图中顶点为 X 中各像素点。 N n 为 X 中样本 x n 的 K 近邻集，将每个顶点 x n 与属于它的 N n 连接，并定义图权重矩阵 <math id="M7"><mi mathvariant="bold-italic">W</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mi>N</mi><mo>×</mo><mi>N</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994859&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994856&type= 15.91733265 3.38666677 ，原空间中顶点 x n 和 x m 间的权重 <math id="M8"><msub><mrow><mi>W</mi></mrow><mrow><mi>n</mi><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994869&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994864&type= 6.51933336 3.72533321 越大，在子空间中的 s n 和 s m 距离也越近。通过权重矩阵 W 构造图正则项： <math display="block" id="M9"><mtable columnalign="left"><mtr><mtd><msup><mrow><mi>O</mi></mrow><mrow><mi mathvariant="normal">G</mi></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="bold-italic">S</mi><mo>,</mo><mi mathvariant="bold-italic">W</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>,</mo><mi>m</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>N</mi></mrow></munderover></mstyle><mrow><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><msub><mrow><mi>W</mi></mrow><mrow><mi>n</mi><mi>m</mi></mrow></msub><mo>=</mo></mtd></mtr><mtr><mtd><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">S</mi><mi mathvariant="bold-italic">D</mi><msup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">)</mo><mo>-</mo><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">S</mi><mi mathvariant="bold-italic">W</mi><msup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">S</mi><mi mathvariant="bold-italic">L</mi><msup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">)</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994881&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994874&type= 58.33533096 14.22399998 ， 其中： s n 和 s m 为像元 x n 和 x m 在端元基向量上的编码向量，即丰度； <math id="M10"><msub><mrow><mi>W</mi></mrow><mrow><mi>n</mi><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994890&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994885&type= 6.51933336 3.72533321 为 x n 和 x m 间的权重； S 为丰度矩阵； D 为对角矩阵， <math id="M11"><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>n</mi><mi>n</mi></mrow></msub><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>m</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>N</mi></mrow></msubsup><mrow><msub><mrow><mi>W</mi></mrow><mrow><mi>n</mi><mi>m</mi></mrow></msub></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994900&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994895&type= 26.24666595 4.74133301 ； <math id="M12"><mi mathvariant="bold-italic">L</mi><mo>=</mo><mi mathvariant="bold-italic">D</mi><mo>-</mo><mi mathvariant="bold-italic">W</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994911&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994905&type= 18.54199982 2.96333337 是图拉普拉斯矩阵；“tr（·）”表示矩阵的对角线元素之和。常用定义权重矩阵 W 的方法有：0-1加权、热核加权和点积加权等。 将 式（3） 式（3） 与（2）结合，得到GNMF的目标函数： <math display="block" id="M13"><mtable columnalign="left"><mtr><mtd><msup><mrow><mi>O</mi></mrow><mrow><mi mathvariant="normal">G</mi><mi mathvariant="normal">N</mi><mi mathvariant="normal">M</mi><mi mathvariant="normal">F</mi></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="bold-italic">A</mi><mo>,</mo><mi mathvariant="bold-italic">S</mi><mo>,</mo><mi mathvariant="bold-italic">W</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">)</mo><mo>-</mo><mn mathvariant="normal">2</mn><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo></mtd></mtr><mtr><mtd><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">S</mi><msup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><mi>α</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">S</mi><mi mathvariant="bold-italic">L</mi><msup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">)</mo><mo>,</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994923&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994918&type= 71.37400055 9.14400005 其中： α 为权重参数，GNMF求解约束最小化问题 <math id="M14"><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msup><mrow><mi>O</mi></mrow><mrow><mi mathvariant="normal">G</mi><mi mathvariant="normal">N</mi><mi mathvariant="normal">M</mi><mi mathvariant="normal">F</mi></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="bold-italic">A</mi><mo>,</mo><mi mathvariant="bold-italic">S</mi><mo>,</mo><mi mathvariant="bold-italic">W</mi><mo stretchy="false">)</mo><mtext>    </mtext><mi mathvariant="normal">s</mi><mo>.</mo><mi mathvariant="normal">t</mi><mo>.</mo><mtext> </mtext><mi mathvariant="bold-italic">A</mi><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo><mtext> </mtext><mi mathvariant="bold-italic">S</mi><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994932&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994928&type= 62.14533615 3.80999994 。 2.3　多图正则非负矩阵分解MGNMF 2 根据不同定义权重矩阵的模型，可计算相应的图权重矩阵和图拉普拉斯矩阵。MGNMF采用不同数量的最近邻构建图并进行加权，经过算法自动选择，得到最优多图正则项。若已知一组 M 种模型的图权重矩阵｛ W 1 ， W 2 ，…， W M ｝和相应的图拉普拉斯矩阵｛ L 1 ， L 2 ，…， L M ｝，将这 M 个图权重矩阵线性组合，则相应的图拉普拉斯矩阵也进行相同的线性组合： <math display="block" id="M15"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi mathvariant="bold-italic">W</mi><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>m</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>M</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>m</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">L</mi><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>m</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>M</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>m</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mi mathvariant="normal">s</mi><mo>.</mo><mi mathvariant="normal">t</mi><mo>.</mo><mtext> </mtext><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>m</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>M</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><mtext> </mtext><msub><mrow><mi>γ</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>≥</mo><mn mathvariant="normal">0</mn></mrow></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994942&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994937&type= 36.23733139 32.34266663 ， 其中， γ m 是第 M 个图权重矩阵和图拉普拉斯矩阵的权重。