{"id":75274,"date":"2026-02-11T21:58:52","date_gmt":"2026-02-11T13:58:52","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/75274.html"},"modified":"2026-02-11T21:58:52","modified_gmt":"2026-02-11T13:58:52","slug":"%e3%80%90%e6%b7%b1%e5%ba%a6%e5%ad%a6%e4%b9%a0%e6%95%99%e7%a8%8b-05_%e7%94%9f%e6%88%90%e6%a8%a1%e5%9e%8bgenerative%e3%80%9123_vae%e5%a6%82%e4%bd%95%e5%9c%a8%e6%bd%9c%e7%a9%ba","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/75274.html","title":{"rendered":"\u3010\u6df1\u5ea6\u5b66\u4e60\u6559\u7a0b\u2014\u201405_\u751f\u6210\u6a21\u578b(Generative)\u301123_VAE\u5982\u4f55\u5728\u6f5c\u7a7a\u95f4\u63d2\u503c\uff1f\u53d8\u5206\u63a8\u65ad\u7684\u6982\u7387\u89c6\u89d2"},"content":{"rendered":"

23_VAE\u5982\u4f55\u5728\u6f5c\u7a7a\u95f4\u63d2\u503c？\u53d8\u5206\u63a8\u65ad\u7684\u6982\u7387\u89c6\u89d2<\/h2>\n

\u672c\u7ae0\u76ee\u6807：\u89e3\u51b3 Autoencoder \u6f5c\u7a7a\u95f4\u4e0d\u8fde\u7eed\u7684\u95ee\u9898\u3002\u5f15\u5165 VAE (Variational Autoencoder)，\u8ba9\u6a21\u578b\u5b66\u4f1a\u5b66\u4e60"\u5206\u5e03"\u800c\u4e0d\u662f"\u70b9"，\u4ece\u800c\u80fd\u591f\u751f\u6210\u5e73\u6ed1\u53d8\u5316\u7684\u56fe\u50cf\u3002<\/p>\n

\n

\u76ee\u5f55<\/h3>\n
  • Autoencoder \u7684\u7f3a\u9677：\u6f5c\u7a7a\u95f4\u4e0d\u8fde\u7eed<\/li>\n
  • VAE \u7684\u6838\u5fc3\u601d\u60f3：\u5b66\u4e60\u5206\u5e03<\/li>\n
  • \u91cd\u53c2\u6570\u5316\u6280\u5de7 (Re-parameterization Trick)<\/li>\n
  • KL \u6563\u5ea6：\u9632\u6b62\u65b9\u5dee\u4e3a 0<\/li>\n
  • \u5b9e\u6218：PyTorch \u5b9e\u73b0 VAE \u751f\u6210\u624b\u5199\u6570\u5b57<\/li>\n
    \n

    1. Autoencoder \u7684\u7f3a\u9677：\u6f5c\u7a7a\u95f4\u4e0d\u8fde\u7eed<\/h3>\n

    \u666e\u901a\u7684 Autoencoder \u628a\u6bcf\u5f20\u56fe\u538b\u7f29\u6210\u8fd9\u91cc\u7684\u4e00\u4e2a\u70b9\u3002<\/p>\n

      \n
    • \u70b9 A \u662f"\u6708\u4eae"，\u70b9 B \u662f"\u534a\u6708"\u3002<\/li>\n
    • \u4f46\u5982\u679c\u4f60\u53d6 A \u548c B \u7684\u4e2d\u70b9 C，\u89e3\u7801\u51fa\u6765\u53ef\u80fd\u662f\u4e00\u56e2\u4e71\u7801\u3002\u56e0\u4e3a\u7f51\u7edc\u6ca1\u89c1\u8fc7\u4e2d\u95f4\u72b6\u6001\u3002<\/li>\n<\/ul>\n

      \u6211\u4eec\u5e0c\u671b\u6f5c\u7a7a\u95f4\u662f\u8fde\u7eed\u7684：\u4ece\u4e2d\u70b9\u91c7\u6837，\u5e94\u8be5\u80fd\u751f\u6210\u4ecb\u4e8e\u6708\u4eae\u548c\u534a\u6708\u4e4b\u95f4\u7684\u56fe\u3002<\/p>\n

      \n

      2. VAE \u7684\u6838\u5fc3\u601d\u60f3：\u5b66\u4e60\u5206\u5e03<\/h3>\n

      VAE \u4e0d\u518d\u8ba9 Encoder \u8f93\u51fa\u4e00\u4e2a\u56fa\u5b9a\u7684 <\/p>\n

      z<\/p>\n

      z<\/p>\n

      <\/span><\/span>z<\/span><\/span><\/span><\/span><\/span>，\u800c\u662f\u8f93\u51fa\u4e00\u4e2a\u9ad8\u65af\u5206\u5e03 <\/p>\n

