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GAN 基本原理以及数学证明「建议收藏」(通俗理解生成对抗网络GAN)

發布時間：2023/12/19 综合教程 24 生活家

生活随笔收集整理的這篇文章主要介紹了 GAN 基本原理以及数学证明「建议收藏」(通俗理解生成对抗网络GAN) 小編覺得挺不錯的,現在分享給大家,幫大家做個參考.

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@media(min-width:1200px){ .header-inner{width:1160px}
}
@media(min-width:1600px){ .header-inner{width:73%}
}
.site-brand-container{display:flex;flex-shrink:0;padding:0 10px}
.headband{background:rgba(34, 34, 34, 1);height:3px}
.site-meta{flex-grow:1;text-align:center}
@media(max-width:767px){ .site-meta{text-align:center}
}
.brand{border-bottom:none;color:var(--brand-color);display:inline-block;line-height:1.375em;padding:0 40px;position:relative}
.brand:hover{color:var(--brand-hover-color)}
.site-title{font-family:"Lato", "PingFang SC", "Microsoft YaHei", sans-serif;font-size:1.375em;font-weight:normal;margin:0}
.site-subtitle{color:rgba(221, 221, 221, 1);font-size:0.8125em;margin:10px 0}
.use-motion .brand{opacity:0}
.use-motion .site-title, .use-motion .site-subtitle, .use-motion .custom-logo-image{opacity:0;position:relative;top:-10px}
.site-nav-toggle, .site-nav-right{display:none}
@media(max-width:767px){ .site-nav-toggle, .site-nav-right{display:flex;flex-direction:column;justify-content:center}
}
.site-nav-toggle .toggle, .site-nav-right .toggle{color:var(--text-color);padding:10px;width:22px}
.site-nav-toggle .toggle .toggle-line, .site-nav-right .toggle .toggle-line{background:var(--text-color);border-radius:1px}
.site-nav{display:block}
@media(max-width:767px){ .site-nav{clear:both;display:none}
}
.site-nav.site-nav-on{display:block}
.menu{margin-top:20px;padding-left:0;text-align:center}
.menu-item{display:inline-block;list-style:none;margin:0 10px}
@media(max-width:767px){ .menu-item{display:block;margin-top:10px}
.menu-item.menu-item-search{display:none}
}
.menu-item a, .menu-item span.exturl{border-bottom:0;display:block;font-size:0.8125em;transition:border-color 0.2s ease-in-out 0s}
@media (hover: none) { .menu-item a:hover, .menu-item span.exturl:hover{border-bottom-color:rgba(0, 0, 0, 0) !important}
}
.menu-item .fa, .menu-item .fab, .menu-item .far, .menu-item .fas{margin-right:8px}
.menu-item .badge{display:inline-block;font-weight:bold;line-height:1;margin-left:0.35em;margin-top:0.35em;text-align:center;white-space:nowrap}
@media(max-width:767px){ .menu-item .badge{float:right;margin-left:0}
}
.menu-item-active a, .menu .menu-item a:hover, .menu .menu-item span.exturl:hover{background:var(--menu-item-bg-color)}
.use-motion .menu-item{opacity:0}
.book-mark-link{border-bottom:none;display:inline-block;position:fixed;right:30px;top:-10px;transition:0.3s}
@media(max-width:991px){ .book-mark-link{display:none}
}
.book-mark-link::before{color:rgba(232, 112, 78, 1);content:"?";font-size:32px;line-height:1;font-family:"Font Awesome 5 Free";font-weight:900}
.book-mark-link:hover, .book-mark-link-fixed{top:-2px}
.github-corner :hover .octo-arm{animation:560ms ease-in-out octocat-wave}
.github-corner svg{border:0;color:rgba(255, 255, 255, 1);fill:rgba(34, 34, 34, 1);position:absolute;right:0;top:0;z-index:1000}
@media(max-width:991px){ .github-corner{display:none}
.github-corner svg{color:rgba(34, 34, 34, 1);fill:rgba(255, 255, 255, 1)}
.github-corner .github-corner:hover .octo-arm{animation:none}
.github-corner .github-corner .octo-arm{animation:560ms ease-in-out octocat-wave}
}
@-moz-keyframes octocat-wave {
0%,
100%{transform:rotate(0);}
20%,
60%{transform:rotate(-25deg);}
40%,
80%{transform:rotate(10deg);}
}
@-webkit-keyframes octocat-wave {
0%,
100%{transform:rotate(0);}
20%,
60%{transform:rotate(-25deg);}
40%,
80%{transform:rotate(10deg);}
}
@-o-keyframes octocat-wave {
0%,
100%{transform:rotate(0);}
20%,
60%{transform:rotate(-25deg);}
40%,
80%{transform:rotate(10deg);}
}
@keyframes octocat-wave { 0%, 100%{}
20%, 60%{transform:rotate(-25deg)}
40%, 80%{transform:rotate(10deg)}
}
.sidebar{background:rgba(34, 34, 34, 1);bottom:0;position:fixed;top:0}
@media(max-width:991px){ .sidebar{display:none}
}
.sidebar-inner{color:rgba(153, 153, 153, 1);padding:18px 10px;text-align:center}
.cc-license{margin-top:10px;text-align:center}
.cc-license .cc-opacity{border-bottom:none;opacity:0.7}
.cc-license .cc-opacity:hover{opacity:0.9}
.cc-license img{display:inline-block}
.site-author-image{border:1px solid rgba(238, 238, 238, 1);display:block;margin:0 auto;max-width:120px;padding:2px;border-radius:50%}
.site-author-image{transition:transform 1s ease-out}
.site-author-image:hover{transform:rotateY(360deg)}
.site-author-name{color:var(--text-color);font-weight:600;margin:0;text-align:center}
.site-description{color:rgba(153, 153, 153, 1);font-size:0.8125em;margin-top:0;text-align:center}
.links-of-author{margin-top:15px}
.links-of-author a, .links-of-author span.exturl{border-bottom-color:rgba(85, 85, 85, 1);display:inline-block;font-size:0.8125em;margin-bottom:10px;margin-right:10px;vertical-align:middle;transition-delay:0s;transition-duration:0.2s;transition-timing-function:ease-in-out}
.links-of-author a::before, .links-of-author span.exturl::before{background:rgba(124, 255, 154, 1);border-radius:50%;content:" ";display:inline-block;height:4px;margin-right:3px;vertical-align:middle;width:4px}
.sidebar-button{margin-top:15px}
.sidebar-button a{border:1px solid rgba(252, 100, 35, 1);border-radius:4px;color:rgba(252, 100, 35, 1);display:inline-block;padding:0 15px}
.sidebar-button a .fa, .sidebar-button a .fab, .sidebar-button a .far, .sidebar-button a .fas{margin-right:5px}
.sidebar-button a:hover{background:rgba(252, 100, 35, 1);border:1px solid rgba(252, 100, 35, 1);color:rgba(255, 255, 255, 1)}
.sidebar-button a:hover .fa, .sidebar-button a:hover .fab, .sidebar-button a:hover .far, .sidebar-button a:hover .fas{color:rgba(255, 255, 255, 1)}
.links-of-blogroll{font-size:0.8125em;margin-top:10px}
.links-of-blogroll-title{font-size:0.875em;font-weight:600;margin-top:0}
.links-of-blogroll-list{list-style:none;margin:0;padding:0}
#sidebar-dimmer{display:none}
@media(max-width:767px){ #sidebar-dimmer{background:rgba(0, 0, 0, 1);display:block;height:100%;left:100%;opacity:0;position:fixed;top:0;width:100%;z-index:1100}
.sidebar-active+#sidebar-dimmer{opacity:0.7;transform:translateX(-100%);transition:opacity 0.5s}
}
.sidebar-nav{margin:0;padding-bottom:20px;padding-left:0}
.sidebar-nav li{border-bottom:1px solid rgba(0, 0, 0, 0);color:var(--text-color);cursor:pointer;display:inline-block;font-size:0.875em}
.sidebar-nav li.sidebar-nav-overview{margin-left:10px}
.sidebar-nav li:hover{color:rgba(252, 100, 35, 1)}