MGNMF为一组预先计算得到的候选图确定最佳的图权重，而不是先选最佳图矩阵模型并估计参数。MGNMF的多图正则项为： <math display="block" id="M16"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msup><mrow><mi>O</mi></mrow><mrow><mi mathvariant="normal">M</mi><mi mathvariant="normal">G</mi></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="bold-italic">S</mi><mo>,</mo><mi mathvariant="bold-italic">γ</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>m</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>M</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>m</mi></mrow></msub><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">S</mi><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi>m</mi></mrow></msub><msup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr><mtd><mtext>   </mtext><mi mathvariant="normal">s</mi><mo>.</mo><mi mathvariant="normal">t</mi><mo>.</mo><mtext> </mtext><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>m</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>M</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><mtext> </mtext><msub><mrow><mi>γ</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>≥</mo><mn mathvariant="normal">0</mn></mrow></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994952&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994949&type= 49.10666656 20.82799911 . 多图正则项比单图正则项更精准，GNMF的唯一权重不可靠。MGNMF无需选择唯一的图权重矩阵模型，且通过学习所得图权重向量 <math id="M17"><mi mathvariant="bold-italic">γ</mi><mo>=</mo><mo stretchy="false">[</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>γ</mi></mrow><mrow><mi>L</mi></mrow></msub><msup><mrow><mo stretchy="false">]</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994964&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994959&type= 25.99266624 5.07999992 对 M 个图拉普拉斯矩阵进行最优线性组合。 2.4　多核非负矩阵分解MKNMF 2 NMF是线性方法，不能很好地处理数据中的非线性结构，而KNMF通过核方法，将原始数据映射到高维核空间： <math id="M18"><mi mathvariant="bold-italic">X</mi><mo>→</mo><mi mathvariant="bold-italic">Φ</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">[</mo><mi mathvariant="bold-italic">Φ</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>,</mo><mo>…</mo><mo>,</mo><mi mathvariant="bold-italic">Φ</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994973&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994968&type= 53.93266296 4.48733330 ，可以解决数据的非线性问题。核矩阵 <math id="M19"><mi mathvariant="bold-italic">K</mi><mo>=</mo><mi mathvariant="bold-italic">Φ</mi><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mi mathvariant="bold-italic">Φ</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994983&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994977&type= 29.54866600 3.80999994 ， <math id="M20"><mi mathvariant="bold-italic">K</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mi>N</mi><mo>×</mo><mi>N</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994994&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994990&type= 14.98600006 3.38666677 。则在高维核空间，NMF可以表示为 <math id="M21"><mi mathvariant="bold-italic">Φ</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo><mo>≈</mo><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995003&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32994999&type= 19.72733307 3.55599999 。其中 A 为端元矩阵， S 为丰度矩阵。以核空间中样本 <math id="M22"><mi mathvariant="bold-italic">Φ</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995010&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995006&type= 9.99066734 3.55599999 作为基向量，得到端元矩阵 <math id="M23"><mi mathvariant="bold-italic">A</mi><mo>=</mo><mi mathvariant="bold-italic">Φ</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995020&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995017&type= 20.82799911 3.55599999 ， <math id="M24"><mi mathvariant="bold-italic">A</mi><mo>=</mo><mo stretchy="false">[</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi>P</mi></mrow></msub><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995030&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995025&type= 30.05666542 4.48733330 ， P 为端元个数， F 矩阵包含核空间中所有样本对构造各端元的贡献， <math id="M25"><mi mathvariant="bold-italic">F</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mi>N</mi><mo>×</mo><mi>P</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995041&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995035&type= 14.56266594 3.38666677 且 <math id="M26"><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>N</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi><mi>p</mi></mrow></msub><mi mathvariant="bold-italic">Φ</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995051&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995046&type= 26.75466537 8.55133343 ， <math id="M27"><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi><mi>p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995061&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995057&type= 3.21733332 4.57200003 为矩阵 F 中第 n 行 p 列个元素， F 矩阵中的第 p 列为核空间中所有样本对构造第 p 个端元的贡献。当核函数确定后，KNMF有唯一的核空间，这种选择核空间的方式并不准确。 MKNMF算法将 L 个不同核函数对应的核空间联合起来，以构造一个更合适的希尔伯特空间，这 L 个不同核空间对应的核矩阵为 <math id="M28"><mo stretchy="false">[</mo><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi mathvariant="bold-italic">l</mi></mrow></msub><msubsup><mrow><mo stretchy="false">]</mo></mrow><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>L</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995073&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995065&type= 11.59933376 4.65666676 。核函数权重向量 <math id="M29"><mi mathvariant="bold-italic">τ</mi><mo>=</mo><mo stretchy="false">[</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>τ</mi></mrow><mrow><mi>L</mi></mrow></msub><msup><mrow><mo stretchy="false">]</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995083&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995078&type= 24.63800049 4.65666676 将这 L 个不同的核空间线性组合，组合后的多核核空间的核矩阵为： <math display="block" id="M30"><msup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>τ</mi></mrow></msup><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>L</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>l</mi></mrow></msub><msub><mrow><mo>,</mo></mrow><mrow><mtext> </mtext></mrow></msub><mi mathvariant="normal">s</mi><mo>.