      N<\/p>\n

      (<\/p>\n

      \u03bc<\/p>\n

      ,<\/p>\n

      \u03c3<\/p>\n

      2<\/p>\n

      )<\/p>\n

      N(\\\\mu, \\\\sigma^2)<\/p>\n

      <\/span><\/span>N<\/span>(<\/span>\u03bc<\/span>,<\/span><\/span>\u03c3<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

        \n
      • \n

        \u03bc<\/p>\n

        \\\\mu<\/p>\n

        <\/span><\/span>\u03bc<\/span><\/span><\/span><\/span><\/span>：\u5747\u503c（\u5927\u6982\u5728\u54ea\u91cc）\u3002<\/li>\n

      • \n

        \u03c3<\/p>\n

        \\\\sigma<\/p>\n

        <\/span><\/span>\u03c3<\/span><\/span><\/span><\/span><\/span>：\u65b9\u5dee（\u4e0d\u786e\u5b9a\u6027\u6709\u591a\u5927）\u3002<\/li>\n<\/ul>\n

        \u7136\u540e\u6211\u4eec\u4ece\u8fd9\u4e2a\u5206\u5e03\u91cc\u968f\u673a\u91c7\u6837\u4e00\u4e2a <\/p>\n

        z<\/p>\n

        z<\/p>\n

        <\/span><\/span>z<\/span><\/span><\/span><\/span><\/span>，\u6254\u7ed9 Decoder\u3002 \u8fd9\u5c31\u8feb\u4f7f Decoder \u5fc5\u987b\u5bf9 <\/p>\n

        \u03bc<\/p>\n

        \\\\mu<\/p>\n

        <\/span><\/span>\u03bc<\/span><\/span><\/span><\/span><\/span> \u9644\u8fd1\u7684\u566a\u70b9\u5177\u6709\u9c81\u68d2\u6027，\u4ece\u800c\u586b\u8865\u4e86\u6f5c\u7a7a\u95f4\u7684\u7a7a\u9699\u3002<\/p>\n

        \"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\"<\/p>\n

        \n

        3. \u91cd\u53c2\u6570\u5316\u6280\u5de7 (Re-parameterization Trick)<\/h3>\n

        \u95ee\u9898\u6765\u4e86："\u968f\u673a\u91c7\u6837"\u8fd9\u4e2a\u64cd\u4f5c\u662f\u4e0d\u53ef\u5bfc\u7684！ \u53cd\u5411\u4f20\u64ad\u4f1a\u5728\u8fd9\u91cc\u65ad\u6389\u3002<\/p>\n

        \u89e3\u51b3\u529e\u6cd5：\u628a\u968f\u673a\u6027\u5265\u79bb\u51fa\u6765\u3002 \u6211\u4eec\u9700\u8981\u91c7\u6837 <\/p>\n

        z<\/p>\n

        \u223c<\/p>\n

        N<\/p>\n

        (<\/p>\n

        \u03bc<\/p>\n

        ,<\/p>\n

        \u03c3<\/p>\n

        2<\/p>\n

        )<\/p>\n

        z \\\\sim N(\\\\mu, \\\\sigma^2)<\/p>\n

        <\/span><\/span>z<\/span><\/span>\u223c<\/span><\/span><\/span><\/span>N<\/span>(<\/span>\u03bc<\/span>,<\/span><\/span>\u03c3<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002 \u6211\u4eec\u53ef\u4ee5\u5148\u91c7\u6837\u4e00\u4e2a\u6807\u51c6\u6b63\u6001\u5206\u5e03 <\/p>\n

        \u03f5<\/p>\n

        \u223c<\/p>\n

        N<\/p>\n

        (<\/p>\n

        0<\/p>\n

        ,<\/p>\n

        1<\/p>\n

        )<\/p>\n

        \\\\epsilon \\\\sim N(0, 1)<\/p>\n

        <\/span><\/span>\u03f5<\/span><\/span>\u223c<\/span><\/span><\/span><\/span>N<\/span>(<\/span>0<\/span>,<\/span><\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002 \u7136\u540e\u4ee4： <\/p>\n

        z<\/p>\n

        =<\/p>\n

        \u03bc<\/p>\n

        +<\/p>\n

        \u03f5<\/p>\n

        \u22c5<\/p>\n

        \u03c3<\/p>\n

        z = \\\\mu + \\\\epsilon \\\\cdot \\\\sigma <\/p>\n

        <\/span><\/span>z<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03bc<\/span><\/span>+<\/span><\/span><\/span><\/span>\u03f5<\/span><\/span>\u22c5<\/span><\/span><\/span><\/span>\u03c3<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