.sidebar-nav .sidebar-nav-active{border-bottom-color:rgba(252, 100, 35, 1);color:rgba(252, 100, 35, 1)}
.sidebar-nav .sidebar-nav-active:hover{color:rgba(252, 100, 35, 1)}
.sidebar-panel{display:none;overflow-x:hidden;overflow-y:auto}
.sidebar-panel-active{display:block}
.sidebar-toggle{background:rgba(34, 34, 34, 1);bottom:45px;cursor:pointer;height:14px;left:30px;padding:5px;position:fixed;width:14px;z-index:1300}
@media(max-width:991px){ .sidebar-toggle{left:20px;opacity:0.8;display:none}
}
.sidebar-toggle:hover .toggle-line{background:rgba(252, 100, 35, 1)}
.post-toc{font-size:0.875em}
.post-toc ol{list-style:none;margin:0;padding:0 2px 5px 10px;text-align:left}
.post-toc ol>ol{padding-left:0}
.post-toc ol a{transition:all 0.2s ease-in-out 0s}
.post-toc .nav-item{line-height:1.8;overflow:hidden;text-overflow:ellipsis;white-space:nowrap}
.post-toc .nav .nav-child{display:block}
.post-toc .nav .active>.nav-child{display:block}
.post-toc .nav .active-current>.nav-child{display:block}
.post-toc .nav .active-current>.nav-child>.nav-item{display:block}
.post-toc .nav .active>a{border-bottom-color:rgba(252, 100, 35, 1);color:rgba(252, 100, 35, 1)}
.post-toc .nav .active-current>a{color:rgba(252, 100, 35, 1)}
.post-toc .nav .active-current>a:hover{color:rgba(252, 100, 35, 1)}
.site-state{display:flex;justify-content:center;line-height:1.4;margin-top:10px;overflow:hidden;text-align:center;white-space:nowrap}
.site-state-item{padding:0 15px}
.site-state-item:not(:first-child){border-left:1px solid rgba(238, 238, 238, 1)}
.site-state-item a{border-bottom:none}
.site-state-item-count{display:block;font-size:1em;font-weight:600;text-align:center}
.site-state-item-name{color:rgba(153, 153, 153, 1);font-size:0.8125em}
.footer{color:rgba(153, 153, 153, 1);font-size:0.875em;padding:20px 0}
.footer.footer-fixed{bottom:0;left:0;position:absolute;right:0}
.footer-inner{box-sizing:border-box;margin:0 auto;text-align:center;width:calc(100% - 20px)}
@media(min-width:1200px){ .footer-inner{width:1160px}
}
@media(min-width:1600px){ .footer-inner{width:73%}
}
.languages{display:inline-block;font-size:1.125em;position:relative}
.languages .lang-select-label span{margin:0 0.5em}
.languages .lang-select{height:100%;left:0;opacity:0;position:absolute;top:0;width:100%}
.with-love{color:rgba(255, 51, 72, 1);display:inline-block;margin:0 5px;animation:1.33s ease-in-out infinite iconanimate}
.powered-by, .theme-info{display:inline-block}
@-moz-keyframes iconAnimate {
0%,
100%{transform:scale(1);}
10%,
30%{transform:scale(0.9);}
20%,
40%,
60%,
80%{transform:scale(1.1);}
50%,
70%{transform:scale(1.1);}
}
@-webkit-keyframes iconAnimate {
0%,
100%{transform:scale(1);}
10%,
30%{transform:scale(0.9);}
20%,
40%,
60%,
80%{transform:scale(1.1);}
50%,
70%{transform:scale(1.1);}
}
@-o-keyframes iconAnimate {
0%,
100%{transform:scale(1);}
10%,
30%{transform:scale(0.9);}
20%,
40%,
60%,
80%{transform:scale(1.1);}
50%,
70%{transform:scale(1.1);}
}
@keyframes iconAnimate { 0%, 100%{transform:scale(1)}
10%, 30%{transform:scale(0.9)}
20%, 40%, 60%, 80%{transform:scale(1.1)}
50%, 70%{transform:scale(1.1)}
}
.back-to-top{font-size:12px;text-align:center;transition-delay:0s;transition-duration:0.2s;transition-timing-function:ease-in-out}
.back-to-top{background:rgba(0, 0, 0, 0);margin:8px -10px -20px;opacity:0}
.back-to-top.back-to-top-on{cursor:pointer;opacity:0.6}
.back-to-top.back-to-top-on:hover{opacity:0.8}
.reading-progress-bar{background:rgba(232, 112, 78, 1);display:block;height:3px;left:0;position:fixed;width:0;z-index:1500;top:0}
.post-body{font-family:"Lato", "PingFang SC", "Microsoft YaHei", sans-serif;overflow-wrap:break-word;word-wrap:break-word}
@media(min-width:1200px){ .post-body{font-size:1.125em}
}
.post-body .exturl .fa{font-size:0.875em;margin-left:4px}
.post-body .image-caption, .post-body .figure .caption{color:rgba(153, 153, 153, 1);font-size:0.875em;font-weight:bold;line-height:1;margin:-20px auto 15px;text-align:center}
.post-sticky-flag{display:inline-block;transform:rotate(30deg)}
.post-button{margin-top:40px;text-align:center}
.use-motion .post-block, .use-motion .pagination, .use-motion .comments{opacity:0}
.use-motion .post-header{opacity:0}
.use-motion .post-body{opacity:0}
.use-motion .collection-header{opacity:0}
.posts-collapse{margin-left:35px;position:relative}
@media(max-width:767px){ .posts-collapse{margin-left:0;margin-right:0}
}
.posts-collapse .collection-title{font-size:1.125em;position:relative}
.posts-collapse .collection-title::before{background:rgba(153, 153, 153, 1);border:1px solid rgba(255, 255, 255, 1);border-radius:50%;content:" ";height:10px;left:0;margin-left:-6px;margin-top:-4px;position:absolute;top:50%;width:10px}
.posts-collapse .collection-year{font-size:1.5em;font-weight:bold;margin:60px 0;position:relative}
.posts-collapse .collection-year::before{background:rgba(187, 187, 187, 1);border-radius:50%;content:" ";height:8px;left:0;margin-left:-4px;margin-top:-4px;position:absolute;top:50%;width:8px}
.posts-collapse .collection-header{display:block;margin:0 0 0 20px}
.posts-collapse .collection-header small{color:rgba(187, 187, 187, 1);margin-left:5px}
.posts-collapse .post-header{border-bottom:1px dashed rgba(204, 204, 204, 1);margin:30px 0;padding-left:15px;position:relative;transition:border 0.2s ease-in-out 0s}
.posts-collapse .post-header::before{background:rgba(187, 187, 187, 1);border:1px solid rgba(255, 255, 255, 1);border-radius:50%;content:" ";height:6px;left:0;margin-left:-4px;position:absolute;top:0.75em;transition:background 0.2s ease-in-out 0s;width:6px}
.posts-collapse .post-header:hover{border-bottom-color:rgba(102, 102, 102, 1)}
.posts-collapse .post-header:hover::before{background:rgba(34, 34, 34, 1)}
.posts-collapse .post-meta{display:inline;font-size:0.75em;margin-right:10px}
.posts-collapse .post-title{display:inline}
.posts-collapse .post-title a, .posts-collapse .post-title span.exturl{border-bottom:none;color:var(--link-color)}
.posts-collapse .post-title .fa-external-link-alt{font-size:0.875em;margin-left:5px}
.posts-collapse::before{background:rgba(245, 245, 245, 1);content:" ";height:100%;left:0;margin-left:-2px;position:absolute;top:1.25em;width:4px}
.post-eof{background:rgba(204, 204, 204, 1);height:1px;margin:80px auto 60px;text-align:center;width:8%}
.post-block:last-of-type .post-eof{display:none}
.content{padding-top:40px}
@media(min-width:992px){ .post-body{text-align:justify}
}
@media(max-width:991px){ .post-body{text-align:justify}
}
.post-body h1, .post-body h2, .post-body h3, .post-body h4, .post-body h5, .post-body h6{padding-top:10px}
