基于改进Fisher准则的深度卷积生成对抗网络算法

基于改进Fisher准则的深度卷积生成对抗网络算法 Deep convolutional generative adversarial network algorithm based on improved fisher's criterion 1 first-author 1 corresp E-mail： qiguanglei@ccbupt.cn E-mail： qiguanglei@ccbupt.cn 齐光磊（1979-），男，北京人，博士，副教授，2002年、2006年于西安电子科技大学分别获得学士、硕士学位，2017年于北京工业大学获得博士学位，主要从事机器学习、模式识别、计算机视觉等方面的研究。E-mail： qiguanglei@ccbupt.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113450&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113458&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113454&type= 22.01333237 32.00399780 齐光磊 （1979-），男，北京人，博士，副教授，2002年、2006年于西安电子科技大学分别获得学士、硕士学位，2017年于北京工业大学获得博士学位，主要从事机器学习、模式识别、计算机视觉等方面的研究。E-mail： qiguanglei@ccbupt.cn qiguanglei@ccbupt.cn 2 1 北京邮电大学 世纪学院 计算机科学与技术系，北京102101 北京邮电大学 世纪学院 计算机科学与技术系，北京102101 Century College， Beijing University of Posts and Telecommunications， Beijing 102101， China Century College， Beijing University of Posts and Telecommunications， Beijing 102101， China 新疆大学 信息科学与工程学院， 新疆 乌鲁木齐 830046 新疆大学 信息科学与工程学院， 新疆 乌鲁木齐 830046 College of Information Science and Engineering，Xinjiang University，Urumqi 830046， China College of Information Science and Engineering，Xinjiang University，Urumqi 830046， China 针对当训练样本量不足或者迭代次数降低时生成图像质量急剧下降的问题，提出了一种基于改进Fisher准则的深度卷积生成对抗网络算法（FDCGAN，Deep Convolutional Generative Adversarial Network algorithm based on improved Fisher's criterion）。该方法在判别模型中添加线性层，用来提取类别信息。在反向传播中采用基于Fisher的约束准则，结合标签和类别信息，在权值的迭代调整时既考虑误差的最小化，又同时让样本保持类内距离小、类间距离大，从而使权值能更加快速地逼近最优值。通过与最新不同的6个网络模型进行对比实验，FDCGAN模型在FID指标上均取得了较好的效果。此外，通过将该方法运用到目前先进模型上进行泛化测试，实验结果均取得较理想的效果。 An improved Fisher’s criterion-based deep convolutional generative adversarial network algorithm （FDCGAN） is proposed in this study to solve the problem of quality deterioration in generated images when the training sample size is insufficient or number of iterations decreases. In this method， a linear layer is added to the discriminative model to extract category information. Then， Fisher’s criterion is used in backpropagation to combine label and category information. To minimize errors， the weights are adjusted iteratively while maintaining small intra-class and large inter-class distances such that the weights can rapidly approach the optimal value. A comparison of the experimental results of the FDCGAN model with that of the most recent six network models shows that the proposed model achieves better performance in all the FID metrics. In addition， applying the proposed model to the current advanced models in generalization tests yields more satisfactory results. 深度卷积生成对抗网络 Fisher准则 反向传播算法 FID评价指标 Deep Convolutional Generative Adversarial Network（DCGAN） fisher guidelines backward propagation algorithm FID evaluation metrics 1 引 言 1 随着科技的进步和智能化技术的革新，人工智能在信息化社会蓬勃发展，以生成模型为代表的机器学习领域成为研究热点。生成模型学习可观测样本的概率密度，然后从此分布中进行采样，以此随机生成新的数据 ［ 1 1 ］ 。早期的生成模型有受限玻尔兹曼机、深度信念网络、变分自动编码器、深度波尔兹曼机等，然而由于泛化性能较弱且此类方法计算复杂，导致它们的性能和结果不理想 ［ 2 2 ］ 。Goodfellow等人 ［ 3 3 ］ 引入了一种全新的生成模型，即生成对抗网络（Generative Adversarial Networks，GAN），它是一种通过让两个神经网络互相对抗训练，以此提高各自性能，最终使模型达到纳什均衡状态。GAN一经提出便受到人们广泛关注，在理论和应用层面也有飞速发展 ［ 4 4 ］ 。 近年来，GAN的理论研究取得了快速进展，并且可分为基于目标函数进行改进的模型和基于网络体系结构进行改进的模型。基于目标函数改进的GANs中，Nowozin等人 ［ 5 5 ］ 提出f-GANs，将散度估计扩展到模型估计，实验证明了GANs的广泛应用性；Mao等人 ［ 6 6 ］ 提出LSGAN，将目标函数替换成最小二乘损失函数，将生成样本拉向决策边界，提高了模型稳定性；Arjovsky等人 ［ 7 7 ］ 提出WGAN，采用Earth Mover距离来计算生成样本分布和真实样本分布之间的距离，使GAN训练更稳定，同时解决了模型崩塌等问题；Mroueh等人 ［ 8 8 ］ 提出FisherGAN，建立了一种积分概率矩阵框架，使得模型可以稳定训练；Hjelm等人 ［ 9 9 ］ 提出BGAN，可基于离散值训练模型，提高了模型训练的稳定性。基于网络体系结构改进的GANs中，Radford等人 ［ 10 10 ］ 提出DCGAN，将有监督学习的CNNs与无监督学习的GANs结合在一起，提高了生成图像的多样性并优化了模型训练不稳定的问题；Makhzani等人 ［ 11 11 ］ 提出对抗自动编码器AAE，通过在自编码器的隐藏层中增加GAN的思想，使得生成样本分布向真实样本分布靠近，结果更可控；Odena等人 ［ 12 12 ］ 提出ACGAN，附加类别标签输入生成器，判别器给出两个概率输出，使得模型能够生成丰富和高分辨率图像。此外，齐光磊等人 ［ 13 13 ］ 提出FCNN，在处理分类任务时，将Fisher准则应用于普通的卷积神经网络，有效地提高了卷积神经网络的识别率。 尽管上述衍生模型从多方面提高了生成图像的质量和模型训练的稳定性，但往往需要大量的样本进行训练，通常需要上万次更新迭代，生成器才能输出较高质量的图像。而FCNN则存在应用点单一，且未考虑到此方法在其他领域和模型上的泛化性等问题。针对上述问题，为了能够在样本量不足或者在减少训练样本的情况下生成器依然有生成高质量图像的能力。本文提出了基于改进Fisher准则的深度卷积生成对抗网络算法（FDCGAN，Deep Convolutional Generative Adversarial Network algorithm based on improved Fisher's criterion）。 2 基于Fisher准则生成对抗网络模型 1 本文提出一种基于Fisher准则的深度卷积生成对抗网络算法（FDCGAN），在保留DCGAN网络的优点的同时，引入Fisher准则的思想。该方法在判别模型中添加线性层，用来提取类别信息。当利用真实样本训练判别模型时，在反向传播中采用基于Fisher的约束准则，结合标签和类别信息，使得每次迭代训练更新权值时，每层参数的调整不仅保证实际输出值与标签误差尽量小，而且使同类样本距离更近、非同类样本距离更远。沿着这个目标不断更新迭代，从而使权值能更加快速地逼近最优值，当样本量不足或训练迭代次数不多时可有效地提高生成图像的质量。 在FDCGAN的思想下，本文同时提出两种模型结构，分别为单流网络与分流网络，网络模型结构如 图1 图1 所示。单流网络模型不改变原本对抗训练的思想，仅提取网络模型中特定层的输出信息，用作Fisher准则的计算。而分流网络模型在原本对抗训练思想的基础上，再引入分类思想，将网络层中对抗训练的输出和分类的输出拆分为两个单独的处理流，仅提取分类层的输出用作Fisher准则的计算。 2.1　深度卷积生成对抗网络（DCGAN） 2 Radford等人 ［ 10 10 ］ 首次结合CNN的思想对GAN模型进行改进，提出了深度卷积生成对抗网络（Deep Convolutional Generative Adversarial Networks， DCGAN）。其在图像生成方面具有较好的效果。 DCGAN是一种无监督学习模型，模型层面采用了全卷积结构，提高了网络提取特征的能力。由于考虑到图像利用池化层进行下采样会丢失部分信息，因此利用卷积层替代池化层 ［ 14 14 ］ ，其中，G网络取消全连接层，直接使用卷积层连接G网络的输出层和D网络的输入层；G网络除了最后一层层使用Tahn作为激活函数，其余层均采用ReLU激活函数。