        \u8fd9\u6837，\u5bf9\u4e8e <\/p>\n

        \u03bc<\/p>\n

        \\\\mu<\/p>\n

        <\/span><\/span>\u03bc<\/span><\/span><\/span><\/span><\/span> \u548c <\/p>\n

        \u03c3<\/p>\n

        \\\\sigma<\/p>\n

        <\/span><\/span>\u03c3<\/span><\/span><\/span><\/span><\/span> \u6765\u8bf4，\u64cd\u4f5c\u53d8\u6210\u4e86\u52a0\u6cd5\u548c\u4e58\u6cd5，\u5b8c\u7f8e\u53ef\u5bfc！<\/p>\n

        \n

        4. KL \u6563\u5ea6：\u9632\u6b62\u65b9\u5dee\u4e3a 0<\/h3>\n

        \u5982\u679c\u53ea\u7528\u91cd\u5efa\u635f\u5931，\u6a21\u578b\u4f1a\u503e\u5411\u4e8e\u628a <\/p>\n

        \u03c3<\/p>\n

        \\\\sigma<\/p>\n

        <\/span><\/span>\u03c3<\/span><\/span><\/span><\/span><\/span> \u53d8\u6210 0，\u9000\u5316\u6210\u666e\u901a\u7684 Autoencoder\u3002 \u6211\u4eec\u9700\u8981\u52a0\u4e00\u4e2a\u6b63\u5219\u9879：\u8feb\u4f7f\u5b66\u5230\u7684\u5206\u5e03\u63a5\u8fd1\u6807\u51c6\u6b63\u6001\u5206\u5e03 <\/p>\n

        N<\/p>\n

        (<\/p>\n

        0<\/p>\n

        ,<\/p>\n

        1<\/p>\n

        )<\/p>\n

        N(0, 1)<\/p>\n

        <\/span><\/span>N<\/span>(<\/span>0<\/span>,<\/span><\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

        \u8861\u91cf\u4e24\u4e2a\u5206\u5e03\u5dee\u5f02\u7684\u6307\u6807\u53eb KL \u6563\u5ea6 (Kullback-Leibler Divergence)\u3002 <\/p>\n

        L<\/p>\n

        o<\/p>\n

        s<\/p>\n

        s<\/p>\n

        =<\/p>\n

        L<\/p>\n

        o<\/p>\n

        s<\/p>\n

        s<\/p>\n

        r<\/p>\n

        e<\/p>\n

        c<\/p>\n

        o<\/p>\n

        n<\/p>\n

        +<\/p>\n

        \u03b2<\/p>\n

        \u22c5<\/p>\n

        D<\/p>\n

        K<\/p>\n

        L<\/p>\n

        (<\/p>\n

        N<\/p>\n

        (<\/p>\n

        \u03bc<\/p>\n

        ,<\/p>\n

        \u03c3<\/p>\n

        2<\/p>\n

        )<\/p>\n

        \u2223<\/p>\n

        \u2223<\/p>\n

        N<\/p>\n

        (<\/p>\n

        0<\/p>\n

        ,<\/p>\n

        1<\/p>\n

        )<\/p>\n

        )<\/p>\n

        Loss = Loss_{recon} + \\\\beta \\\\cdot D_{KL}(N(\\\\mu, \\\\sigma^2) || N(0, 1)) <\/p>\n

        <\/span><\/span>L<\/span>oss<\/span><\/span>=<\/span><\/span><\/span><\/span>L<\/span>os<\/span>s<\/span><\/span>reco<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>\u03b2<\/span><\/span>\u22c5<\/span><\/span><\/span><\/span>D<\/span><\/span>K<\/span>L<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>N<\/span>(<\/span>\u03bc<\/span>,<\/span><\/span>\u03c3<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>\u2223\u2223<\/span>N<\/span>(<\/span>0<\/span>,<\/span><\/span>1<\/span>))<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