.post-body h1 .header-anchor, .post-body h2 .header-anchor, .post-body h3 .header-anchor, .post-body h4 .header-anchor, .post-body h5 .header-anchor, .post-body h6 .header-anchor{border-bottom-style:none;color:rgba(204, 204, 204, 1);float:right;margin-left:10px;visibility:hidden}
.post-body h1 .header-anchor:hover, .post-body h2 .header-anchor:hover, .post-body h3 .header-anchor:hover, .post-body h4 .header-anchor:hover, .post-body h5 .header-anchor:hover, .post-body h6 .header-anchor:hover{color:inherit}
.post-body h1:hover .header-anchor, .post-body h2:hover .header-anchor, .post-body h3:hover .header-anchor, .post-body h4:hover .header-anchor, .post-body h5:hover .header-anchor, .post-body h6:hover .header-anchor{visibility:visible}
.post-body iframe, .post-body img, .post-body video{margin-bottom:20px}
.post-body .video-container{height:0;margin-bottom:20px;overflow:hidden;padding-top:75%;position:relative;width:100%}
.post-body .video-container iframe, .post-body .video-container object, .post-body .video-container embed{height:100%;left:0;margin:0;position:absolute;top:0;width:100%}
.post-gallery{align-items:center;display:grid;grid-gap:10px;grid-template-columns:1fr 1fr 1fr;margin-bottom:20px}
@media(max-width:767px){ .post-gallery{grid-template-columns:1fr 1fr}
}
.post-gallery a{border:0}
.post-gallery img{margin:0}
.posts-expand .post-header{font-size:1.125em}
.posts-expand .post-title{font-size:1.5em;font-weight:normal;text-align:center;overflow-wrap:break-word;word-wrap:break-word}
.posts-expand .post-title-link{border-bottom:none;color:var(--link-color);display:inline-block;position:relative;vertical-align:top}
.posts-expand .post-title-link::before{background:var(--link-color);bottom:0;content:"";height:2px;left:0;position:absolute;transform:scaleX(0);visibility:hidden;width:100%;transition-delay:0s;transition-duration:0.2s;transition-timing-function:ease-in-out}
.posts-expand .post-title-link:hover::before{transform:scale(1);visibility:visible}
.posts-expand .post-title-link .fa-external-link-alt{font-size:0.875em;margin-left:5px}
.posts-expand .post-meta{color:rgba(153, 153, 153, 1);font-family:"Lato", "PingFang SC", "Microsoft YaHei", sans-serif;font-size:0.75em;margin:3px 0 60px;text-align:center}
.posts-expand .post-meta .post-description{font-size:0.875em;margin-top:2px}
.posts-expand .post-meta time{border-bottom:1px dashed rgba(153, 153, 153, 1);cursor:pointer}
.post-meta .post-meta-item+.post-meta-item::before{content:"|";margin:0 0.5em}
.post-meta-divider{margin:0 0.5em}
.post-meta-item-icon{margin-right:3px}
@media(max-width:991px){ .post-meta-item-icon{display:inline-block}
}
@media(max-width:991px){ .post-meta-item-text{display:none}
}
.post-nav{border-top:1px solid rgba(238, 238, 238, 1);display:flex;justify-content:space-between;margin-top:15px;padding:10px 5px 0}
.post-nav-item{flex:1}
.post-nav-item a{border-bottom:none;display:block;font-size:0.875em;line-height:1.6;position:relative}
.post-nav-item a:active{top:2px}
.post-nav-item .fa{font-size:0.75em}
.post-nav-item:first-child{margin-right:15px}
.post-nav-item:first-child .fa{margin-right:5px}
.post-nav-item:last-child{margin-left:15px;text-align:right}
.post-nav-item:last-child .fa{margin-left:5px}
.rtl.post-body p, .rtl.post-body a, .rtl.post-body h1, .rtl.post-body h2, .rtl.post-body h3, .rtl.post-body h4, .rtl.post-body h5, .rtl.post-body h6, .rtl.post-body li, .rtl.post-body ul, .rtl.post-body ol{direction:rtl;font-family:UKIJ Ekran}
.rtl.post-title{font-family:UKIJ Ekran}
.post-tags{margin-top:40px;text-align:center}
.post-tags a{display:inline-block;font-size:0.8125em}
.post-tags a:not(:last-child){margin-right:10px}
.post-widgets{border-top:1px solid rgba(238, 238, 238, 1);margin-top:15px;text-align:center}
.wp_rating{height:20px;line-height:20px;margin-top:10px;padding-top:6px;text-align:center}
.social-like{display:flex;font-size:0.875em;justify-content:center;text-align:center}
.reward-container{margin:20px auto;padding:10px 0;text-align:center;width:90%}
.reward-container button{background:rgba(0, 0, 0, 0);border:1px solid rgba(252, 100, 35, 1);border-radius:0;color:rgba(252, 100, 35, 1);cursor:pointer;line-height:2;outline:0;padding:0 15px;vertical-align:text-top}
.reward-container button:hover{background:rgba(252, 100, 35, 1);border:1px solid rgba(0, 0, 0, 0);color:rgba(250, 147, 102, 1)}
#qr{padding-top:20px}
#qr a{border:0}
#qr img{display:inline-block;margin:0.8em 2em 0;max-width:100%;width:180px}
#qr p{text-align:center}
.post-copyright{background:var(--card-bg-color);border-left:3px solid rgba(255, 42, 42, 1);list-style:none;margin:2em 0 0;padding:0.5em 1em}
.category-all-page .category-all-title{text-align:center}
.category-all-page .category-all{margin-top:20px}
.category-all-page .category-list{list-style:none;margin:0;padding:0}
.category-all-page .category-list-item{margin:5px 10px}
.category-all-page .category-list-count{color:rgba(187, 187, 187, 1)}
.category-all-page .category-list-count::before{content:" (";display:inline}
.category-all-page .category-list-count::after{content:") ";display:inline}
.category-all-page .category-list-child{padding-left:10px}
.event-list{padding:0}
.event-list hr{background:rgba(34, 34, 34, 1);margin:20px 0 45px}
.event-list hr::after{background:rgba(34, 34, 34, 1);color:rgba(255, 255, 255, 1);content:"NOW";display:inline-block;font-weight:bold;padding:0 5px;text-align:right}
.event-list .event{background:rgba(34, 34, 34, 1);margin:20px 0;min-height:40px;padding:15px 0 15px 10px}
.event-list .event .event-summary{color:rgba(255, 255, 255, 1);margin:0;padding-bottom:3px}
.event-list .event .event-summary::before{animation:1s ease-in-out infinite alternate dot-flash;color:rgba(255, 255, 255, 1);content:"?";display:inline-block;font-size:10px;margin-right:25px;vertical-align:middle;font-family:"Font Awesome 5 Free";font-weight:900}
.event-list .event .event-relative-time{color:rgba(187, 187, 187, 1);display:inline-block;font-size:12px;font-weight:normal;padding-left:12px}
.event-list .event .event-details{color:rgba(255, 255, 255, 1);display:block;line-height:18px;margin-left:56px;padding-bottom:6px;padding-top:3px;text-indent:-24px}
.event-list .event .event-details::before{color:rgba(255, 255, 255, 1);display:inline-block;margin-right:9px;text-align:center;text-indent:0;width:14px;font-family:"Font Awesome 5 Free";font-weight:900}
.event-list .event .event-details.event-location::before{content:"?"}
.event-list .event .event-details.event-duration::before{content:"?"}
.event-list .event-past{background:rgba(245, 245, 245, 1)}