</mo><mi mathvariant="normal">t</mi><mo>.</mo><mtext> </mtext><msub><mrow><mi>τ</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo><mtext> </mtext><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>L</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995090&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995086&type= 54.10200119 8.55133343 . 通过将学习好的参数 <math id="M31"><mi>τ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995102&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995094&type= 1.60866666 2.79399991 带入上式，避免了不同核函数权重分配的问题；而多个核函数可构造出更适合原数据的核空间，比单核更可靠。将 式（4） 式（4） 代入KNMF目标函数中，得MKNMF的目标函数： <math display="block" id="M32"><mtable><mtr><mtd><msup><mrow><mi>O</mi></mrow><mrow><mi mathvariant="normal">M</mi><mi mathvariant="normal">K</mi><mi mathvariant="normal">N</mi><mi mathvariant="normal">M</mi><mi mathvariant="normal">F</mi></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="bold-italic">F</mi><mo>,</mo><mi mathvariant="bold-italic">S</mi><mo>,</mo><mi mathvariant="bold-italic">τ</mi><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mo stretchy="false">[</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>L</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi mathvariant="bold-italic">F</mi><mi mathvariant="bold-italic">S</mi><mo stretchy="false">)</mo><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi mathvariant="bold-italic">F</mi><mi mathvariant="bold-italic">S</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">]</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995111&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995106&type= 50.71533585 14.13933277 . 3 MGMKNMF 1 本文提出的MGMKNMF在多核空间构造多图，为更新多图，给定参数 τ ，用欧几里德距离的平方重新定义多核空间中 x n 的 K 近邻集 <math id="M33"><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>τ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995118&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995114&type= 4.40266657 4.06400013 ： <math display="block" id="M34"><mtable><mtr><mtd><msub><mrow><mi>d</mi></mrow><mrow><mi>τ</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">Φ</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mi mathvariant="bold-italic">Φ</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mi>τ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>=</mo></mtd></mtr><mtr><mtd><msup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>τ</mi></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>τ</mi></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mn mathvariant="normal">2</mn><msup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>τ</mi></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>L</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">[</mo><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mn mathvariant="normal">2</mn><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mo>.</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995126&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995122&type= 73.66000366 21.59000015 多核空间中的多图正则化项为： <math display="block" id="M35"><mtable><mtr><mtd><msup><mrow><mi>O</mi></mrow><mrow><mi mathvariant="normal">M</mi><mi mathvariant="normal">G</mi></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="bold-italic">S</mi><mo>,</mo><msup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>τ</mi></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>,</mo><mi>m</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>N</mi></mrow></munderover></mstyle><mrow><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>n</mi><mi>m</mi></mrow><mrow><mi>τ</mi></mrow></msubsup></mrow><mo>=</mo></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>m</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>M</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">S</mi><msubsup><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>τ</mi></mrow></msubsup><msup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">)</mo><mo>,</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995135&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995131&type= 60.62133789 19.21933365 其中， <math id="M36"><msubsup><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>τ</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>τ</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>τ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995143&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995139&type= 23.53733444 4.06400013 是核空间中的图拉普拉斯矩阵。和MGNMF类似，为一组预先计算所得图拉普拉斯矩阵确定最佳权重，精准的多图正则项应用在多核空间将更加合理。 最终将 式（8） 式（8） 与 式（10） 式（10） 结合，得到MGMKNMF的目标函数： <math display="block" id="M37"><mtable><mtr><mtd><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow><mrow><mi mathvariant="bold-italic">F</mi><mo>,</mo><mi mathvariant="bold-italic">S</mi><mo>,</mo><mi mathvariant="bold-italic">τ</mi><mo>,</mo><mi mathvariant="bold-italic">γ</mi></mrow></munder><msup><mrow><mi>O</mi></mrow><mrow><mi mathvariant="normal">M</mi><mi mathvariant="normal">G</mi><mi mathvariant="normal">M</mi><mi mathvariant="normal">K</mi><mi mathvariant="normal">N</mi><mi mathvariant="normal">M</mi><mi mathvariant="normal">F</mi></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="bold-italic">F</mi><mo>,</mo><mi mathvariant="bold-italic">S</mi><mo>,</mo><mi mathvariant="bold-italic">τ</mi><mo>,</mo><mi mathvariant="bold-italic">γ</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi>O</mi></mrow><mrow><mi mathvariant="normal">M</mi><mi mathvariant="normal">K</mi><mi mathvariant="normal">N</mi><mi mathvariant="normal">M</mi><mi mathvariant="normal">F</mi></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="bold-italic">F</mi><mo>,</mo><mi mathvariant="bold-italic">S</mi><mo>,</mo><mi mathvariant="bold-italic">τ</mi><mo stretchy="false">)</mo><mo>+</mo><mi>α</mi><msup><mrow><mi>O</mi></mrow><mrow><mi mathvariant="normal">M</mi><mi