        \u516c\u5f0f\u5316\u7b80\u540e： <\/p>\n

        D<\/p>\n

        K<\/p>\n

        L<\/p>\n

        =<\/p>\n

        \u2212<\/p>\n

        0.5<\/p>\n

        \u22c5<\/p>\n

        \u2211<\/p>\n

        (<\/p>\n

        1<\/p>\n

        +<\/p>\n

        ln<\/p>\n

        \u2061<\/p>\n

        (<\/p>\n

        \u03c3<\/p>\n

        2<\/p>\n

        )<\/p>\n

        \u2212<\/p>\n

        \u03bc<\/p>\n

        2<\/p>\n

        \u2212<\/p>\n

        \u03c3<\/p>\n

        2<\/p>\n

        )<\/p>\n

        D_{KL} = -0.5 \\\\cdot \\\\sum (1 + \\\\ln(\\\\sigma^2) – \\\\mu^2 – \\\\sigma^2) <\/p>\n

        <\/span><\/span>D<\/span><\/span>K<\/span>L<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>0.5<\/span><\/span>\u22c5<\/span><\/span><\/span><\/span>\u2211<\/span>(<\/span>1<\/span><\/span>+<\/span><\/span><\/span><\/span>ln<\/span>(<\/span>\u03c3<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03bc<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03c3<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

        \n

        5. \u5b9e\u6218：PyTorch \u5b9e\u73b0 VAE \u751f\u6210\u624b\u5199\u6570\u5b57<\/h3>\n

        import<\/span> torch
        \nimport<\/span> torch.<\/span>nn as<\/span> nn
        \nimport<\/span> torch.<\/span>nn.<\/span>functional as<\/span> F<\/p>\n

        class<\/span> VAE<\/span>(<\/span>nn.<\/span>Module)<\/span>:<\/span>
        \n def<\/span> __init__<\/span>(<\/span>self)<\/span>:<\/span>
        \n super<\/span>(<\/span>VAE,<\/span> self)<\/span>.<\/span>__init__(<\/span>)<\/span>
        \n # Encoder<\/span>
        \n self.<\/span>fc1 =<\/span> nn.<\/span>Linear(<\/span>784<\/span>,<\/span> 400<\/span>)<\/span>
        \n self.<\/span>fc21 =<\/span> nn.<\/span>Linear(<\/span>400<\/span>,<\/span> 20<\/span>)<\/span> # mu<\/span>
        \n self.<\/span>fc22 =<\/span> nn.<\/span>Linear(<\/span>400<\/span>,<\/span> 20<\/span>)<\/span> # logvar (\u9884\u6d4blogvar\u66f4\u6570\u503c\u7a33\u5b9a)<\/span><\/p>\n

        # Decoder<\/span>
        \n self.<\/span>fc3 =<\/span> nn.<\/span>Linear(<\/span>20<\/span>,<\/span> 400<\/span>)<\/span>
        \n self.<\/span>fc4 =<\/span> nn.<\/span>Linear(<\/span>400<\/span>,<\/span> 784<\/span>)<\/span><\/p>\n

        def<\/span> encode<\/span>(<\/span>self,<\/span> x)<\/span>:<\/span>
        \n h1 =<\/span> F.<\/span>relu(<\/span>self.<\/span>fc1(<\/span>x)<\/span>)<\/span>
        \n return<\/span> self.<\/span>fc21(<\/span>h1)<\/span>,<\/span> self.<\/span>fc22(<\/span>h1)<\/span><\/p>\n

        def<\/span> reparameterize<\/span>(<\/span>self,<\/span> mu,<\/span> logvar)<\/span>:<\/span>
        \n std =<\/span> torch.<\/span>exp(<\/span>0.5<\/span> *<\/span> logvar)<\/span>
        \n eps =<\/span> torch.<\/span>randn_like(<\/span>std)<\/span>
        \n return<\/span> mu +<\/span> eps *<\/span> std # z = mu + eps * std<\/span><\/p>\n

        def<\/span> decode<\/span>(<\/span>self,<\/span> z)<\/span>:<\/span>
        \n h3 =<\/span> F.<\/span>relu(<\/span>self.<\/span>fc3(<\/span>z)<\/span>)<\/span>
        \n return<\/span> torch.<\/span>sigmoid(<\/span>self.<\/span>fc4(<\/span>h3)<\/span>)<\/span><\/p>\n

        def<\/span> forward<\/span>(<\/span>self,<\/span> x)<\/span>:<\/span>
        \n mu,<\/span> logvar =<\/span> self.<\/span>encode(<\/span>x.<\/span>view(<\/span>–<\/span>1<\/span>,<\/span> 784<\/span>)<\/span>)<\/span>
        \n z =<\/span> self.<\/span>reparameterize(<\/span>mu,<\/span> logvar)<\/span>
        \n return<\/span> self.<\/span>decode(<\/span>z)<\/span>,<\/span> mu,<\/span> logvar<\/p>\n