.event-list .event-past .event-summary, .event-list .event-past .event-details{color:rgba(187, 187, 187, 1);opacity:0.9}
.event-list .event-past .event-summary::before, .event-list .event-past .event-details::before{animation:none;color:rgba(187, 187, 187, 1)}
@-moz-keyframes dot-flash {
from{opacity:1;transform:scale(1);}
to{opacity:0;transform:scale(0.8);}
}
@-webkit-keyframes dot-flash {
from{opacity:1;transform:scale(1);}
to{opacity:0;transform:scale(0.8);}
}
@-o-keyframes dot-flash {
from{opacity:1;transform:scale(1);}
to{opacity:0;transform:scale(0.8);}
}
@keyframes dot-flash { 0%{opacity:1;transform:scale(1)}
100%{opacity:0;transform:scale(0.8)}
}
ul.breadcrumb{font-size:0.75em;list-style:none;margin:1em 0;padding:0 2em;text-align:center}
ul.breadcrumb li{display:inline}
ul.breadcrumb li+li::before{content:"/?";font-weight:normal;padding:0.5em}
ul.breadcrumb li+li:last-child{font-weight:bold}
.tag-cloud{text-align:center}
.tag-cloud a{display:inline-block;margin:10px}
.tag-cloud a:hover{color:var(--link-hover-color) !important}
.gt-header a, .gt-comments a, .gt-popup a{border-bottom:none}
.gt-container .gt-popup .gt-action.is--active::before{top:0.7em}
.search-pop-overlay{background:rgba(0, 0, 0, 0);height:100%;left:0;position:fixed;top:0;transition:visibility 0s linear 0.2s, background 0.2s;visibility:hidden;width:100%;z-index:1400}
.search-pop-overlay.search-active{background:rgba(0, 0, 0, 0.3);transition:background 0.2s;visibility:visible}
.search-popup{background:var(--card-bg-color);border-radius:5px;height:80%;left:calc(50% - 350px);position:fixed;top:10%;transform:scale(0);transition:transform 0.2s;width:700px;z-index:1500}
.search-active .search-popup{transform:scale(1)}
@media(max-width:767px){ .search-popup{border-radius:0;height:100%;left:0;margin:0;top:0;width:100%}
}
.search-popup .search-icon, .search-popup .popup-btn-close{color:rgba(153, 153, 153, 1);font-size:18px;padding:0 10px}
.search-popup .popup-btn-close{cursor:pointer}
.search-popup .popup-btn-close:hover .fa{color:rgba(34, 34, 34, 1)}
.search-popup .search-header{background:rgba(238, 238, 238, 1);border-top-left-radius:5px;border-top-right-radius:5px;display:flex;padding:5px}
.search-popup input.search-input{background:rgba(0, 0, 0, 0);border:0;outline:0;width:100%}
.search-popup input.search-input::-webkit-search-cancel-button{display:none}
.search-popup .search-input-container{flex-grow:1;padding:2px}
.search-popup ul.search-result-list{margin:0 5px;padding:0;width:100%}
.search-popup p.search-result{border-bottom:1px dashed rgba(204, 204, 204, 1);padding:5px 0}
.search-popup a.search-result-title{font-weight:bold}
.search-popup .search-keyword{border-bottom:1px dashed rgba(255, 42, 42, 1);color:rgba(255, 42, 42, 1);font-weight:bold}
.search-popup #search-result{display:flex;height:calc(100% - 55px);overflow:auto;padding:5px 25px}
.search-popup #no-result{color:rgba(204, 204, 204, 1);margin:auto}
.popular-posts-header{border-bottom:1px solid rgba(238, 238, 238, 1);display:block;font-size:1.125em;margin-bottom:10px;margin-top:60px}
.popular-posts{padding:0}
.popular-posts .popular-posts-item{margin-left:2em}
.popular-posts .popular-posts-item .popular-posts-title{font-size:0.875em;font-weight:normal;line-height:2.4;margin:0}
mjx-container[jax="CHTML"][display="true"], .has-jax{}
mjx-container+br{display:none}
.header{margin:0 auto;position:relative;width:calc(100% - 20px)}
@media(min-width:1200px){ .header{width:1160px}
}
@media(min-width:1600px){ .header{width:73%}
}
@media(max-width:991px){ .header{width:auto}
}
.header-inner{background:var(--content-bg-color);box-shadow:0 2px 2px rgba(0, 0, 0, 0.12), 0 3px 1px -2px rgba(0, 0, 0, 0.06), 0 1px 5px rgba(0, 0, 0, 0.12);overflow:hidden;padding:0;position:absolute;top:0;width:240px}
@media(min-width:1200px){ .header-inner{width:240px}
}
@media(max-width:991px){ .header-inner{position:relative;width:auto}
}
.main-inner{align-items:flex-start;display:flex;justify-content:space-between}
@media(max-width:991px){ .main-inner{width:auto}
}
.content-wrap{background:var(--content-bg-color);box-shadow:0 2px 2px rgba(0, 0, 0, 0.12), 0 3px 1px -2px rgba(0, 0, 0, 0.06), 0 1px 5px rgba(0, 0, 0, 0.12);box-sizing:border-box;padding:40px;width:calc(100% - 252px)}
@media(max-width:991px){ .content-wrap{padding:20px;width:100%}
}
.header-inner{right:0}
.book-mark-link{left:30px}
.footer-inner{padding-right:260px}
@media(max-width:991px){ .footer-inner{padding-left:0;padding-right:0;width:auto}
}
.site-brand-container{background:rgba(34, 34, 34, 1)}
@media(max-width:991px){ .site-brand-container{box-shadow:0 0 16px rgba(0, 0, 0, 0.5)}
}
.site-meta{padding:20px 0}
.brand{padding:0}
.site-subtitle{margin:10px 10px 0}
.custom-logo-image{margin-top:20px}
@media(max-width:991px){ .custom-logo-image{display:none}
}
@media(min-width:768px) and (max-width:991px){ .site-nav-toggle, .site-nav-right{display:flex;flex-direction:column;justify-content:center}
}
.site-nav-toggle .toggle, .site-nav-right .toggle{color:rgba(255, 255, 255, 1)}
.site-nav-toggle .toggle .toggle-line, .site-nav-right .toggle .toggle-line{background:rgba(255, 255, 255, 1)}
@media(min-width:768px) and (max-width:991px){ .site-nav{display:none}
}
.menu .menu-item{display:block;margin:0}
.menu .menu-item a, .menu .menu-item span.exturl{padding:5px 20px;position:relative;text-align:left;transition-property:background-color}
@media(max-width:991px){ .menu .menu-item.menu-item-search{display:none}
}
.menu .menu-item .badge{background:rgba(204, 204, 204, 1);border-radius:10px;color:rgba(255, 255, 255, 1);float:right;padding:2px 5px;text-shadow:1px 1px rgba(0, 0, 0, 0.1);vertical-align:middle}
.sub-menu{background:var(--content-bg-color);border-bottom:1px solid rgba(221, 221, 221, 1);margin:0;padding:6px 0}
.sub-menu .menu-item{display:inline-block}
.sub-menu .menu-item a, .sub-menu .menu-item span.exturl{background:rgba(0, 0, 0, 0);margin:5px 10px}
.sub-menu .menu-item a:hover, .sub-menu .menu-item span.exturl:hover{background:rgba(0, 0, 0, 0);color:rgba(252, 100, 35, 1)}
.sub-menu .menu-item-active a{border-bottom-color:rgba(252, 100, 35, 1);color:rgba(252, 100, 35, 1)}
.sub-menu .menu-item-active a:hover{border-bottom-color:rgba(252, 100, 35, 1)}
.sidebar{background:var(--body-bg-color);box-shadow:none;margin-top:100%;position:static;width:240px}
@media(max-width:991px){ .sidebar{display:none}
}
.sidebar-toggle{display:none}
.sidebar-inner{background:var(--content-bg-color);box-shadow:0 2px 2px rgba(0, 0, 0, 0.12), 0 3px 1px -2px rgba(0, 0, 0, 0.06), 0 1px 5px rgba(0, 0, 0, 0.12), 0 -1px 0.5px rgba(0, 0, 0, 0.09);box-sizing:border-box;color:var(--text-color);width:240px;opacity:0}