D网络与G网络模型结构基本对称，所有层采用LeakyReLU激活函数。此外，除了G网络的输出层及其对应的D网络的输入层外，其他层上都采用了Batch Normalization（BN） ［ 15 15 ］ ，用于处理初始化不理想导致的训练问题。 DCGAN将更多的注意力放在应用上，这使DCGAN能够应用到图像生成、风格迁移和监督任务等多个领域 ［ 16 16 ］ 。但DCGAN仍然存在某些形式的模型不稳定性，当样本量减少或者迭代次数不多时，其生成图像的质量并不理想。 2.2　FDCGAN-单流网络 2 如 图2 图2 所示，单流网络在保持原判别模型的基础上，在其模型最终输出之前添加一层全连接网络，网络层的神经元个数为数据集类别数 n ，并且采用LeakyRuLU作为激活函数得到输出 <math id="M1"><msub><mrow><mi mathvariant="bold-italic">h</mi></mrow><mrow><mi>w</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi mathvariant="normal">F</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">e</mi><msub><mrow><mi mathvariant="normal">r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><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=38113498&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113479&type= 18.79599953 3.89466691 ： <math display="block" id="M2"><msub><mrow><mi mathvariant="bold-italic">h</mi></mrow><mrow><mi>w</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi mathvariant="normal">F</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">e</mi><msub><mrow><mi mathvariant="normal">r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi mathvariant="bold-italic">x</mi><mo>></mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mi>c</mi><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi mathvariant="bold-italic">x</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=38113521&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113516&type= 38.94666672 9.31333351 ， 式中： w 为各层单元之间的连接参数，即权值； b 为偏置项； c 为常数，通常为0.01。 此外，模型的最终输出采用Sigmoid作为激活函数得到输出 <math id="M3"><msub><mrow><mi mathvariant="bold-italic">h</mi></mrow><mrow><mi>w</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi mathvariant="normal">f</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">l</mi></mrow></msub><mo 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http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113555&type= 5.84200001 3.21733332 ，（ <math id="M7"><msup><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo></mrow></msup><mo>,</mo><msup><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113573&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113567&type= 14.30866718 4.23333359 ｝，它们分别属于 n 个类别， <math id="M8"><msup><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113583&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113579&type= 4.57200003 3.89466691 是样本 <math id="M9"><msup><mrow><mi 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mathvariant="normal">1</mn></mrow><mrow><mi>m</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113632&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113629&type= 46.31266785 9.05933285 ， 式中： <math id="M14"><msub><mrow><mi mathvariant="bold-italic">h</mi></mrow><mrow><mi>w</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi mathvariant="normal">F</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113649&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113636&type= 20.48933220 4.65666676 为网络特定层的输出，作为Fisher模块的输入； <math id="M15"><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113660&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113655&type= 6.34999990 3.13266683 为第 i 类样本的均值，即： <math display="block" id="M16"><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi mathvariant="bold-italic">h</mi></mrow><mrow><mi>w</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi mathvariant="normal">F</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo></mrow></mrow><mrow><mi>n</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113671&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113666&type= 38.60800171 13.20800018 . 因此，当判别器训练真实样本时，单流网络判别模型的代价函数模型为： <math display="block" id="M17"><mtable><mtr><mtd><mi>J</mi><mo>=</mo><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>γ</mi><msub><mrow><mi>J</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mi>η</mi><msub><mrow><mi>J</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>m</mi></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>m</mi></mrow></munderover></mstyle><mrow><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">h</mi></mrow><mrow><mi>w</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi mathvariant="normal">f</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">l</mi></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced><mo>+</mo></mrow></mtd></mtr><mtr><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi>γ</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>m</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</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">h</mi></mrow><mrow><mi>w</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi mathvariant="normal">F</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">e</mi><msub><mrow><mi