        # Loss Function<\/span>
        \ndef<\/span> loss_function<\/span>(<\/span>recon_x,<\/span> x,<\/span> mu,<\/span> logvar)<\/span>:<\/span>
        \n BCE =<\/span> F.<\/span>binary_cross_entropy(<\/span>recon_x,<\/span> x.<\/span>view(<\/span>–<\/span>1<\/span>,<\/span> 784<\/span>)<\/span>,<\/span> reduction=<\/span>'sum'<\/span>)<\/span>
        \n # KL Divergence Formula<\/span>
        \n KLD =<\/span> –<\/span>0.5<\/span> *<\/span> torch.<\/span>sum<\/span>(<\/span>1<\/span> +<\/span> logvar –<\/span> mu.<\/span>pow<\/span>(<\/span>2<\/span>)<\/span> –<\/span> logvar.<\/span>exp(<\/span>)<\/span>)<\/span>
        \n return<\/span> BCE +<\/span> KLD<\/p>\n

        \n

        \u4e0b\u4e00\u7ae0\u9884\u544a： VAE \u751f\u6210\u7684\u56fe\u7247\u603b\u662f\u6709\u70b9\u6a21\u7cca（\u56e0\u4e3a Loss \u662f\u7b97\u50cf\u7d20\u5747\u65b9\u5dee）\u3002\u6709\u6ca1\u6709\u529e\u6cd5\u8ba9\u751f\u6210\u7684\u56fe\u7247\u6781\u5176\u903c\u771f，\u8fde\u6bdb\u5b54\u90fd\u6e05\u6670\u53ef\u89c1？ \u6211\u4eec\u9700\u8981\u4e24\u4e2a\u7f51\u7edc\u4e92\u76f8\u6253\u67b6 \u2014\u2014 GAN (\u751f\u6210\u5bf9\u6297\u7f51\u7edc)\u3002<\/p>\n

        \u4e0b\u4e00\u7ae0：24_GAN\u7684\u535a\u5f08\u5982\u4f55\u8fbe\u5230\u7eb3\u4ec0\u5747\u8861？\u751f\u6210\u5bf9\u6297\u7f51\u7edc\u539f\u7406<\/p>\n","protected":false},"excerpt":{"rendered":"

        23_VAE\u5982\u4f55\u5728\u6f5c\u7a7a\u95f4\u63d2\u503c？\u53d8\u5206\u63a8\u65ad\u7684\u6982\u7387\u89c6\u89d2 \u672c\u7ae0\u76ee\u6807：\u89e3\u51b3 Autoencoder \u6f5c\u7a7a\u95f4\u4e0d\u8fde\u7eed\u7684\u95ee\u9898\u3002\u5f15\u5165 VAE (Variational Autoencoder)，\u8ba9\u6a21\u578b\u5b66\u4f1a\u5b66\u4e60\\”\u5206\u5e03\\”\u800c\u4e0d\u662f\\”\u70b9\\”，\u4ece\u800c\u80fd\u591f\u751f\u6210\u5e73\u6ed1\u53d8\u5316\u7684\u56fe\u50cf\u3002 \u76ee\u5f55
        \nAutoencoder \u7684\u7f3a\u9677：\u6f5c\u7a7a\u95f4\u4e0d\u8fde\u7eedVAE \u7684\u6838\u5fc3\u601d\u60f3：\u5b66\u4e60\u5206\u5e03\u91cd\u53c2\u6570\u5316\u6280\u5de7 (Re-parameterization Trick)KL <\/p>\n","protected":false},"author":2,"featured_media":75273,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[50,86],"topic":[],"class_list":["post-75274","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-server","tag-50","tag-86"],"yoast_head":"\n\u3010\u6df1\u5ea6\u5b66\u4e60\u6559\u7a0b\u2014\u201405_\u751f\u6210\u6a21\u578b(Generative)\u301123_VAE\u5982\u4f55\u5728\u6f5c\u7a7a\u95f4\u63d2\u503c\uff1f\u53d8\u5206\u63a8\u65ad\u7684\u6982\u7387\u89c6\u89d2 - \u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.wsisp.com\/helps\/75274.html\" \/>\n<meta property=\"og:locale\" content=\"zh_CN\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u3010\u6df1\u5ea6\u5b66\u4e60\u6559\u7a0b\u2014\u201405_\u751f\u6210\u6a21\u578b(Generative)\u301123_VAE\u5982\u4f55\u5728\u6f5c\u7a7a\u95f4\u63d2\u503c\uff1f\u53d8\u5206\u63a8\u65ad\u7684\u6982\u7387\u89c6\u89d2 - 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