.sidebar-inner.affix{position:fixed;top:12px}
.sidebar-inner.affix-bottom{position:absolute}
.site-state-item{padding:0 10px}
.sidebar-button{border-bottom:1px dotted rgba(204, 204, 204, 1);border-top:1px dotted rgba(204, 204, 204, 1);margin-top:10px;text-align:center}
.sidebar-button a{border:0;color:rgba(252, 100, 35, 1);display:block}
.sidebar-button a:hover{background:none;border:0;color:rgba(227, 70, 3, 1)}
.sidebar-button a:hover .fa, .sidebar-button a:hover .fab, .sidebar-button a:hover .far, .sidebar-button a:hover .fas{color:rgba(227, 70, 3, 1)}
.links-of-author{display:flex;flex-wrap:wrap;margin-top:10px;justify-content:center}
.links-of-author-item{margin:5px 0 0}
.links-of-author-item a, .links-of-author-item span.exturl{box-sizing:border-box;display:inline-block;margin-bottom:0;margin-right:0;max-width:216px;overflow:hidden;padding:0 5px;text-overflow:ellipsis;white-space:nowrap}
.links-of-author-item a, .links-of-author-item span.exturl{border-bottom:none;display:block;text-decoration:none}
.links-of-author-item a::before, .links-of-author-item span.exturl::before{display:none}
.links-of-author-item a:hover, .links-of-author-item span.exturl:hover{background:var(--body-bg-color);border-radius:4px}
.links-of-author-item .fa, .links-of-author-item .fab, .links-of-author-item .far, .links-of-author-item .fas{margin-right:2px}
.links-of-blogroll-item{padding:0}
.back-to-top{background:var(--body-bg-color);margin:-4px -10px -18px}
.back-to-top.back-to-top-on{margin-top:16px}
.content-wrap{box-shadow:initial}
.post-block{background:var(--content-bg-color);box-shadow:0 2px 2px rgba(0, 0, 0, 0.12), 0 3px 1px -2px rgba(0, 0, 0, 0.06), 0 1px 5px rgba(0, 0, 0, 0.12);padding:40px}
.post-block+.post-block{box-shadow:0 2px 2px rgba(0, 0, 0, 0.12), 0 3px 1px -2px rgba(0, 0, 0, 0.06), 0 1px 5px rgba(0, 0, 0, 0.12), 0 -1px 0.5px rgba(0, 0, 0, 0.09);margin-top:12px}
.comments{background:var(--content-bg-color);box-shadow:0 2px 2px rgba(0, 0, 0, 0.12), 0 3px 1px -2px rgba(0, 0, 0, 0.06), 0 1px 5px rgba(0, 0, 0, 0.12), 0 -1px 0.5px rgba(0, 0, 0, 0.09);margin-top:12px;padding:40px}
.tabs-comment{margin-top:1em}
.content{padding-top:initial}
.post-eof{display:none}
.pagination{background:var(--content-bg-color);box-shadow:0 2px 2px rgba(0, 0, 0, 0.12), 0 3px 1px -2px rgba(0, 0, 0, 0.06), 0 1px 5px rgba(0, 0, 0, 0.12), 0 -1px 0.5px rgba(0, 0, 0, 0.09);margin:12px 0 0;padding:10px 0}
.pagination .prev, .pagination .next, .pagination .page-number{margin-bottom:initial;top:initial}
.main{padding-bottom:initial}
.footer{bottom:auto}
.sub-menu{box-shadow:0 2px 2px rgba(0, 0, 0, 0.12), 0 3px 1px -2px rgba(0, 0, 0, 0.06), 0 1px 5px rgba(0, 0, 0, 0.12)}
.sub-menu+.content .post-block{box-shadow:0 2px 2px rgba(0, 0, 0, 0.12), 0 3px 1px -2px rgba(0, 0, 0, 0.06), 0 1px 5px rgba(0, 0, 0, 0.12), 0 -1px 0.5px rgba(0, 0, 0, 0.09);margin-top:12px}
@media(min-width:768px) and (max-width:991px){ .sub-menu+.content .post-block{margin-top:10px}
}
@media(max-width:767px){ .sub-menu+.content .post-block{margin-top:8px}
}
.post-body h1, .post-body h2{border-bottom:1px solid rgba(238, 238, 238, 1)}
.post-body h3{border-bottom:1px dotted rgba(238, 238, 238, 1)}
@media(min-width:768px) and (max-width:991px){ .content-wrap{padding:10px}
.posts-expand .post-button{margin-top:20px}
.post-block{box-shadow:0 2px 2px rgba(0, 0, 0, 0.12), 0 3px 1px -2px rgba(0, 0, 0, 0.06), 0 1px 5px rgba(0, 0, 0, 0.12), 0 -1px 0.5px rgba(0, 0, 0, 0.09);padding:20px}
.post-block+.post-block{margin-top:10px}
.comments{margin-top:10px;padding:10px 20px}
.pagination{margin:10px 0 0}
}
@media(max-width:767px){ .content-wrap{padding:8px}
.posts-expand .post-button{margin:12px 0}
.post-block{box-shadow:0 2px 2px rgba(0, 0, 0, 0.12), 0 3px 1px -2px rgba(0, 0, 0, 0.06), 0 1px 5px rgba(0, 0, 0, 0.12), 0 -1px 0.5px rgba(0, 0, 0, 0.09);padding:12px}
.post-block+.post-block{margin-top:8px}
.comments{margin-top:8px;padding:10px 12px}
.pagination{margin:8px 0 0}
}
.fa, .fab, .fad, .fal, .far, .fas{-moz-osx-font-smoothing:grayscale;-webkit-font-smoothing:antialiased;display:inline-block;font-style:normal;font-variant:normal;text-rendering:auto;line-height:1}
.fa-lg{font-size:1.33333em;line-height:0.75em;vertical-align:-0.0667em}
.fa-xs{font-size:0.75em}
.fa-sm{font-size:0.875em}
.fa-1x{font-size:1em}
.fa-2x{font-size:2em}
.fa-3x{font-size:3em}
.fa-4x{font-size:4em}
.fa-5x{font-size:5em}
.fa-6x{font-size:6em}
.fa-7x{font-size:7em}
.fa-8x{font-size:8em}
.fa-9x{font-size:9em}
.fa-10x{font-size:10em}
.fa-fw{text-align:center;width:1.25em}
.fa-ul{list-style-type:none;margin-left:2.5em;padding-left:0}
.fa-ul>li{position:relative}
.fa-li{left:-2em;position:absolute;text-align:center;width:2em;line-height:inherit}
.fa-border{border:0.08em solid rgba(238, 238, 238, 1);border-radius:0.1em;padding:0.2em 0.25em 0.15em}
.fa-pull-left{float:left}
.fa-pull-right{float:right}
.fa.fa-pull-left, .fab.fa-pull-left, .fal.fa-pull-left, .far.fa-pull-left, .fas.fa-pull-left{margin-right:0.3em}
.fa.fa-pull-right, .fab.fa-pull-right, .fal.fa-pull-right, .far.fa-pull-right, .fas.fa-pull-right{margin-left:0.3em}
.fa-spin{-webkit-animation:fa-spin 2s linear infinite;animation:2s linear infinite fa-spin}
.fa-pulse{-webkit-animation:fa-spin 1s steps(8) infinite;animation:1s steps(8) infinite fa-spin}
@-webkit-keyframes fa-spin {
0%{-webkit-transform:rotate(0deg);transform:rotate(0deg)}
to{-webkit-transform:rotate(1turn);transform:rotate(1turn)}
}
@keyframes fa-spin { 0%{-webkit-transform:rotate(0deg);transform:rotate(0deg)}
100%{-webkit-transform:rotate(1turn);transform:rotate(1turn)}
}
.fa-rotate-90{-ms-filter:"progid:DXImageTransform.Microsoft.BasicImage(rotation=1)";-webkit-transform:rotate(90deg);transform:rotate(90deg)}
.fa-rotate-180{-ms-filter:"progid:DXImageTransform.Microsoft.BasicImage(rotation=2)";-webkit-transform:rotate(180deg);transform:rotate(180deg)}
.fa-rotate-270{-ms-filter:"progid:DXImageTransform.Microsoft.BasicImage(rotation=3)";-webkit-transform:rotate(270deg);transform:rotate(270deg)}
.fa-flip-horizontal{-ms-filter:"progid:DXImageTransform.Microsoft.BasicImage(rotation=0, mirror=1)";-webkit-transform:scaleX(-1);transform:scaleX(-1)}
.fa-flip-vertical{-webkit-transform:scaleY(-1);transform:scaleY(-1)}
.fa-flip-both, .fa-flip-horizontal.fa-flip-vertical, .fa-flip-vertical{-ms-filter:"progid:DXImageTransform.Microsoft.BasicImage(rotation=2, mirror=1)"}