mathvariant="normal">r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo></mrow></mrow></mtd></mtr><mtr><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi>η</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>m</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mrow></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113688&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113681&type= 57.40399933 36.32199860 ， 式中： <math id="M18"><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113702&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113694&type= 3.89466691 3.72533321 为MSE（Mean Squared Error）均方误差代价函数； <math id="M19"><msub><mrow><mi mathvariant="bold-italic">h</mi></mrow><mrow><mi>w</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi mathvariant="normal">f</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">l</mi></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113709&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113707&type= 18.79599953 4.65666676 为网络层的最终输出，即判别器对图像真假的判断； <math id="M20"><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113720&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113714&type= 3.64066648 3.72533321 是判别器鉴别真实样本时的标签，即为1； <math id="M21"><mi mathvariant="normal">γ</mi><mi mathvariant="normal">和</mi><mi mathvariant="normal">η</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113729&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113724&type= 7.45066643 3.80999994 为权重，取值范围在0~1之间；总体代价函数为J。 2.3　FDCGAN-分流网络 2 如 图3 图3 所示，分流网络共享经典DCGAN中判别模型所提取的特征，在输出结果时拆分成两个单独的处理流。其中一条支线连接神经元个数为1的线性层，并且采用Sigmoid作为激活函数得到输出 <math id="M22"><msub><mrow><mi mathvariant="bold-italic">h</mi></mrow><mrow><mi>w</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi mathvariant="normal">f</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">l</mi></mrow></msub><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=38113741&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113735&type= 16.42533302 4.48733330 ；另一条支线连接神经元个数为类别数 n 的线性层，并且采用Softmax作为激活函数得到输出 <math id="M23"><msub><mrow><mi mathvariant="bold-italic">h</mi></mrow><mrow><mi>w</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi mathvariant="normal">F</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">e</mi><msub><mrow><mi mathvariant="normal">r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113750&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113747&type= 19.47333336 4.82600021 ： <math display="block" id="M24"><msub><mrow><mi mathvariant="bold-italic">h</mi></mrow><mrow><mi>w</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi mathvariant="normal">F</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">e</mi><msub><mrow><mi mathvariant="normal">r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msup><mrow><mi>e</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msup></mrow><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>e</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msup></mrow></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113759&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113755&type= 33.18933105 13.88533401 . 因此，当训练真实样本时，分流网络判别模型的代价函数模型如 式（7） 式（7） 所示： <math display="block" id="M25"><mtable><mtr><mtd><mi>J</mi><mo>=</mo><mi>α</mi><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>β</mi><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi>γ</mi><msub><mrow><mi>J</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mi>η</mi><msub><mrow><mi>J</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>m</mi></mrow></mfrac><mi>α</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>m</mi></mrow></munderover></mstyle><mrow><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">h</mi></mrow><mrow><mi>w</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi mathvariant="normal">f</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">l</mi></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced><mo>+</mo></mrow></mtd></mtr><mtr><mtd><mi>β</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mi>i</mi></mrow></msub><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">g</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">h</mi></mrow><mrow><mi>w</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi mathvariant="normal">F</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">e</mi><msub><mrow><mi mathvariant="normal">r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mo>+</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi>γ</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>m</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</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">h</mi></mrow><mrow><mi>w</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi mathvariant="normal">F</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">e</mi><msub><mrow><mi