mathvariant="normal">G</mi></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="bold-italic">S</mi><mo>,</mo><msup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>τ</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><mi>β</mi><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">τ</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>μ</mi><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">γ</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo></mtd></mtr><mtr><mtd><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mo stretchy="false">[</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>L</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mi>l</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi mathvariant="bold-italic">F</mi><mi mathvariant="bold-italic">S</mi><mo stretchy="false">)</mo><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi mathvariant="bold-italic">F</mi><mi mathvariant="bold-italic">S</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></mrow><mo stretchy="false">]</mo><mo>+</mo><mi>α</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>m</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>M</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>m</mi></mrow></msub><mi>t</mi><mi>r</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">S</mi><msubsup><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>τ</mi></mrow></msubsup><msup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>T</mi></mrow></msup><mo stretchy="false">)</mo></mrow><mo>+</mo><mi>β</mi><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">τ</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>μ</mi><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">γ</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo></mtd></mtr><mtr><mtd><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mo stretchy="false">[</mo><msup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>τ</mi></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi mathvariant="bold-italic">F</mi><mi mathvariant="bold-italic">S</mi><mo stretchy="false">)</mo><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi mathvariant="bold-italic">F</mi><mi mathvariant="bold-italic">S</mi><mo stretchy="false">)</mo></mrow><mrow><mi>T</mi></mrow></msup><mo stretchy="false">]</mo><mo>+</mo><mi>α</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">S</mi><msup><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi>τ</mi></mrow></msup><msup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><mi>β</mi><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">τ</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>μ</mi><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">γ</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mtd></mtr><mtr><mtd><mtext> </mtext><mi mathvariant="normal">s</mi><mo>.</mo><mi mathvariant="normal">t</mi><mo>.</mo><mtext> </mtext><mi mathvariant="bold-italic">F</mi><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi mathvariant="bold-italic">S</mi><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>τ</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>L</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><mtext> </mtext><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>m</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>M</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><mtext> </mtext><msub><mrow><mi>γ</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>≥</mo><mn mathvariant="normal">0</mn></mrow></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995150&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995146&type= 128.35466003 38.26933289 . 多核空间可更好地挖掘数据间的非线性关系，在多核空间嵌入多图能更好地表达数据的非线性流形结构。 式（11） 式（11） 中的‖ τ ‖ 2 可防止参数 <math id="M38"><mi mathvariant="bold-italic">τ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995338&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995334&type= 1.69333339 2.79399991 过度偏向到一个核函数中；‖ γ ‖ 2 项可避免 γ 偏向到一个权重构造函数中； α ， β ， μ 均是权衡上式所用的权衡参数，其值均为非负。因 α 是约束多图正则项的参数，与约束核权重和图权重的 β ， μ 相比， α 应大于 β ， μ 。根据参考文献［ 21 21 ］和多次实验的结果，本文实验中：Cuprite数据 α =100，Jasper Ridge数据 α =20，HAPKE模拟数据 α =20，GBM模拟数据 α =20；所有数据的 β 均为10， μ 均为10。 根据参考文献［ 21 21 ］和［22］， 式（11） 式（11） 中的MGMKNMF目标函数是非凸的，无法得到全局最小值。MGMKNMF采用分步迭代策略优化目标函数，可得到局部最小值 ［ 21 21 - 22 22 21-22 ］ ，具体如下： （1）固定 <math id="M39"><mi mathvariant="bold-italic">τ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995338&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995334&type= 1.69333339 2.79399991 和 <math id="M40"><mi mathvariant="bold-italic">γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995345&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995341&type= 2.11666679 3.64066648 ，更新 F ， S 。 式（11） 式（11） 可写为： <math display="block" id="M41"><mtable><mtr><mtd><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow><mrow><mi mathvariant="bold-italic">F</mi><mo>,</mo><mi mathvariant="bold-italic">S</mi></mrow></munder><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mo stretchy="false">[</mo><msup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>τ</mi></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi mathvariant="bold-italic">F</mi><mi mathvariant="bold-italic">S</mi><mo stretchy="false">)</mo><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi mathvariant="bold-italic">F</mi><mi mathvariant="bold-italic">S</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">]</mo><mo>+</mo><mi>α</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">S</mi><msup><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi>τ</mi></mrow></msup><msup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mtext>                              </mtext><mi mathvariant="normal">s</mi><mo>.</mo><mi mathvariant="normal">t</mi><mo>.</mo><mtext> </mtext><mi mathvariant="bold-italic">F</mi><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi mathvariant="bold-italic">S</mi><mo>≥</mo><mn mathvariant="normal">0</mn><mo>.