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scruel authored 2019-05-25 13:51 . images path in htmls
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<div id='write' class = 'is-node'><div class='md-toc' mdtype='toc'><p class="md-toc-content"><span class="md-toc-item md-toc-h1" data-ref="n2"><a class="md-toc-inner" href="#header-n2">8 神经网络:表达(Neural Networks: Representation)</a></span><span class="md-toc-item md-toc-h2" data-ref="n3"><a class="md-toc-inner" href="#header-n3">8.1 非线性假设(Non-linear Hypotheses)</a></span><span class="md-toc-item md-toc-h2" data-ref="n9"><a class="md-toc-inner" href="#header-n9">8.2 神经网络和大脑(Neurons and the Brain)</a></span><span class="md-toc-item md-toc-h2" data-ref="n17"><a class="md-toc-inner" href="#header-n17">8.3 模型表示1(Model Representation I)</a></span><span class="md-toc-item md-toc-h2" data-ref="n57"><a class="md-toc-inner" href="#header-n57">8.4 模型表示2(Model Representation II)</a></span><span class="md-toc-item md-toc-h2" data-ref="n87"><a class="md-toc-inner" href="#header-n87">8.5 例子和直观理解1(Examples and Intuitions I)</a></span><span class="md-toc-item md-toc-h2" data-ref="n99"><a class="md-toc-inner" href="#header-n99">8.6 例子和直观理解2(Examples and Intuitions II)</a></span><span class="md-toc-item md-toc-h2" data-ref="n107"><a class="md-toc-inner" href="#header-n107">8.7 多类别分类(Multiclass Classification)</a></span></p></div><h1><a name='header-n2' class='md-header-anchor '></a>8 神经网络:表达(Neural Networks: Representation)</h1><h2><a name='header-n3' class='md-header-anchor '></a>8.1 非线性假设(Non-linear Hypotheses)</h2><p>理论上我们可以用多项式函数去近似任意函数(泰勒极数(Taylor series)),从而可得到任意问题的拟合曲线。</p><p>在实际处理时,特征量通常会很多,如果再构造高阶多项式等,特征数量将会急剧增加,这使得回归模型的复杂度太高,可见并不合适。神经网络无需构造高阶多项式,在特征量很大时也可以处理的很好。</p><p>那特征能有多大呢?下面是一个计算机视觉中的例子:</p><p><img src='images/20180115_084326.png' alt='' referrerPolicy='no-referrer' /></p><p>如上图,如果选取一小块 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.839ex" height="1.994ex" viewBox="0 -755.9 2944.4 858.4" role="img" focusable="false" style="vertical-align: 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y="0"></use></g></g></svg></span><script type="math/tex">50 * 50</script> 像素的灰度图片(一个像素只有亮度一个值),选择每个像素点作为特征,则特征总量 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.136ex" height="1.994ex" viewBox="0 -755.9 3933.6 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E276-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E276-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E276-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path stroke-width="0" id="E276-MJMAIN-35" d="M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z"></path><path stroke-width="0" id="E276-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E276-MJMATHI-6E" x="0" y="0"></use><use xlink:href="#E276-MJMAIN-3D" x="877" y="0"></use><g transform="translate(1933,0)"><use xlink:href="#E276-MJMAIN-32"></use><use xlink:href="#E276-MJMAIN-35" x="500" y="0"></use><use xlink:href="#E276-MJMAIN-30" x="1000" y="0"></use><use xlink:href="#E276-MJMAIN-30" x="1500" y="0"></use></g></g></svg></span><script type="math/tex">n=2500</script>(换成 RGB(一个像素有三个值),则 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.136ex" height="1.994ex" viewBox="0 -755.9 3933.6 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E277-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E277-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E277-MJMAIN-37" d="M55 458Q56 460 72 567L88 674Q88 676 108 676H128V672Q128 662 143 655T195 646T364 644H485V605L417 512Q408 500 387 472T360 435T339 403T319 367T305 330T292 284T284 230T278 162T275 80Q275 66 275 52T274 28V19Q270 2 255 -10T221 -22Q210 -22 200 -19T179 0T168 40Q168 198 265 368Q285 400 349 489L395 552H302Q128 552 119 546Q113 543 108 522T98 479L95 458V455H55V458Z"></path><path stroke-width="0" id="E277-MJMAIN-35" d="M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z"></path><path stroke-width="0" id="E277-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E277-MJMATHI-6E" x="0" y="0"></use><use xlink:href="#E277-MJMAIN-3D" x="877" y="0"></use><g transform="translate(1933,0)"><use xlink:href="#E277-MJMAIN-37"></use><use xlink:href="#E277-MJMAIN-35" x="500" y="0"></use><use xlink:href="#E277-MJMAIN-30" x="1000" y="0"></use><use xlink:href="#E277-MJMAIN-30" x="1500" y="0"></use></g></g></svg></span><script type="math/tex">n = 7500</script>),如果将其两两组合作为新特征,则特征数量将为 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="17.563ex" height="3.044ex" viewBox="0 -906.7 7561.8 1310.7" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="0" id="E278-MJMATHI-43" d="M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z"></path><path stroke-width="0" id="E278-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path stroke-width="0" id="E278-MJMAIN-35" d="M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 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411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E278-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E278-MJMATHI-6C" d="M117 59Q117 26 142 26Q179 26 205 131Q211 151 215 152Q217 153 225 153H229Q238 153 241 153T246 151T248 144Q247 138 245 128T234 90T214 43T183 6T137 -11Q101 -11 70 11T38 85Q38 97 39 102L104 360Q167 615 167 623Q167 626 166 628T162 632T157 634T149 635T141 636T132 637T122 637Q112 637 109 637T101 638T95 641T94 647Q94 649 96 661Q101 680 107 682T179 688Q194 689 213 690T243 693T254 694Q266 694 266 686Q266 675 193 386T118 83Q118 81 118 75T117 65V59Z"></path><path stroke-width="0" id="E278-MJMATHI-6F" d="M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z"></path><path stroke-width="0" id="E278-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E278-MJMATHI-43" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E278-MJMAIN-32" x="1093" y="487"></use><g transform="translate(715,-307)"><use transform="scale(0.707)" xlink:href="#E278-MJMAIN-32"></use><use transform="scale(0.707)" xlink:href="#E278-MJMAIN-35" x="500" y="0"></use><use transform="scale(0.707)" xlink:href="#E278-MJMAIN-30" x="1000" y="0"></use><use transform="scale(0.707)" xlink:href="#E278-MJMAIN-30" x="1499" y="0"></use></g><use xlink:href="#E278-MJMAIN-2248" x="2506" y="0"></use><use xlink:href="#E278-MJMAIN-33" x="3562" y="0"></use><use xlink:href="#E278-MJMATHI-6D" x="4312" y="0"></use><use xlink:href="#E278-MJMATHI-69" x="5190" y="0"></use><use xlink:href="#E278-MJMATHI-6C" x="5535" y="0"></use><use xlink:href="#E278-MJMATHI-6C" x="5833" y="0"></use><use xlink:href="#E278-MJMATHI-69" x="6131" y="0"></use><use xlink:href="#E278-MJMATHI-6F" x="6476" y="0"></use><use xlink:href="#E278-MJMATHI-6E" x="6961" y="0"></use></g></svg></span><script type="math/tex">C_{2500}^{2} \approx 3\ million</script></p><h2><a name='header-n9' class='md-header-anchor '></a>8.2 神经网络和大脑(Neurons and the Brain)</h2><p>脑科学家通过对动物实验,发现大脑中专用于处理听觉信号的脑皮层也能处理其他诸如视觉等信号,即如果切断其与耳朵的联系,将其与眼睛相连,则这块负责听觉的脑皮层区域也能接受并处理视觉信号,从而学会“看”。脑科学家通过这类换源实验,就推论假设大脑的学习算法只有一种(“one learning algorithm” hypothesis)。那么如果能找出这种学习算法并应用于计算机中,那梦想中和人一样的人工智能就成真了。</p><p>神经网络就源于<strong>模拟人类大脑</strong>,但其需要的计算量很大。随着计算机硬件性能的提高,神经网络逐渐从衰落变为流行,如今已广泛地被应用在各行各业中。</p><p>下图是根据研究做的一些应用(有兴趣可回顾视频):</p><p><img src='images/20180115_101441.png' alt='' referrerPolicy='no-referrer' /></p><p>BrainPort 系统:帮助失明人士通过摄像头以及舌尖感官“看”东西</p><p><img src='images/20180115_101442.png' alt='' referrerPolicy='no-referrer' /></p><p>触觉皮带:在朝北时蜂鸣器会发出声响,可使人拥有方向感(声音信号转换为方向信号)。</p><h2><a name='header-n17' class='md-header-anchor '></a>8.3 模型表示1(Model Representation I)</h2><p>既然神经网络模仿的是大脑神经元,那就先看一下大脑的神经元长什么样吧:</p><p><img src='images/20141213201613758.jpg' alt='来源: http://blog.csdn.net/zzwu/article/details/574931' referrerPolicy='no-referrer' /></p><p>想象一下印刷厂中流水线的工人(机器人也算哦),每个工人都有特定的任务,比如装订,塑封,贴防伪标识等等,工人们看到书本并处理完自己的任务后,就回放回传送带,紧接着传送带就传给下一个环节的工人,如此不断重复从而完成一个又一个环节,直到一本书印制完成。</p><p>那么类比一下,把上图中的<strong>细胞核(nucleus)</strong>类比成工人,<strong>轴突(axon)</strong>类比传送带,<strong>树突(dendrite)</strong>则比类比成工人的双眼。一个又一个细胞体,从树突接收需要处理的信息,对其进行处理后,再经由轴突通过电信号把处理完的信息传递出去,直到理解信息的内容。当然啦,我们大脑的实际上还要更为复杂,而且一个人的神经元数目就比地球上所有流水线的工人之和还要多呢~</p><p>人工神经网络中,树突对应<strong>输入(input)</strong>,细胞核对应<strong>激活单元(activation unit)</strong>,轴突对应<strong>输出(output)</strong></p><p>我们一般把神经网络划分为三部分(注意,不是只有三层!),即输入层(input layer),隐藏层(hidden layer)和输出层(output layer)。</p><p><img src='images/20180116_001543.png' alt='' referrerPolicy='no-referrer' /></p><p>图中的一个圈表示神经网络中的一个激活单元,输入层对应输入单元,隐藏层对应中间单元,输出层则对应输出单元。中间激活单元应用<strong>激活函数</strong>(<a href='https://en.wikipedia.org/wiki/Activation_function'>activation_function</a>)处理数据。