mathvariant="normal">r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo></mrow></mrow></mtd></mtr><mtr><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi>η</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>m</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mrow></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113796&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113791&type= 57.40399933 46.73600006 ， 式中： <math id="M26"><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113806&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113802&type= 3.89466691 3.72533321 为交叉熵损失函数； <math id="M27"><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113815&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113811&type= 2.96333337 4.57200003 指标签的真实概率分布，即为1或0； <math id="M28"><mi>α</mi><mi mathvariant="normal">和</mi><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113826&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113820&type= 7.61999989 3.89466691 为权重，一般赋予 <math id="M29"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113835&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113830&type= 1.94733346 2.79399991 较大权重，赋予 <math id="M30"><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114058&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114053&type= 2.28600001 3.64066648 较小权重。 2.4　基于判别信息的深度卷积算法 2 当使用 <math id="M31"><msub><mrow><mi>J</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113852&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113848&type= 2.87866688 3.72533321 作为代价函数进行梯度算法计算时，每一步迭代，使得样本预测值到样本所属类别的平均值更小。当使用 <math id="M32"><msub><mrow><mi>J</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113862&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113857&type= 2.87866688 3.72533321 作为代价函数进行梯度算法计算时，每一步迭代，使得不同类别间的距离更大 ［ 13 13 ］ 。 为了使网络每层学习到的特征更有利于生成更高质量的图像，本文提出加入类内类间约束的代价函数模型，即： <math display="block" id="M33"><mi>J</mi><mo>=</mo><mi>R</mi><mo>+</mo><mi>γ</mi><msub><mrow><mi>J</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mi>η</mi><msub><mrow><mi>J</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113871&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113867&type= 28.02466965 4.57200003 ， 式中 R 为判别器在训练真实样本时的代价函数。 在考虑误差的同时，使样本类内间距小，类间间距大。这样，在调整权值时，能使每层权值考虑到类别信息的因素，从而使得当样本数量较少或训练迭代次数较少时，权值能更快地接近最优值。 用BP算法调整权值时，关键的问题是求得输出单元的残差，每项子函数可分别求解其对应的输出单元的残差 ［ 13 13 ］ 。对于类内约束目标函数 <math id="M34"><msub><mrow><mi>J</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113882&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113876&type= 2.87866688 3.72533321 ，其输出层每个输出单元的残差计算公式为： <math display="block" id="M35"><mtable><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">δ</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mi>l</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mfrac><mrow><mo>∂</mo><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mo>∂</mo><msubsup><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mi>l</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></mfrac><mo>=</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mo>∂</mo><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mo>∂</mo><msubsup><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mi>l</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></mfrac><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>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>m</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</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">h</mi></mrow><mrow><mi>w</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi mathvariant="normal">F</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mrow></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113890&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113885&type= 59.68999863 19.72733307 ， 式中： <math id="M36"><msup><mrow><msub><mrow><mi mathvariant="bold-italic">δ</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mi>l</mi><mo stretchy="false">)</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113899&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113894&type= 6.77333355 3.97933316 为 <math id="M37"><msub><mrow><mi mathvariant="bold-italic">h</mi></mrow><mrow><mi>w</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi mathvariant="normal">F</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113907&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113903&type= 22.01333237 4.65666676 与实际标签直接的差距（ nl 表示输出层）； <math id="M38"><msup><mrow><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mi>l</mi><mo stretchy="false">)</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113916&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113912&type= 7.11199999 3.97933316 表示输出层第 i 单元输入加权和，即： <math display="block" id="M39"><msubsup><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi><mi>l</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">b</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113923&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113921&type= 39.70866776 8.97466660 ， 式中： <math id="M40"><msup><mrow><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113936&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113929&type= 7.28133297 3.97933316 为第 l 层中第 j 个单元与第 l +1层中第 i 个单元直接的连接参数即连接权， s 代表网络中单元个数， sl 表示第 l 层的节点数， <math id="M41"><msup><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113946&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113941&type= 8.89000034 3.97933316 表示第 l 层中第 i 个单元的偏置。 