.fa-flip-both, .fa-flip-horizontal.fa-flip-vertical{-webkit-transform:scale(-1);transform:scale(-1)}:root .fa-flip-both, :root .fa-flip-horizontal, :root .fa-flip-vertical, :root .fa-rotate-90, :root .fa-rotate-180, :root .fa-rotate-270{-webkit-filter:none;filter:none}
.fa-stack{display:inline-block;height:2em;line-height:2em;position:relative;vertical-align:middle;width:2.5em}
.fa-stack-1x, .fa-stack-2x{left:0;position:absolute;text-align:center;width:100%}
.fa-stack-1x{line-height:inherit}
.fa-stack-2x{font-size:2em}
.fa-inverse{color:rgba(255, 255, 255, 1)}
.fa-500px:before{content:"?"}
.fa-accessible-icon:before{content:"?"}
.fa-accusoft:before{content:"?"}
.fa-acquisitions-incorporated:before{content:"?"}
.fa-ad:before{content:"?"}
.fa-address-book:before{content:"?"}
.fa-address-card:before{content:"?"}
.fa-adjust:before{content:"?"}
.fa-adn:before{content:"?"}
.fa-adobe:before{content:"?"}
.fa-adversal:before{content:"?"}
.fa-affiliatetheme:before{content:"?"}
.fa-air-freshener:before{content:"?"}
.fa-airbnb:before{content:"?"}
.fa-algolia:before{content:"?"}
.fa-align-center:before{content:"?"}
.fa-align-justify:before{content:"?"}
.fa-align-left:before{content:"?"}
.fa-align-right:before{content:"?"}
.fa-alipay:before{content:"?"}
.fa-allergies:before{content:"?"}
.fa-amazon:before{content:"?"}
.fa-amazon-pay:before{content:"?"}
.fa-ambulance:before{content:"?"}
.fa-american-sign-language-interpreting:before{content:"?"}
.fa-amilia:before{content:"?"}
.fa-anchor:before{content:"?"}
.fa-android:before{content:"?"}
.fa-angellist:before{content:"?"}
.fa-angle-double-down:before{content:"?"}
.fa-angle-double-left:before{content:"?"}
.fa-angle-double-right:before{content:"?"}
.fa-angle-double-up:before{content:"?"}
.fa-angle-down:before{content:"?"}
.fa-angle-left:before{content:"?"}
.fa-angle-right:before{content:"?"}
.fa-angle-up:before{content:"?"}
.fa-angry:before{content:"?"}
.fa-angrycreative:before{content:"?"}
.fa-angular:before{content:"?"}
.fa-ankh:before{content:"?"}
.fa-app-store:before{content:"?"}
.fa-app-store-ios:before{content:"?"}
.fa-apper:before{content:"?"}
.fa-apple:before{content:"?"}
.fa-apple-alt:before{content:"?"}
.fa-apple-pay:before{content:"?"}
.fa-archive:before{content:"?"}
.fa-archway:before{content:"?"}
.fa-arrow-alt-circle-down:before{content:"?"}
.fa-arrow-alt-circle-left:before{content:"?"}
.fa-arrow-alt-circle-right:before{content:"?"}
.fa-arrow-alt-circle-up:before{content:"?"}
.fa-arrow-circle-down:before{content:"?"}
.fa-arrow-circle-left:before{content:"?"}
.fa-arrow-circle-right:before{content:"?"}
.fa-arrow-circle-up:before{content:"?"}
.fa-arrow-down:before{content:"?"}
.fa-arrow-left:before{content:"?"}
.fa-arrow-right:before{content:"?"}
.fa-arrow-up:before{content:"?"}
.fa-arrows-alt:before{content:"?"}
.fa-arrows-alt-h:before{content:"?"}
.fa-arrows-alt-v:before{content:"?"}
.fa-artstation:before{content:"?"}
.fa-assistive-listening-systems:before{content:"?"}
.fa-asterisk:before{content:"?"}
.fa-asymmetrik:before{content:"?"}
.fa-at:before{content:"?"}
.fa-atlas:before{content:"?"}
.fa-atlassian:before{content:"?"}
.fa-atom:before{content:"?"}
.fa-audible:before{content:"?"}
.fa-audio-description:before{content:"?"}
.fa-autoprefixer:before{content:"?"}
.fa-avianex:before{content:"?"}
.fa-aviato:before{content:"?"}
.fa-award:before{content:"?"}
.fa-aws:before{content:"?"}
.fa-baby:before{content:"?"}
.fa-baby-carriage:before{content:"?"}
.fa-backspace:before{content:"?"}
.fa-backward:before{content:"?"}
.fa-bacon:before{content:"?"}
.fa-bahai:before{content:"?"}
.fa-balance-scale:before{content:"?"}
.fa-balance-scale-left:before{content:"?"}
.fa-balance-scale-right:before{content:"?"}
.fa-ban:before{content:"?"}
.fa-band-aid:before{content:"?"}
.fa-bandcamp:before{content:"?"}
.fa-barcode:before{content:"?"}
.fa-bars:before{content:"?"}
.fa-baseball-ball:before{content:"?"}
.fa-basketball-ball:before{content:"?"}
.fa-bath:before{content:"?"}
.fa-battery-empty:before{content:"?"}
.fa-battery-full:before{content:"?"}
.fa-battery-half:before{content:"?"}
.fa-battery-quarter:before{content:"?"}
.fa-battery-three-quarters:before{content:"?"}
.fa-battle-net:before{content:"?"}
.fa-bed:before{content:"?"}
.fa-beer:before{content:"?"}
.fa-behance:before{content:"?"}
.fa-behance-square:before{content:"?"}
.fa-bell:before{content:"?"}
.fa-bell-slash:before{content:"?"}
.fa-bezier-curve:before{content:"?"}
.fa-bible:before{content:"?"}
.fa-bicycle:before{content:"?"}
.fa-biking:before{content:"?"}
.fa-bimobject:before{content:"?"}
.fa-binoculars:before{content:"?"}
.fa-biohazard:before{content:"?"}
.fa-birthday-cake:before{content:"?"}
.fa-bitbucket:before{content:"?"}
.fa-bitcoin:before{content:"?"}
.fa-bity:before{content:"?"}
.fa-black-tie:before{content:"?"}
.fa-blackberry:before{content:"?"}
.fa-blender:before{content:"?"}
.fa-blender-phone:before{content:"?"}
.fa-blind:before{content:"?"}
.fa-blog:before{content:"?"}
.fa-blogger:before{content:"?"}
.fa-blogger-b:before{content:"?"}
.fa-bluetooth:before{content:"?"}
.fa-bluetooth-b:before{content:"?"}
.fa-bold:before{content:"?"}
.fa-bolt:before{content:"?"}
.fa-bomb:before{content:"?"}
.fa-bone:before{content:"?"}
.fa-bong:before{content:"?"}
.fa-book:before{content:"?"}
.fa-book-dead:before{content:"?"}
.fa-book-medical:before{content:"?"}
.fa-book-open:before{content:"?"}
.fa-book-reader:before{content:"?"}
.fa-bookmark:before{content:"?"}
.fa-bootstrap:before{content:"?"}
.fa-border-all:before{content:"?"}
.fa-border-none:before{content:"?"}
.fa-border-style:before{content:"?"}
.fa-bowling-ball:before{content:"?"}
.fa-box:before{content:"?"}
.fa-box-open:before{content:"?"}
.fa-box-tissue:before{content:"拏"}
.fa-boxes:before{content:"?"}
.fa-braille:before{content:"?"}
.fa-brain:before{content:"?"}
.fa-bread-slice:before{content:"?"}
.fa-briefcase:before{content:"?"}
.fa-briefcase-medical:before{content:"?"}
.fa-broadcast-tower:before{content:"?"}
.fa-broom:before{content:"?"}
.fa-brush:before{content:"?"}
.fa-btc:before{content:"?"}
.fa-buffer:before{content:"?"}
.fa-bug:before{content:"?"}
.fa-building:before{content:"?"}
.fa-bullhorn:before{content:"?"}
.fa-bullseye:before{content:"?"}
.fa-burn:before{content:"?"}
.fa-buromobelexperte:before{content:"?"}
.fa-bus:before{content:"?"}
.fa-bus-alt:before{content:"?"}
.fa-business-time:before{content:"?"}
.fa-buy-n-large:before{content:"?"}
.fa-buysellads:before{content:"?"}
.fa-calculator:before{content:"?"}
.fa-calendar:before{content:"?"}
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.fa-calendar-check:before{content:"?"}
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本文標題:GAN 基本原理以及數學證明