</mo><mtext>                        </mtext><mo stretchy="false">(</mo><mn mathvariant="normal">12</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995183&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995178&type= 75.43800354 10.83733368 假设 <math id="M42"><msub><mrow><mi>φ</mi></mrow><mrow><mi>l</mi><mi>p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995195&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995188&type= 4.06400013 4.65666676 和 <math id="M43"><msub><mrow><mi>ψ</mi></mrow><mrow><mi>p</mi><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995205&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995198&type= 4.23333359 4.48733330 分别是 <math id="M44"><msub><mrow><mi>f</mi></mrow><mrow><mi>l</mi><mi>p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995213&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995209&type= 2.87866688 4.57200003 和 <math id="M45"><msub><mrow><mi>s</mi></mrow><mrow><mi>p</mi><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995222&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995217&type= 3.38666677 3.72533321 的拉格朗日乘子，令 <math id="M46"><msub><mrow><mi>φ</mi></mrow><mrow><mi>l</mi><mi>p</mi></mrow></msub><mo>=</mo><mi>φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995234&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995230&type= 11.26066685 4.65666676 ， <math id="M47"><msub><mrow><mi>ψ</mi></mrow><mrow><mi>p</mi><mi>n</mi></mrow></msub><mo>=</mo><mi>ψ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995242&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995238&type= 11.17599964 4.48733330 ，则式（12）的拉格朗日函数为： <math display="block" id="M48"><mtable><mtr><mtd><mi>λ</mi><mo>=</mo><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow><mrow><mi mathvariant="bold-italic">F</mi><mo>,</mo><mi mathvariant="bold-italic">S</mi></mrow></munder><mtext> </mtext><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mo stretchy="false">[</mo><msup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>τ</mi></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi mathvariant="bold-italic">F</mi><mi mathvariant="bold-italic">S</mi><mo stretchy="false">)</mo><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi mathvariant="bold-italic">F</mi><mi mathvariant="bold-italic">S</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">]</mo><mo>+</mo></mtd></mtr><mtr><mtd><mi>α</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">S</mi><msup><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi>τ</mi></mrow></msup><msup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mo stretchy="false">(</mo><mi>φ</mi><msup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mo stretchy="false">(</mo><mi>ψ</mi><msup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">)</mo><mo>.</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995249&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995245&type= 59.94400024 11.34533310 式（13） 式（13） 分别对 F 和 S 求偏导得： <math display="block" id="M49"><mtable><mtr><mtd><mfrac><mrow><mo>∂</mo><mi>λ</mi></mrow><mrow><mo>∂</mo><mi mathvariant="bold-italic">F</mi></mrow></mfrac><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>τ</mi></mrow></msup><mi mathvariant="bold-italic">F</mi><mi mathvariant="bold-italic">S</mi><msup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>τ</mi></mrow></msup><msup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>T</mi></mrow></msup><mo>+</mo><mi>φ</mi></mtd></mtr><mtr><mtd><mfrac><mrow><mo>∂</mo><mi>λ</mi></mrow><mrow><mo>∂</mo><mi mathvariant="bold-italic">S</mi></mrow></mfrac><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>T</mi></mrow></msup><msup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>T</mi></mrow></msup><msup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>τ</mi></mrow></msup><mi mathvariant="bold-italic">F</mi><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>τ</mi></mrow></msup><mi mathvariant="bold-italic">F</mi><mo>+</mo><mi>α</mi><mi mathvariant="bold-italic">S</mi><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi>τ</mi></mrow></msup><mo>+</mo><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi>τ</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow/></msup><mo>+</mo><mi>ψ</mi></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995256&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995253&type= 73.82933807 17.69533348 . 结合卡罗需-库恩-塔克（KKT）条件： <math id="M50"><msub><mrow><mi>φ</mi></mrow><mrow><mi>l</mi><mi>p</mi></mrow></msub><msub><mrow><mi>f</mi></mrow><mrow><mi>l</mi><mi>p</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995262&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995258&type= 14.64733315 4.65666676 ， <math id="M51"><msub><mrow><mi>ψ</mi></mrow><mrow><mi>p</mi><mi>n</mi></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mi>p</mi><mi>n</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995270&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995266&type= 14.73200035 4.48733330 有： <math display="block" id="M52"><mtable><mtr><mtd><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>τ</mi></mrow></msup><mi mathvariant="bold-italic">F</mi><mi mathvariant="bold-italic">S</mi><msup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">)</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>l</mi><mi>p</mi></mrow></msub><mo>-</mo><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>τ</mi></mrow></msup><msup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">)</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>l</mi><mi>p</mi></mrow></msub><mo>+</mo><mi>φ</mi><msub><mrow><mi>f</mi></mrow><mrow><mi>l</mi><mi>p</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mtext> </mtext></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>τ</mi></mrow></msup><mi mathvariant="bold-italic">F</mi><mo stretchy="false">)</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>p</mi><mi>n</mi></mrow></msub><mo>-</mo><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>τ</mi></mrow></msup><mi mathvariant="bold-italic">F</mi><mo stretchy="false">)</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>p</mi><mi>n</mi></mrow></msub><mo>+</mo><mo stretchy="false">(</mo><mi>α</mi><mi mathvariant="bold-italic">S</mi><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi>τ</mi></mrow></msup><mo>+</mo><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi>τ</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>p</mi><mi>n</mi></mrow></msub><mo>+</mo><mi>ψ</mi><msub><mrow><mi>s</mi></mrow><mrow><mi>p</mi><mi>n</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995276&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995272&type= 94.99600220 10.83733368 . 