对于单流网络，由 式（9） 式（9） 继续推导得到： <math display="block" id="M42"><mtable><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">δ</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mi>l</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mfrac><mrow><mo>∂</mo><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mo>∂</mo><msubsup><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mi>l</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></mfrac><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>m</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">h</mi></mrow><mrow><mi>w</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi mathvariant="normal">F</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">e</mi><msub><mrow><mi mathvariant="normal">r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo></mrow><mo>·</mo></mrow></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">h</mi></mrow><mrow><mi>w</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi mathvariant="normal">F</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">e</mi><msub><mrow><mi mathvariant="normal">r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>'</mi><mo>-</mo><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mi>'</mi><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>m</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mi>l</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo></mrow><mo>·</mo><mi>k</mi><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></mfenced></mrow><mo>,</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113958&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113951&type= 71.37400055 26.24666595 式中：当 <math id="M43"><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113967&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113962&type= 6.26533365 3.04800010 > 0时， k =1；当 <math id="M44"><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113975&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113971&type= 6.26533365 3.04800010 ≤0时， k = c 。 而对于分流网络，由 式（9） 式（9） 继续推导得到： <math display="block" id="M45"><mtable><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">δ</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mi>l</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mfrac><mrow><mo>∂</mo><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mo>∂</mo><msubsup><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mi>l</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></mfrac><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>m</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">h</mi></mrow><mrow><mi>w</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi mathvariant="normal">F</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">e</mi><msub><mrow><mi mathvariant="normal">r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo></mrow><mo>·</mo></mrow></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">h</mi></mrow><mrow><mi>w</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi mathvariant="normal">F</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">e</mi><msub><mrow><mi mathvariant="normal">r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>'</mi><mo>-</mo><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mi>'</mi><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>m</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mi>l</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo></mrow><mo>·</mo><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mi>l</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>·</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mi>l</mi><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">)</mo><mo>-</mo></mrow></mtd></mtr><mtr><mtd><mfrac><mrow><msubsup><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mi>l</mi><mo 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stretchy="false">)</mo></mrow></msubsup><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo></mrow><mo>·</mo><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mi>l</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>·</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mi>l</mi><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></mfenced></mrow><mo>.