</p><p>下面列出一些已有概念在神经网络中的别称:</p><ul><li><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.382ex" height="1.76ex" viewBox="0 -504.6 1025.6 757.9" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E100-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E100-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E100-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E100-MJMAIN-30" x="808" y="-213"></use></g></svg></span><script type="math/tex">x_0</script>: 偏置单元(bias unit),<span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.382ex" height="1.76ex" viewBox="0 -504.6 1025.6 757.9" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E100-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E100-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" 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stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E279-MJMAIN-398" x="0" y="0"></use></g></svg></span><script type="math/tex">\Theta</script>: 权重(weight),即参数。</li><li>激活函数: <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.115ex" height="1.877ex" viewBox="0 -504.6 480 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E160-MJMATHI-67" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 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94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E27-MJMATHI-79" x="0" y="0"></use></g></svg></span><script type="math/tex">y</script></li></ul><blockquote><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.415ex" height="3.511ex" 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y="-429"></use></g></svg></span><script type="math/tex">a^{(j)}_i</script>: 第 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.985ex" height="2.461ex" viewBox="-12 -755.9 424 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex; margin-left: -0.028ex;"><defs><path stroke-width="0" id="E57-MJMATHI-6A" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g 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style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E286-MJMATHI-76" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E286-MJMATHI-76" x="0" y="0"></use></g></svg></span><script type="math/tex">v</script> 个单元的权重</p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.998ex" height="1.994ex" viewBox="0 -504.6 860.3 858.4" 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type="math/tex">h_\Theta(x) = a_1^{(3)} = g(\Theta_{10}^{(2)}a_0^{(2)} + \Theta_{11}^{(2)}a_1^{(2)} + \Theta_{12}^{(2)}a_2^{(2)} + \Theta_{13}^{(2)}a_3^{(2)})</script></p><p>上面的计算过程被称为<strong>前向传播(Forward propagation)</strong>,即从输入层开始,一层一层地向下计算并传递结果。</p><p>&nbsp;</p><p>再回顾一下逻辑回归:</p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="36.504ex" height="2.577ex" viewBox="0 -806.1 15716.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E298-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 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y="-4231"></use></g></g></g></svg></span><script type="math/tex">z^{(2)}=\left[ \begin{matrix}z_1^{(2)}\\ z_1^{(2)} \\ z_1^{(2)}\end{matrix} \right]</script></p><p>则有 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="26.716ex" height="2.928ex" viewBox="0 -956.9 11502.8 1260.5" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E304-MJMATHI-61" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 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transform="translate(3735,0)"><use xlink:href="#E304-MJMAIN-398" x="0" y="0"></use><g transform="translate(778,362)"><use transform="scale(0.707)" xlink:href="#E304-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E304-MJMAIN-31" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E304-MJMAIN-29" x="888" y="0"></use></g></g><g transform="translate(5516,0)"><use xlink:href="#E304-MJMATHI-61" x="0" y="0"></use><g transform="translate(529,362)"><use transform="scale(0.707)" xlink:href="#E304-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E304-MJMAIN-31" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E304-MJMAIN-29" x="888" y="0"></use></g></g><use xlink:href="#E304-MJMAIN-29" x="7049" y="0"></use><use xlink:href="#E304-MJMAIN-3D" x="7716" y="0"></use><use xlink:href="#E304-MJMATHI-67" x="8772" y="0"></use><use xlink:href="#E304-MJMAIN-28" x="9252" y="0"></use><g transform="translate(9641,0)"><use 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375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E308-MJMATHI-61" x="0" y="0"></use><g transform="translate(529,362)"><use transform="scale(0.707)" xlink:href="#E308-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E308-MJMATHI-6A" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E308-MJMAIN-29" x="801" y="0"></use></g><use xlink:href="#E308-MJMAIN-3D" x="1748" y="0"></use><use xlink:href="#E308-MJMATHI-67" x="2804" y="0"></use><use xlink:href="#E308-MJMAIN-28" x="3284" y="0"></use><g 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