文章作者:gyro永不抽風

發布時間:2020年08月14日 – 19:08

最后更新:2020年09月22日 – 21:09

原始鏈接:http://hexo.gyrojeff.moe/2020/08/14/GAN-%E5%9F%BA%E6%9C%AC%E5%8E%9F%E7%90%86%E4%BB%A5%E5%8F%8A%E6%95%B0%E5%AD%A6%E8%AF%81%E6%98%8E/

許可協議: 署名-非商業性使用-相同方式共享 4.0 國際 (CC BY-NC-SA 4.0) 轉載請保留原文鏈接及作者！

Preface

今天在PD Lib和DL斗智斗勇時，突然想起了自己非常想學的GAN，機緣巧合下便百度了，得到了以下兩篇文章：

https://zhuanlan.zhihu.com/p/72279816
https://blog.csdn.net/jizhidexiaoming/article/details/96485095

于是便對GAN有了初步的了解（以前肯定是心不在焉才沒有理解的（劃掉）），隨后又在五樓生命科學的書架上找到了相關資料，遂學了一波。

GAN概述

2014 年，Ian Goodfellow 和他在蒙特利爾大學的同事發表了一篇震撼學界的論文。沒錯，我說的就是《Generative Adversarial Nets》，這標志著生成對抗網絡（GAN）的誕生，而這是通過對計算圖和博弈論的創新性結合。他們的研究展示，給定充分的建模能力，兩個博弈模型能夠通過簡單的反向傳播（backpropagation）來協同訓練。

這兩個模型的角色定位十分鮮明。給定真實數據集 R，G 是生成器（generator），它的任務是生成能以假亂真的假數據；而 D 是判別器（discriminator），它從真實數據集或者 G 那里獲取數據，然后做出判別真假的標記。Ian Goodfellow 的比喻是，G 就像一個贗品作坊，想要讓做出來的東西盡可能接近真品，蒙混過關。而 D 就是文物鑒定專家，要能區分出真品和高仿（但在這個例子中，造假者 G 看不到原始數據，而只有 D 的鑒定結果——前者是在盲干）。

理想情況下，D 和 G 都會隨著不斷訓練，做得越來越好——直到 G 基本上成為了一個“贗品制造大師”，而 D 因無法正確區分兩種數據分布輸給 G。

實踐中，Ian Goodfellow 展示的這項技術在本質上是：G 能夠對原始數據集進行一種無監督學習，找到以更低維度的方式（lower-dimensional manner）來表示數據的某種方法。而無監督學習之所以重要，就好像 Yann LeCun 的那句話：“無監督學習是蛋糕的糕體”。這句話中的蛋糕，指的是無數學者、開發者苦苦追尋的“真正的 AI”。
——pytorch實現GAN

GAN – Generative Adversarial Nets, 生成對抗網絡，簡單來講其有兩個組成部分：

D (Discriminator) – 判別器，判斷輸入時捏造的還是真實的
G (Generator) – 生成器，從隨機噪聲中生成我們想要的數據

隨著訓練的進行，我們要提高D的辨析能力，但同時也要G的能力來騙過D，因為我們的最終目的是要讓G來生成可以騙過D的信息。總結來說，通過對這兩個模型的訓練，我們就可以找到隨機噪聲與有意義數據的映射，達到創作的目的。

GAN的流程和目標函數

GAN的目標函數

GAN的目標函數如下：

$$ V(D, G) = \mathbb E_{x \sim P_{\text{data}} } [\log D(x)] + \mathbb E_{\boldsymbol {z} \sim P_z}[\log(1 – D(G(\boldsymbol {z})))] $$

其中，$D$為Discriminator的模型函數，$G$為Generator的模型函數，隨機變量$x$服從原來正確的數據集的分布$P_\text {data}$，隨機變量（這里可能是高維隨機變量，取決于模型具體實現）$\boldsymbol {z}$服從分布$P_z$（生成噪音），$\mathbb E$代表期望。

GAN的流程

$$ G^* = \arg \min _G \max _D V(D, G) $$

即，可以分為兩步理解：

在$G$為常數的情況下，選擇合適的$D$使得$V(D,G)$能夠最大化。
在這之后，選取合適的$G$來最小化$V(D, G)$，這個$G$就是我們想要的生成模型。

在每一步的訓練中：

取$m$個真實數據：
$$ \{x^{(1)}, x^{(2)}, x^{(3)}, \cdots, x^{(m)}\} $$

使用$G$和$m$組隨機數（服從于噪音分布$P_G$，一般使用服從正態分布的隨機數）

$$ \{\boldsymbol {z}^{(1)}, \boldsymbol {z}^{(2)}, \boldsymbol {z}^{(3)}, \cdots, \boldsymbol {z}^{(m)}\} $$

生成$m$個假數據，其中

$$ \forall i \in [1, m], i \in \mathbb Z \Rightarrow x^{(i)} \sim P_{\text{data}}, \boldsymbol {z} ^ {(i)} \sim P_z $$
根據$\max$部分的目標使用隨機梯度上升（Stochastic Gradient Ascent）更新$D$的參數，提高$D$的分辨能力
$$ \theta_d := \theta_d + \alpha_d \nabla_{\theta_d} \frac 1 m \sum _ {i = 1} ^ m \Big [ \log D\big ( x^{(i)} \big) + \log \big ( 1 – D \big ( G(\boldsymbol{z}^{(i)}) \big ) \big ) \Big] $$
根據$\min$部分的目標使用隨機梯度下降（Stochastic Gradient Descent）更新$G$的參數，使$G$生成的數據更有迷惑性
$$ \theta_g := \theta_g – \alpha_g\nabla_{\theta_g} \frac 1 m \sum _{i = 1} ^ m\log \Big ( 1 – D\big ( G(\boldsymbol{z} ^ {(i)})\big ) \Big ) $$

GAN的數學原理

Prerequisites

信息量(自信息)

信息量是指信息多少的量度，即，對于一條信息，傳達這條信息所需的最少信息長度為自信息。

信息論創始人C.E.Shannon，1938年首次使用比特（bit）概念：1（bit）= $\log_2 2$。它相當于對二個可能結局所作的一次選擇量。信息論采用對隨機分布概率取對數的辦法，解決了不定度的度量問題。

定義：符合分布$P$的某一事件$x$出現，傳達出這條信息的信息量記為：

$$ I = \log \frac 1 {P(x)} = – \log P(x) $$

香農熵

從離散分布$P$中隨機抽選一個事件，傳達這條信息所需的最優平均信息長度為香農熵，表達為：

$$ H(P) = \sum_x P(x) \log \frac 1 {P(x)} = – \sum_x P(x) \log P(x) $$

若分布是連續的，則：

$$ H(P) = \int_x P(x) \log \frac 1 {P(x)} \mathrm dx = -\int_x P(x) \log P(x) \mathrm dx $$

交叉熵

用分布$P$的最佳信息傳遞方式來傳達分布$Q$中隨機抽選的一個事件，所需的平均信息長度為交叉熵，表達為

$$ H_P(Q) = \sum_x Q(x) \log \frac 1 {P(x)} = – \sum_x Q(x) \log P(x) $$

$$ H_P(Q) = \int_x Q(x) \log \frac 1 {P(x)} \mathrm dx = – \int_x Q(x) \log P(x) \mathrm dx $$

$KL$ Divergence

$KL$散度：用分布$P$的最佳信息傳遞方式來傳達分布$Q$，比用分布$Q$自己的最佳信息傳遞方式來傳達分布$Q$，平均多耗費的信息長度為$KL$散度，表達為$D_P(Q)$或$D_{KL}(Q||P)$，$KL$散度衡量了兩個分布之間的差異。

$$ \begin{aligned} D_{KL}(Q||P) = D_P(Q) &= H_P(Q) – H(Q) \\ &= \sum_x Q(x) \log \frac 1 {P(x)} – \sum _x Q(x) \log \frac 1 {Q(x)} \\ &= \sum_x Q(x) \log \frac {Q(x)} {P(x)} \end{aligned} $$

對于連續分布：

$$ D_{KL}(Q||P) = D_P(Q) = \int_{-\infty} ^ {\infty} P(x)\log \frac {P(x)}{Q(x)} \mathrm dx $$