</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113983&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113979&type= 72.30532837 47.15933609 在 式（11） 式（11） 和 式（12） 式（12） 中，为了避免推导过程冗长，使用 <math id="M46"><msup><mrow><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mi>l</mi><mo stretchy="false">)</mo></mrow></msup><mi mathvariant="normal">代替</mi><msub><mrow><mi mathvariant="bold-italic">h</mi></mrow><mrow><mi>w</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi mathvariant="normal">F</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">e</mi><msub><mrow><mi mathvariant="normal">r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo 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http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38113997&type= 22.69066620 4.14866638 。 对于类间约束目标函数 <math id="M48"><msub><mrow><mi>J</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114009&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114005&type= 2.87866688 3.72533321 ，其输出层每个输出单元的残差计算公式为： <math display="block" id="M49"><mtable><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">δ</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mi>l</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mfrac><mrow><mo>∂</mo><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mo>∂</mo><msubsup><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mi>l</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>∂</mo></mrow><mrow><mo>∂</mo><msubsup><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mi>l</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></mfrac><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>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>m</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mo>=</mo></mrow></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mo>·</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mi>'</mi><mo>-</mo><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mi>'</mi><mo stretchy="false">]</mo><mo>=</mo></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mo>·</mo><mfenced open="[" close="" separators="|"><mrow><mi>n</mi><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup><mo>-</mo><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mfenced><mo>-</mo></mrow></mfenced></mtd></mtr><mtr><mtd><mfenced open="" close="]" separators="|"><mrow><mi>n</mi><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msup><mo>·</mo><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mfenced></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114017&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114014&type= 75.94599915 37.93066406 在模型中，每项子函数求得最后一层的残差以后，就可以通过BP算法进行迭代并求得全部权值。 式（8） 式（8） 中，针对不同的数据集和不同的网络结构，参数 <math id="M50"><mi>γ</mi><mo>,</mo><mi>η</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114027&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114021&type= 5.58799982 3.64066648 的数值选择略有不同。通常选择范围在0~1之间。本文针对FDCGAN-单流网络，选取 <math id="M51"><mi>γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114066&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114062&type= 1.94733346 3.64066648 =0.009， <math id="M52"><mi>η</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114074&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114070&type= 1.86266661 3.64066648 =0.009；针对FDCGAN-分流网络，选取 <math id="M53"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114049&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114045&type= 1.94733346 2.79399991 =0.999， <math id="M54"><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114058&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114053&type= 2.28600001 3.64066648 =0.001， <math id="M55"><mi>γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114066&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114062&type= 1.94733346 3.64066648 =0.002， <math id="M56"><mi>η</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114074&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114070&type= 1.86266661 3.64066648 =0.002。 3 实验结果及分析 1 在后续实验中，将本文所提出的模型FDCGAN与DCGAN，ACGAN，AAE，BGAN，LSGAN，WGAN进行对比，可发现：FDCGAN的FID值 ［ 17 17 ］ 大部分情况下明显低于其他方法。此外，本文还将此方法增加在上述所提出的5种先进模型中，以验证本文方法的泛化能力，可发现：改进后的模型的FID值均明显低于原方法。实验结果表明：当样本量不足或训练迭代次数较少时，生成器也可达到比较理想的效果。 3.1　实验数据与评价指标 2 为了验证本文提出的FDCGAN算法的有效性，在MNIST数据集和Fashion-MNIST数据集上进行对比实验。MNIST手写字数据集，总计60 000个训练样本和10 000个测试样本，其中类别标签为数字0~9。Fashion-MNIST数据集，共记60 000个训练样本以及10 000个测试样本。不同于MNIST手写字数据集，Fashion-MNIST标签包含如：T恤、裙子、运动鞋、包等10个不同的类别。 在两种数据集上，本文采用FID（Fréchet Inception Distance） ［ 17 17 ］ 作为GAN网络生成图像的评价指标。该方法将生成图片放入分类器中，并且提取了输出层之前的全连接层进行计算。当真实样本分布和生成样本分布越接近时，GAN网络性能越好，反之亦然。