得到 F 和 S 的乘法更新法则分别为： <math display="block" id="M53"><msub><mrow><mi>f</mi></mrow><mrow><mi>l</mi><mi>p</mi></mrow></msub><mo>←</mo><mfrac><mrow><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>τ</mi></mrow></msup><msup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>l</mi><mi>p</mi></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>τ</mi></mrow></msup><mi mathvariant="bold">F</mi><mi mathvariant="bold">S</mi><msup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">)</mo></mrow></mfrac><msub><mrow><mi>f</mi></mrow><mrow><mi>l</mi><mi>p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995284&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995280&type= 29.97200012 8.97466660 ， <math display="block" id="M54"><msub><mrow><mi>s</mi></mrow><mrow><mi>p</mi><mi>n</mi></mrow></msub><mo>←</mo><mfrac><mrow><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>τ</mi></mrow></msup><mo>+</mo><mi>α</mi><mi mathvariant="bold-italic">S</mi><msup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>τ</mi></mrow></msup><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>τ</mi></mrow></msup><mi mathvariant="bold">F</mi><mi mathvariant="bold">S</mi><mo>+</mo><mi>α</mi><mi mathvariant="bold-italic">S</mi><msup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>τ</mi></mrow></msup><mo stretchy="false">)</mo></mrow></mfrac><msub><mrow><mi>s</mi></mrow><mrow><mi>p</mi><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995291&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995288&type= 43.43399811 8.97466660 . （2）固定 F ， S 和 <math id="M55"><mi mathvariant="bold-italic">γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995345&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995341&type= 2.11666679 3.64066648 ，更新 <math id="M56"><mi mathvariant="bold-italic">τ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995338&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995334&type= 1.69333339 2.79399991 ， 式（11） 式（11） 可写为： <math display="block" id="M57"><mtable><mtr><mtd><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow><mrow><mi mathvariant="bold-italic">τ</mi></mrow></munder><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mo stretchy="false">[</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>L</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi mathvariant="bold-italic">F</mi><mi mathvariant="bold-italic">S</mi><mo stretchy="false">)</mo><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi mathvariant="bold-italic">F</mi><mi mathvariant="bold-italic">S</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">]</mo><mo>+</mo><mi>β</mi><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">τ</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo></mtd></mtr><mtr><mtd><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mo stretchy="false">[</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>L</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>l</mi></mrow></msub><mi mathvariant="bold-italic">Z</mi><msup><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>T</mi></mrow></msup><mo stretchy="false">]</mo><mo>+</mo><mi>β</mi><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">τ</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>L</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>+</mo><mi>β</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>L</mi></mrow></munderover></mstyle><mrow><msup><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mtd></mtr><mtr><mtd><mtext> </mtext><mi mathvariant="normal">s</mi><mo>.</mo><mi mathvariant="normal">t</mi><mo>.</mo><mtext> </mtext><msub><mrow><mi>τ</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo><mtext> </mtext><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>L</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></mrow></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995314&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995311&type= 73.15200043 29.20999908 ， 其中， <math id="M58"><mi mathvariant="bold-italic">Z</mi><mo>=</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi mathvariant="bold-italic">F</mi><mi mathvariant="bold-italic">S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995324&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995320&type= 18.37266731 2.79399991 ， <math id="M59"><msub><mrow><mi>g</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>=</mo><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mo stretchy="false">[</mo><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>l</mi></mrow></msub><mi mathvariant="bold-italic">Z</mi><msup><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995330&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995326&type= 25.31533241 5.07999992 。 （3）固定 F 和 S 和 <math id="M60"><mi mathvariant="bold-italic">τ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995338&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995334&type= 1.69333339 2.79399991 ，更新 <math id="M61"><mi mathvariant="bold-italic">γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995345&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995341&type= 2.11666679 3.64066648 ， 式（11） 式（11） 可以写为： <math display="block" id="M62"><mtable><mtr><mtd><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow><mrow><mi mathvariant="bold-italic">γ</mi></mrow></munder><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>m</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>M</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">S</mi><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi>m</mi></mrow></msub><msup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><mi>μ</mi><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">γ</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mtd></mtr><mtr><mtd><mi mathvariant="normal">s</mi><mo>.