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  223. <div  id='write'  class = 'is-node show-fences-line-number'><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" style="cursor: pointer;" 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" style="cursor: pointer;" 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="n14"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n14">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="n29"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n29">8.3 模型表示1(Model Representation I)</a></span><span class="md-toc-item md-toc-h2" data-ref="n30"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n30">8.4 模型表示2(Model Representation II)</a></span><span class="md-toc-item md-toc-h2" data-ref="n31"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n31">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="n32"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n32">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="n33"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n33">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，从而可得到任意问题的拟合曲线，即泰勒极数(Taylor series)。</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，这使得线性模型的复杂度太高，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='image/20180115_084326.png' alt='' /></p><p>如上图，如果选取一小块 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-284-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.845ex" height="1.994ex" viewBox="0 -755.9 2946.9 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E290-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="1" id="E290-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><path stroke-width="1" id="E290-MJMAIN-2217" d="M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E290-MJMAIN-35"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E290-MJMAIN-30" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E290-MJMAIN-2217" x="1223" y="0"></use><g transform="translate(1945,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E290-MJMAIN-35"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E290-MJMAIN-30" x="500" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-284">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_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-297-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.143ex" height="1.994ex" viewBox="0 -755.9 3936.6 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E312-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="1" id="E312-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="1" id="E312-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="1" id="E312-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="1" id="E312-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 xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E312-MJMATHI-6E" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E312-MJMAIN-3D" x="878" y="0"></use><g transform="translate(1934,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E312-MJMAIN-32"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E312-MJMAIN-35" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E312-MJMAIN-30" x="1001" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E312-MJMAIN-30" x="1501" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-297">n=2500</script>（换成 RGB（一个像素有三个值），则 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-308-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.143ex" height="1.994ex" viewBox="0 -755.9 3936.6 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E328-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="1" id="E328-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="1" id="E328-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="1" id="E328-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="1" id="E328-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 xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E328-MJMATHI-6E" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E328-MJMAIN-3D" x="878" y="0"></use><g transform="translate(1934,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E328-MJMAIN-37"></use><use 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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="1" id="E430-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="1" id="E430-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 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338T555 289Q514 289 468 310T388 357T308 404T224 426Q164 426 125 393T83 318Q81 289 69 289Q55 289 55 319ZM55 85Q55 126 72 159T114 210T163 238T205 248Q207 248 213 248T223 249Q262 249 296 234T393 179L443 147Q502 112 553 112Q609 112 649 141T694 220Q694 249 708 249T722 217Q722 153 675 104T555 55Q514 55 468 76T388 123T308 170T224 192Q164 192 125 159T83 84Q80 55 69 55Q55 55 55 85Z"></path><path stroke-width="1" id="E430-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 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284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E430-MJMATHI-43" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E430-MJMAIN-32" x="1094" y="488"></use><g transform="translate(715,-308)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E430-MJMAIN-32"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E430-MJMAIN-35" x="500" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E430-MJMAIN-30" x="1001" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E430-MJMAIN-30" x="1501" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E430-MJMAIN-2248" x="2508" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E430-MJMAIN-33" x="3565" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E430-MJMATHI-6D" x="4315" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E430-MJMATHI-69" x="5194" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E430-MJMATHI-6C" x="5539" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E430-MJMATHI-6C" x="5838" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E430-MJMATHI-69" x="6136" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E430-MJMATHI-6F" x="6482" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E430-MJMATHI-6E" x="6967" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-364">C_{2500}^{2} \approx 3\ million</script>。</p><h2><a name='header-n14' 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='image/20180115_101441.png' alt='' /></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='image/20180115_101442.png' alt='' /></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-n29' class='md-header-anchor '></a>8.3 模型表示1(Model Representation I)</h2><p>&nbsp;</p><h2><a name='header-n30' class='md-header-anchor '></a>8.4 模型表示2(Model Representation II)</h2><h2><a name='header-n31' class='md-header-anchor '></a>8.5 例子和直观理解1(Examples and Intuitions I)</h2><h2><a name='header-n32' class='md-header-anchor '></a>8.6 例子和直观理解2(Examples and Intuitions II)</h2><h2><a name='header-n33' class='md-header-anchor '></a>8.7 多类别分类(Multiclass Classification)</h2></div>

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