FID是生成图像与真实图像在特征层之间距离的度量，其公式如式（14）： <math display="block" id="M57"><mi>F</mi><mi>I</mi><mi>D</mi><mo>=</mo><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mrow><mi>g</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>t</mi><mi>r</mi><mo stretchy="false">(</mo><munder><mo movablelimits="false">∑</mo><mrow><mi mathvariant="bold-italic">r</mi></mrow></munder><mrow><munder><mo movablelimits="false">∑</mo><mrow><mi mathvariant="bold-italic">g</mi></mrow></munder><mrow><mo>-</mo><mn mathvariant="normal">2</mn></mrow></mrow><mo stretchy="false">(</mo><munder><mo movablelimits="false">∑</mo><mrow><mi mathvariant="bold-italic">r</mi></mrow></munder><mrow><munder><mo movablelimits="false">∑</mo><mrow><mi mathvariant="bold-italic">g</mi></mrow></munder><mrow><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup></mrow></mrow><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114084&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114078&type= 70.78133392 8.04333401 ， （14） 其中：记真实样本和生成样本特征均值分别为 <math id="M58"><msub><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mrow><mi>r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114092&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114088&type= 3.38666677 4.57200003 和 <math id="M59"><msub><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mrow><mi>g</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114100&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114097&type= 3.55599999 4.57200003 ， <math id="M60"><munder><mo movablelimits="false">∑</mo><mrow><mi mathvariant="bold-italic">r</mi></mrow></munder><mrow/></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114110&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114105&type= 5.92666674 6.60400009 和 <math id="M61"><munder><mo movablelimits="false">∑</mo><mrow><mi mathvariant="bold-italic">g</mi></mrow></munder><mrow/></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114118&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114113&type= 5.92666674 7.11199999 分别为两者的协方差矩阵。计算得到的FID数值越大，表明真实样本分布与生成样本分布关系越疏远，生成图像的质量越低。经实践证明：FID相较于其他评价指标具有较好的鲁棒性，也可以解决模式坍塌的问题 ［ 14 14 ］ 。 在实验中，本文使用Pytorch深度学习框架。模型训练的学习率为0.002，使用自适应矩估计（Adaptive Moment Estimation，Adam）进行优化，batch_size设置为20，同时将原始图像大小调整为32×32作为模型的输入。在计算FID评价指标时，使用pytorch_fid接口，选用生成器随机生成的1 400张图像和测试集中前1 400张图像作为接口输入。实验数据均为3次随机实验取均值。 3.2　训练样本减少时生成图片质量的情况 2 本次实验采用MNIST数据集，将训练样本从10 000个依次增加到60 000个，对经典的DCGAN和FDCGAN进行对比实验，训练次数均为25次。其中，最大提高量为DCGAN的生成图像的质量减去FDCGAN-单流网络和FDCGAN-分流网络中的最小值。观察 表1 表1 可发现：当训练样本只有10 000时，最高提高量较高，即FDCGAN方法可有效提升生成图像的质量；但当训练样本只有20 000时，本文方法与对比方法效果接近；当训练样本多于30 000时，最高提高量较小，FDCGAN方法均小幅度优于传统方法。 注： 黑色加粗为单行指标最好结果。 3 Training Sample 3 Experimental method 3 Maximum Improvement 2 DCGAN FDCGAN FDCGAN single-stream network shunt network 10 000 99.79 91.76 95.17 8.03 20 000 92.25 92.14 92.42 0.11 30 000 88.39 83.57 83.15 5.24 40 000 80.13 79.72 77.38 2.75 50 000 78.08 73.84 78.81 4.24 60 000 75.42 74.73 73.70 1.72 在标签样本量减少的情况下，FDCGAN方法可比不加判别约束的经典方法达到更好的效果。因此，在带标签样本量不太充足时，FDCGAN模型对于生成图像质量提升显著。 3.3　训练次数不同时生成图片质量的情况 2 在MNIST数据集上实验结果如 表2 表2 所示，其中，提高量为训练次数为15时的FID值与训练次数为35时的FID值之差。观察实验数据，可以看出，FDCGAN-单流模型和FDCGAN-分流模型在FID指标上优于原方法以及其他先进的方法。同时，观察提高量数据可以发现，FDCGAN方法的提高量较小，即：在FDCGAN模型训练时，生成器与判别器性能波动幅度不大。因此，FDCGAN方法能够在训练迭代次数较少时就达到比较理想的效果，且模型训练时较为稳定。 注： 黑色加粗为单列指标最好结果。 2 Comparison Method 5 Number of training sessions 2 Improvement 15 20 25 30 35 AAE ［ 11 11 ］ 96.34 <math id="M64"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 0.22 94.34 <math id="M65"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 1.31 95.71 <math id="M66"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 0.41 93.64 <math id="M67"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 1.35 92.50 <math id="M68"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 0.25 3.83 BGAN ［ 9 9 ］ 203.88 <math id="M69"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 0.43 180.43 <math id="M70"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 1.08 168.45 <math id="M71"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 1.61 161.91 <math id="M72"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 1.05 136.67 <math id="M73"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 0.72 67.21 LSGAN ［ 6 6 ］ 90.60 <math id="M74"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 0.20 90.61 <math id="M75"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 0.12 84.30 <math id="M76"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 0.20 83.59 <math id="M77"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 1.10 82.26 <math id="M78"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 0.63 8.34 ACGAN ［ 12 12 ］ 86.48 <math id="M79"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 0.41 84.78 <math id="M80"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 0.95 80.19 <math id="M81"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 1.15 76.90 <math