</mo><mi mathvariant="normal">t</mi><mo>.</mo><mtext> </mtext><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>m</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>M</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><mtext> </mtext><msub><mrow><mi>γ</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>≥</mo><mn mathvariant="normal">0</mn></mrow></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995353&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995349&type= 47.41333389 18.88066673 . 式（18） 式（18） 和 式（19） 式（19） 的约束二次规划问题可根据文献［ 23 23 ］中的方法来解决。 MGMKNMF解混算法总结如下： MGMKNMF解混算法 输入：原始高光谱数据 X ， L 个核矩阵 <math id="M63"><mo>{</mo><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>L</mi></mrow></msub><mo>}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995358&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995356&type= 26.50066566 3.89466691 ， M 个图拉普拉斯矩阵 <math id="M64"><mo>{</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995364&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995361&type= 26.58533096 3.89466691 ，最大迭代次数 T Step1.初始化矩阵 F 0 和 S 0 ， Step2.初始化核权重变量 <math id="M65"><msubsup><mrow><mi>τ</mi></mrow><mrow><mi>l</mi></mrow><mrow><mn>0</mn></mrow></msubsup><mo>=</mo><mrow><mrow><mn>1</mn></mrow><mo>/</mo><mrow><mi>L</mi></mrow></mrow><mo>,</mo><mi>l</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>L</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995369&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995367&type= 28.70199966 3.13266683 ，初始化图权重 <math id="M66"><msubsup><mrow><mi>γ</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn>0</mn></mrow></msubsup><mo>=</mo><mrow><mrow><mn>1</mn></mrow><mo>/</mo><mrow><mi>M</mi></mrow></mrow><mo>,</mo><mi>m</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>M</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995374&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995372&type= 32.25799942 3.38666677 for t =1 to T do 通过 式（7） 式（7） 更新图 <math id="M67"><msup><mrow><mi>G</mi></mrow><mrow><msup><mrow><mi>τ</mi></mrow><mrow><mi>t</mi></mrow></msup></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995378&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995376&type= 3.80999994 2.96333337 和相应的拉普拉斯矩阵 <math id="M68"><msup><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><msup><mrow><mi>τ</mi></mrow><mrow><mi>t</mi></mrow></msup></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995382&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995379&type= 3.30200005 2.96333337 ； 通过 式（11） 式（11） 和式（12）更新 <math id="M69"><msup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>t</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995389&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995387&type= 3.04800010 2.62466669 和 <math id="M70"><msup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>t</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995395&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995392&type= 2.62466669 2.62466669 ； 通过 式（13） 式（13） 和 式（14） 式（14） 更新核权重 <math id="M71"><msup><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mrow><mi>t</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995400&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995398&type= 2.20133328 2.62466669 和图权重 <math id="M72"><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mi>t</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995407&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995403&type= 2.45533323 3.30200005 ； end 输出： <math id="M73"><mi mathvariant="bold-italic">F</mi><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>t</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>,</mo><mtext> </mtext><mi mathvariant="bold-italic">S</mi><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>t</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995413&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995409&type= 28.44799995 2.96333337 <math id="M74"><mtext> </mtext><mi mathvariant="bold-italic">τ</mi><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mrow><mi>t</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>,</mo><mtext> </mtext><mi mathvariant="bold-italic">γ</mi><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mi>t</mi><mo>-</mo><mn>1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995416&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995414&type= 25.56933212 3.30200005 4 实验结果与分析 1 将所提MGMKNMF算法与GNMF、npKNMF、KSNMF、KDP和MGKNMF解混算法进行对比，使用SAD和RMSE作为评估指标，采用HAPKE和GBM模拟数据，以及Cuprite和Jasper Ridge真实数据验证该算法的有效性。所有实验中，我们选择0-1加权图、热核加权图和点积加权图的图权重矩阵模型构成最终图正则项。多核函数选择高斯核函数，所选取的核参数为1/32，1/16，1/8，1/4，1/2，1，2，4，8，16，32以构成不同的核函数。各算法的迭代次数 T 均设置为200。 4.1　评价标准 2 SAD值反映了解混所得端元光谱与原端元光谱之间相似性，定义为 <math id="M75"><msub><mrow><mi>V</mi></mrow><mrow><mi mathvariant="normal">S</mi><mi mathvariant="normal">A</mi><mi mathvariant="normal">D</mi></mrow></msub><mo>=</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mi mathvariant="bold-italic">b</mi><mo>/</mo><mo stretchy="false">|</mo><mo stretchy="false">|</mo><mi mathvariant="bold-italic">a</mi><mo stretchy="false">|</mo><mo stretchy="false">|</mo><mtext> </mtext><mo stretchy="false">|</mo><mo stretchy="false">|</mo><mi mathvariant="bold-italic">b</mi><mo stretchy="false">|</mo><mo stretchy="false">|</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995424&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995422&type= 42.67200089 4.82600021 ，其中， a ， b 是两个端元光谱。 