KL Divergence越大，兩個分布差異越大，反之差異越小。

數學原理

看完Prerequisites，我們回歸正題討論GAN的原理。我們現在想要做的事情，其實就是將一個服從$P_G$的隨機噪聲$\boldsymbol z$通過一個生成網絡$G$得到一個和真實數據分布$P_{\text {data}}(x)$差不多的生成分布$P_G(x;\theta_g)$，其中$\theta_g$為生成網絡$G$的參數。我們希望找到一個$\theta_g$使得兩個分布$P_{\text {data}}(x)$與$P_G(x;\theta)$盡可能地相似（使得他們地KL散度盡可能得小）。

我們從真實數據分布$P_\text{data}(x)$中取$m$個樣本，記作：

$$ \{x^{(1)}, x^{(2)}, x^{(3)}, \cdots, x^{(m)}\} $$

根據生成網絡的參數$\theta_g$，我們可以計算出這$m$個真實樣本在生成網絡中出現的概率$P_G(x^{(i)}; \theta_g)$，那么生成這樣的$m$個樣本數據的似然（likelihood）為：

$$ L = \prod_{i = 1} ^ m P_G(x^{(i)}; \theta_g) $$

由于我們想要兩個分布盡量相似，那么我們肯定希望這個似然$L$盡量大，即生成這樣的真實數據的概率盡量大，遂我們最大化這個似然，找到$\theta_g^*$：

$$ \begin{aligned} \theta_g^* &= \arg \max _ {\theta_g} \prod _ {i = 1} ^ m P_G(x^{(i)}; \theta_g) \\ &\Leftrightarrow \arg \max _ {\theta_g} \log \prod _ {i = 1} ^ m P_G(x^{(i)}; \theta_g) \\ &= \arg \max _ {\theta_g} \sum_{i = 1} ^ m \log P_G(x^{(i)}; \theta_g) \\ &\approx \arg \max _ {\theta_g} \mathbb E_{x \sim P_{\text {data}}} [\log P_G(x; \theta_g)] \\ &= \arg \max _ {\theta_g} \int _ x P_\text{data}(x) \log P_G(x; \theta_g)\mathrm dx \\ &= \arg \max _ {\theta_g} \int _ x P_\text{data}(x) \log P_G(x; \theta_g)\mathrm dx \\&~~~~- \int _ x P_\text{data}(x) \log P_\text {data}(x)\mathrm dx \\ &= \arg \max _ {\theta_g} \int _ x P_\text {data} (x) \log \frac {P_G(x; \theta_g)} {P_\text {data}(x)} \mathrm dx \\ &= \arg \max _ {\theta_g} – \int _ x P_\text {data} (x) \log \frac {P_\text {data}(x)} {P_G(x; \theta_g)} \mathrm dx \\ &= \arg \min _ {\theta_g} KL(P_\text {data} || P_G(x; \theta)) \end{aligned} $$

所以可見，其實最大化這個似然，和最小化KL散度是基本相同的。

上述式子中，$P_G(x;\theta_g)$代表在生成分布中出現$x$的概率，也可以如下計算得到：

$$ P_G(x) = \int _ \boldsymbol{z} P_z(z)\cdot 1\{G(z) = x\} \mathrm dz $$

注：$1{\cdot}$的含義是若打括號內的邏輯運算為真則取$1$，假則取$0$. 即

$$1\{\text {True}\} = 1, 1 \{\text {False}\} = 0 $$

但是我們發現，上述的過程是難以進行計算的，甚至完全沒辦法求$P_G(x)$，這只是模型的想法而已。

現在我們看回之前我們提到的目標函數：

$$ V(D, G) = \mathbb E_{x \sim P_{\text{data}} } [\log D(x)] + \mathbb E_{\boldsymbol {z} \sim P_z}[\log(1 – D(G(\boldsymbol {z})))] $$

與最優化生成模型：

$$ G^* = \arg \min _G \max _D V(D, G) $$

我們接下來分步解釋。

首先，我們不妨解釋一下$\max_D V(G, D)$，這部分的含義之前也解釋過，是在給定$G$的情況下，最大化$V(G, D)$。觀察發現，其形式其實與交叉熵損失函數非常相似：

$$ \mathcal L(\hat y, y) = -(y\log \hat y + (1 – y) \log (1 – \hat y)) $$

其實他們表達的目的也差不多。我們先化簡一下$V(G, D)$看看能得到什么結果：

$$ \begin{aligned} V(G, D) &= \mathbb E_{x \sim P_{\text{data}} } [\log D(x)] + \mathbb E_{\boldsymbol {z} \sim P_z}[\log(1 – D(G(\boldsymbol {z})))] \\ &= \int_x P_\text {data}(x) \log D(x) \mathrm dx + \int_\boldsymbol{z} P_z(\boldsymbol{z}) \log (1 – D(G(\boldsymbol{z})))\mathrm dz \\ &= \int_x P_\text {data}(x) \log D(x) \mathrm dx + \int_x P_G(x)\log (1 – D(x)) \mathrm dx \\ &= \int_x [P_\text {data}(x) \log D(x) + P_G(x)\log (1 – D(x))] \mathrm dx \end{aligned} $$

讓我們考察積分內部的項，我們可以對它做指數運算，即：

$$ e^{P_\text {data}(x) \log D(x) + P_G(x)\log (1 – D(x))} = D(x) ^ {P_\text {data}}\times(1- D(x))^{P_G(x)} $$

其想表達什么便不言而喻了，它表達的就是判別器判別是真的的正確率和判別是假的的正確率，總體來說就是衡量$D$的能力，所以我們想要最大化$V$，提高$D$的判別能力。

令

$$ f(D) = P_\text {data}(x) \log D(x) \mathrm dx + P_G(x)\log (1 – D(x)) $$

因為這里$P_\text {data} (x)$和$P_G(x)$都可以看作常數，所以

$$ \arg \max _ D \int_x f(D) \mathrm dx = \arg \max _D f(D) $$

最大化$f(D)$，即令其導數為$0$：

$$ \frac {\mathrm d f(D)} {\mathrm dD } = \frac {P_\text {data}(x)}{D} – \frac {P_G(x)} {1 – D} = 0 $$

則：

$$ \frac {P_\text {data}(x)}{D} = \frac {P_G(x)} {1 – D} $$

$$ D^*(x) = \frac {P_\text {data}(x)}{P_\text {data}(x) + P_G(x)} $$

這樣，我們就得到了那個狀態下最優的$D^*$的表達式。我們將這個能夠最大化$V$的$D$代入回$V(G, D)$：

$$ \begin{aligned} \max V(G, D) &= V(G, D^*) \\ &= \mathbb E_{x \sim P_\text {data}} \Big [ \log \frac {P_\text {data}(x)}{P_\text {data}(x) + P_G(x)} \Big] + \mathbb E_{x \sim P_G} \Big [ \frac {P_G(x)}{P_\text {data}(x) + P_G(x)}\Big ] \\ &= \int_x P_\text {data} (x) \log \frac {\frac 1 2 P_\text {data}}{\frac 1 2 (P_\text{data}(x) + P_G(x))} \mathrm dx + \int _ x P_G(x) \log \frac {\frac 1 2 P_G(x)} {\frac 1 2 (P_\text {data} (x) + P_G(x))} \mathrm dx \\ &= -2\log 2 + D_{KL} (P_\text {data} || \frac 1 2 [P_\text{data}(x) + P_G(x)]) + D_{KL} (P_G(x) || \frac 1 2 [P_\text {data}(x) + P_G(x)]) \\ &= -2\log 2 + JSD (P_\text{data}(x) || P_G(x)) \end{aligned} $$

其中，我們引入了$JS$ Divergence，定義如下：

$$ JSD(P||Q) = \frac 1 2 D_{KL} (P||M) + \frac 1 2 D_{KL} (Q || M), M = \frac 1 2 (P + Q) $$

容易得到，KL Divergence是不對稱的，而JS Divergence是對稱的。他們都可以衡量兩組分布建的差異。這里我們想要兩組分布差異最小，故取$\min$

所以，這也就解釋了為什么：

$$ \arg \min _G \max _D V(G, D) $$

是我們的目標過程。

總結

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