id="M82"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 0.89 73.92 <math id="M83"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 0.60 12.55 WGAN ［ 7 7 ］ 152.60 <math id="M84"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 2.35 149.58 <math id="M85"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 0.01 125.97 <math id="M86"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 0.05 120.23 <math id="M87"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 0.39 124.03 <math id="M88"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 1.03 28.57 DCGAN ［ 10 10 ］ 85.47 <math id="M89"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 0.57 81.68 <math id="M90"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 0.44 84.06 <math id="M91"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 0.78 80.71 <math id="M92"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 0.98 81.77 <math id="M93"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 0.65 3.7 FDCGAN single-stream network 81.21 <math id="M94"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 0.38 77.51 <math id="M95"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 0.43 74.73 <math id="M96"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 1.39 76.05 <math id="M97"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 1.01 75.94 <math id="M98"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 0.58 5.27 FDCGAN shunt network 80.36 <math id="M99"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 0.21 78.72 <math id="M100"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 1.16 73.70 <math id="M101"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 1.24 72.16 <math id="M102"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 0.58 73.20 <math id="M103"><mo>±</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114430&type= 2.53999996 2.45533323 0.01 7.16 在Fashion-MNIST数据集上实验结果如 图4 图4 所示，可以发现：当训练次数为5时，LSGAN模型的FID最小，其次为DCGAN模型；当训练次数为10时，FDCGAN模型效果逐渐提升；当训练次数大于等于15时，FDCGAN模型的FID值最小。观察FDCGAN模型数据走势，可发现：FDCGAN模型的FID值总体呈现下降趋势，且收敛时较为稳定。 由此验证了FDCGAN方法的有效性，即：在反向传播时，基于改进Fisher准则的模型能够使网络权值能更加快速地逼近最优值，当样本量不足或训练迭代次数不多时可有效地提高生成图像的质量。 3.4　模型效果展示 2 如 图5 图5 所示，本文所提出的方法不仅在评价指标上取得良好的效果，在视觉观感上也有较为明显的提升。本次实验采用MNIST库和Fashion-MNIST库，针对经典DCGAN模型与FDCGAN模型的生成图像进行效果展示，训练次数均为25次。观察效果图可以发现：经典DCGAN模型所生成的图像存在图像模糊、图像所属类别不确定等问题。然而，由于FDCGAN-单流网络采用了基于Fisher准则的约束，在计算类内类间距离时考虑到了类别信息和标签，因而其生成的图像更有边界感，不易产生图像所属类别模糊的情况。FDCGAN-分流网络在更新迭代权重时则不止考虑了Fisher约束，还考虑交叉熵损失函数的约束，使模型考虑到了分类的性能，因此分流网络模型同样能够在一定程度上提升图像的质量，使其生成的图像所属类别更确切。 3.5　方法的泛化性测试 2 为验证本文方法的有效性以及其泛化能力，将基于改进Fisher准则的单流网络与分流网络方法引入AAE，BGAN，LSGAN，WGAN，DCGAN，ACGAN等先进模型中。 以MNIST数据集和Fashion-MNIST数据集作为训练数据，各网络的性能如 图6 图6 ~ 图7 图7 所示。其中，图例中注明了原方法、单流网络（ <math id="M104"><mi>γ</mi><mo>,</mo><mi>η</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114482&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114479&type= 5.58799982 3.64066648 ），分流网络（ <math id="M105"><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>γ</mi><mo>,</mo><mi>η</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114494&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38114488&type= 13.12333298 3.64066648 ）。可以看出，对各网络增加Fisher方法后，其生成器所生成的图像质量均有一定程度的提升。 4 结 论 1 （1）本文提出了FDCGAN方法，该方法结合网络特定层输出的类别信息和标签计算出类内距离和类间距离，在权值的迭代调整时既考虑误差的最小化，同时又让样本保持类内距离小，类间距离大，从而使权值能更加快速地逼近最优值，改善了判别器与生成器的性能，最终使得生成器在样本量不足或训练次数较少的情况下能够生成更高质量的图像。 （2）在FDCGAN的思想下，本文提出两种模型结构，分别为单流网络和分流网络。单流网络采用了基于Fisher准则约束的代价函数，而分流网络则在此基础上又加入了交叉熵损失，使网络模型在互相对抗训练时的同时考虑到分类的性能。 （3）在MNIST库和Fashion-MNIST库中进行实验，结果表明本文方法在指标FID上优于其他方法。并且在视觉观感上，由FDCGAN模型所生成的图像同样优于其他方法。 （4）本文将基于Fisher准则约束的方法加到先进的GANs模型中，验证了FDCGAN方法的有效性和较强的泛化能力。 参考文献 陈佛计 ， 朱枫 ， 吴清潇 ， 等 . 生成对抗网络及其在图像生成中的应用研究综述 ［J］. 计算机学报 ， 2021 ， 44 （ 2 ）： 347 - 369 . doi: 10.11897/SP.J.1016.2021.00347 http://dx.doi.org/10.11897/SP.J.1016.2021.00347 生成对抗网络及其在图像生成中的应用研究综述 CHEN F J ， ZHU F ， WU Q X ， et al . A survey about image generation with generative adversarial nets ［J］. Chinese Journal of Computers ， 2021 ， 44 （ 2 ）： 347 - 369 . （in Chinese） . doi: 10.11897/SP.J.1016.2021.00347 http://dx.doi.org/10.11897/SP.J.1016.2021.00347 A survey about image generation with generative adversarial nets ge Generation with Generative Adversarial Nets ［J］. Chinese Journal of Computers ， 2021 ， 44 （ 02 ）： 347 - 369 . （in Chinese） . doi: 10.11897/SP.J.1016.2021.00347 http://dx.doi.org/10.11897/SP.J.1016.2021.00347 ge Generation with Generative Adversarial Nets 汪美琴 ， 袁伟伟 ， 张继业 . 生成对抗网络GAN的研究综述 ［J］. 计算机工程与设计 ， 2021 ， 42 （ 12 ）： 3389 - 3395 . 生成对抗网络GAN的研究综述 WANG M Q ， YUAN W W ， ZHANG J Y . Overview of research on generative adversarial network GAN ［J］. Computer Engineering and Design ， 2021 ， 42 （ 12 ）： 3389 - 3395 . （in Chinese） Overview of research on generative adversarial network GAN GOODFELLOW I ， POUGET-ABADIE J ， MIRZA M ， et al . Generative adversarial nets ［J］. 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