RMSE值反映解混所得丰度与实验室测量的实际丰度间的差别，定义为 <math id="M76"><msub><mrow><mi>V</mi></mrow><mrow><mi mathvariant="normal">R</mi><mi mathvariant="normal">M</mi><mi mathvariant="normal">S</mi><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>/</mo><mi>N</mi><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>̂</mo></mover></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995429&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995426&type= 40.80933380 5.16466665 ，其中， <math id="M77"><msub><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995432&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995430&type= 2.11666679 3.72533321 和 <math id="M78"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>̂</mo></mover></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995437&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995435&type= 2.11666679 3.80999994 分别是实际丰度和解混所得丰度。 4.2　模拟数据实验与分析 2 本文采用HAPKE和GBM两种非线性模型生成模拟数据集。HAPKE模型是一种紧密混合模型；GBM是一种双线性混合模型，是LMM线性结构与端元间的二次散射项的加权组合。本文从美国光谱库（USGS）随机选择6种地物光谱作为端元，如 图1 图1 。并且丰度矢量满足丰度非负和丰度和为一的约束。最后模拟生成的高光谱每个大小为20 <math id="M79"><mo>×</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995441&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995439&type= 2.53999996 2.79399991 20，每个像素波段为224，并加入不同信噪比（Signal to Noise Ratio， SNR）的零均值高斯噪声来更好地接近真实数据。 4.3　模拟数据实验结果及分析 2 表 1 1 ~ 4 4 分别显示了端元数目为6，SNR不同时，各算法在HAPKE和GBM模拟数据上的SAD、RMSE值。从表中可以看出，随着SNR值的增加，SAD和RMSE的值均呈下降趋势，而 MGMKNMF算法与其他6种算法相比具有较好的准确度。NMF和GNMF因为属于线性解混算法，对非线性数据解混精度都很差，但GNMF算法在NMF的基础上增加了图正则约束，揭示了数据内在固有的流形结构，相比NMF算法有进一步提升；剩下的5种算法均利用了核函数的概念，更适合于非线性数据。npKNMF算法在核空间中应用了NMF，相比NMF准确度有所提高；而KSNMF在丰度中加了L 1 范数使丰度更加稀疏，得到了更好的准确率；KDP在进行端元选择时使用了大型光谱库，相比其余两者更加准确，MGKNMF在单核空间中构造多图正则项。而提出的MGMKNMF算法不仅用多图来刻画数据内在流形结构，更使用多核学习找到合适的核参数与核函数，进一步增加了算法的准确度。 SNR/dB NMF GNMF npKNMF KSNMF KDP MGKNMF MGMKNMF 10 0.494 8 0.456 7 0.170 2 0.169 3 0.167 1 0.190 1 0.213 3 20 0.382 9 0.309 4 0.164 8 0.160 4 0.154 3 0.157 6 0.154 1 30 0.341 7 0.296 0 0.162 1 0.153 1 0.148 2 0.129 4. 0.124 8 40 0.308 4 0.251 9 0.159 6 0.149 2 0.142 7 0.146 6 0.137 5 SNR/dB NMF GNMF npKNMF KSNMF KDP MGKNMF MGMKNMF 10 0.401 1 0.283 9 0.180 3 0.172 3 0.171 4 0.169 3 0.258 8 20 0.395 9 0.236 4 0.174 2 0.166 8 0.164 3 0.145 5 0.121 9 30 0.341 3 0.214 2 0.168 1 0.161 1 0.160 8 0.139 0 0.113 1 40 0.300 1 0.199 8 0.161 3 0.159 1 0.157 7 0.149 9 0.145 6 SNR/dB NMF GNMF npKNMF KSNMF KDP MGKNMF MGMKNMF 10 0.185 0 0.186 9 0.079 2 0.068 9 0.067 2 0.099 0 0.091 0 20 0.162 4 0.183 9 0.076 9 0.063 7 0.061 1 0.061 8 0.060 1 30 0.149 5 0.180 1 0.071 3 0.058 7 0.058 0 0.052 3 0.049 0 40 0.129 9 0.178 3 0.066 2 0.052 3 0.051 8 0.051 5 0.050 6 SNR/dB NMF GNMF npKNMF KSNMF KDP MKGNMF MGMKNMF 10 0.321 7 0.290 6 0.081 7 0.071 1 0.070 8 0.079 0 0.112 3 20 0.307 4 0.253 1 0.079 4 0.062 4 0.061 8 0.060 6 0.059 7 30 0.292 3 0.233 7 0.073 6 0.060 3 0.059 3 0.052 2 0.050 1 40 0.258 6 0.201 2 0.069 1 0.058 4 0.058 0 0.057 7 0.057 0 表 5 5 ~ 8 8 分别显示了SNR为40 dB，端元数目 P 不同时，各算法在这2个模型的模拟数据上的SAD与RMSE值。因每次生成模拟数据时是在USGS库随机选取端元，导致表5~8的结果在 P =6时，与表 1 1 ~ 4 4 中SNR为40 dB时的结果不同。可以看出，因为模拟数据非线性程度高，端元数的增加会导致算法解混能力的下降。相比NMF算法，MGMKNMF算法在HAPKE数据中，端元为6时，SAD值和RMSE值分别减少了约0.17和0.13；在GBM数据中，分别减少了约0.12和0.17。和其余算法相比，MGMKNMF算法也基本保持着最优结果。 Endmember NMF GNMF npKNMF KSNMF KDP MGKNMF MGMKNMF P =6 0.308 4 0.281 7 0.159 6 0.149 0 0.142 2 0.145 7 0.141 1 P =5 0.251 0 0.218 7 0.124 0 0.118 2 0.107 7 0.096 8 0.081 4 P =4 0.198 5 0.195 4 0.100 4 0.089 8 0.081 2 0.080 2 0.078 9 P =3 0.172 9 0.143 8 0.073 5 0.068 2 0.060 0 0.044 3 0.034 3 Endmember NMF GNMF npKNMF KSNMF KDP MGKNMF MGMKNMF P =6 0.271 1 0.278 9 0.169 5 0.159 8 0.158 0 0.148 3 0.147 1 P =5 0.268 3 0.241 9 0.125 1 0.121 5 0.113 6 0.100 2 0.095 1 P =4 0.209 2 0.202 1 0.091 4 0.085 4 0.081 7 0.070 0 0.063 2 P =3 0.182 0 0.143 5 0.067 9 0.060 5 0.057 2 0.042 3 0.029 9 Endmember NMF GNMF npKNMF KSNMF KDP MGKNMF MGMKNMF P =6 0.189 9 0.176 4 0.084 1 0.068 1 0.066 9 0.060 3 0.059 4 P =5 0.175 3 0.162 1 0.078 3 0.062 3 0.061 1 0.060 1 0.056 3 P =4 0.151 5 0.160 1 0.074 5 0.058 9 0.057 4 0.054 7 0.052 3 P =3 0.203 0 0.157 7 0.066 2 0.055 3 0.051 8 0.051 4 0.050 8 Endmember NMF GNMF npKNMF KSNMF KDP MGKNMF MGMKNMF P =6 0.227 2 0.200 1 0.086 3 0.079 2 0.077 5 0.060 1 0.061 0 P =5 0.180 4 0.173 4 0.081 2 0.068 3 0.066 2 0，0 622 0.060 3 P =4 0.153 4 0.149 7 0.079 9 0.065 1 0.064 3 0.064 0 0.063 3 P =3 0.119 6 0.139 6 0.069 1 0.058 4 0.058 0 0.041 2 0.035 4 4.4　真实数据实验结果及分析 2 本文采用真实地物Cuprite和Jasper Ridge数据集对MGMKNMF的有效性进行验证，两个数据集均可以在 https：//rslab.ut.ac.ir/data https://rslab.ut.ac.ir/data 中下载。 Cuprite是高光谱解混研究常用的数据集，包含美国内华州Cuprite矿区。在除去低信噪比和吸水通道后，留有188个通道可以使用。每张图像大小为250×191，共有12种类别。 表9 表9 总结了在Cuprite数据集各类算法的SAD值。可以看出，MGMKNMF算法的平均SAD值是最优的。 Item NMF GNMF npKNMF KSNMF KDP MGKNMF MGMKNMF Alunite 0.297 9 0.243 4 0.063 2 0.085 9 0.062 9 0.078 7 0.069 5 Andradite 0.383 9 0.262 6 0.079 5 0.101 4 0.204 5 0.066 0 0.079 4 Buddingtonite 0.367 2 0.266 6 0.079 6 0.073 7 0.116 1 0.088 5 0.119 6 Dumortierite 0.283 9 0.574 3 0.156 2 0.071 7 0.071 2 0.084 8 0.082 9 Kaolinite_1 0.235 9 0.336 2 0.080 2 0.061 3 0.081 3 0.082 1 0.087 5 Kaolinite_2 0.309 5 0.344 3 0.262 2 0.189 0 0.114 0 0.056 5 0.073 3 Muscovite 0.379 4 0.318 3 0.113 7 0.052 1 0.147 2 0.102 4 0.104 1 Montmorillonite 0.473 1 0.355 7 0.131 0 0.140 2 0.107 0 0.054 5 0.055 6 Nontronite 0.451 6 0.451 6 0.129 3 0.064 7 0.078 8 0.121 4 0.104 3 Pyrope 0.292 7 0.455 0 0.054 1 0.142 0 0.074 5 0.117 8 0.065 5 Sphene 0.282 1 0.287 9 0.091 3 0.279 8 0.097 6 0.201 5 0.197 1 Chalcedony 0.318 6 0.318 6 0.213 2 0.149 5 0.132 7 0.071 5 0.065 9 Average 0.339 6 0.337 1 0.120 8 0.117 6 0.107 8 0.093 8 0.092 1 图2 图2 为MGMKNMF算法在Cuprite数据集上的丰度图。结合 表9 表9 ，在此算法下，12种端元对应的丰度图应是最优的，可清晰地看出各类别的划分情况。 由于Jasper Ridge数据集太复杂无法处理，我们仅考虑100×100像素的子图像，子图像的第一个像素从原始图像中的第（105，269）像素开始，且去除了低噪声和水吸收波段，保留198个通道。所以，该高光谱遥感数据中只有树、水、土壤和道路这4类端元。 从 表10 表10 可以看到，提出的MGMKNMF算法在Jasper Ridge数据集上仍然有效，平均SAD值依旧是最优的。KDP算法在KSNMF算法的基础上利用光谱库挑选端元，而MGKNMF在单核空间应用多图，精度仅次于MGMKNMF。 Item NMF GNMF npKNMF KSNMF KDP MGKNMF MGMKNMF Tree 0.267 7 0.238 1 0.139 7 0.144 6 0.095 3 0.079 3 0.096 4 Water 0.319 2 0.338 8 0.118 6 0.114 3 0.102 5 0.091 1 0.120 3 Soil 0.428 9 0.081 9 0.131 7 0.105 3 0.123 7 0.124 2 0.097 1 Road 0.410 8 0.309 3 0.127 2 0.094 2 0.086 9 0.097 4 0.074 1 Average 0.356 6 0.242 0 0.129 3 0.114 6 0.102 1 0.098 0 0.097 0 图3 图3 为Jasper Ridge数据集各算法解混出的丰度图，从左至右分别对应树、水、土壤和道路这4个端元。 从 图3 图3 可看出，NMF算法与真实地物丰度相差很大，尤其在水这一端元上；GNMF算法比NMF算法的丰度图清晰。而其余的5种算法均在核空间进行，丰度图明显优于NMF和GNMF，其中水的丰度图更接近真实地物。相比其他算法，在核空间进行的算法尽管其丰度图的区别甚微，但结合 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