[{"data":1,"prerenderedAt":1002},["ShallowReactive",2],{"content-query-OcJgf3NSsq":3},{"_path":4,"_dir":5,"_draft":6,"_partial":6,"_locale":7,"title":8,"description":9,"date":10,"cover":11,"type":12,"category":13,"body":14,"_type":996,"_id":997,"_source":998,"_file":999,"_stem":1000,"_extension":1001},"/technology-blogs/zh/698","zh",false,"","【跟着小Mi一起机器学习吧！】支持向量机（中）","什么是机器学习，什么是机器学习，如果你想知道什么是机器学习，那么小Mi带你一起研究！","2021-08-28","https://obs-mindspore-file.obs.cn-north-4.myhuaweicloud.com/file/2021/08/28/b464021858d1416fb79e93e8a4a6b6ab.png","technology-blogs","基础知识",{"type":15,"children":16,"toc":991},"root",[17,25,31,42,50,88,151,231,270,315,322,380,404,409,416,421,428,457,496,525,532,567,615,644,651,658,674,681,686,693,722,729,788,826,833,871,901,911,918,951,958,975,986],{"type":18,"tag":19,"props":20,"children":22},"element","h1",{"id":21},"跟着小mi一起机器学习吧支持向量机中",[23],{"type":24,"value":8},"text",{"type":18,"tag":26,"props":27,"children":28},"p",{},[29],{"type":24,"value":30},"上周小Mi带大家学习了支持向量机的优化问题以及如何得到大间距分类器，而这背后的数学原理大家是否略知一二呢？今天跟着小Mi后面慢慢推理，帮助大家对支持向量机中的优化问题，以及如何得到大间距分类器，产生更好的直观理解。",{"type":18,"tag":32,"props":33,"children":35},"h2",{"id":34},"_3-大边界分类背后的数学",[36],{"type":18,"tag":37,"props":38,"children":39},"strong",{},[40],{"type":24,"value":41},"3 大边界分类背后的数学",{"type":18,"tag":26,"props":43,"children":44},{},[45],{"type":18,"tag":46,"props":47,"children":49},"img",{"alt":7,"src":48},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/0943459xxcxhxkplvotdaw.png",[],{"type":18,"tag":26,"props":51,"children":52},{},[53,55,59,61,65,67,71,73,77,79,82,83,86],{"type":24,"value":54},"首先，我们需要了解下向量内积这个知识点（敲黑板了哦~）。假设有两个向量",{"type":18,"tag":46,"props":56,"children":58},{"alt":7,"src":57},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/143854io9xdvxertbbgfse.png",[],{"type":24,"value":60},"和",{"type":18,"tag":46,"props":62,"children":64},{"alt":7,"src":63},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/143942qicbm5ujuxq7uy9e.png",[],{"type":24,"value":66},"，两者都是二维向量，需要计算一下",{"type":18,"tag":46,"props":68,"children":70},{"alt":7,"src":69},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/144003rbwbrytxqytwepke.png",[],{"type":24,"value":72},"的结果，这里的",{"type":18,"tag":46,"props":74,"children":76},{"alt":7,"src":75},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/144007g88qzd46mwccdkns.png",[],{"type":24,"value":78},"就叫做向量",{"type":18,"tag":46,"props":80,"children":81},{"alt":7,"src":57},[],{"type":24,"value":60},{"type":18,"tag":46,"props":84,"children":85},{"alt":7,"src":63},[],{"type":24,"value":87},"之间的内积。",{"type":18,"tag":26,"props":89,"children":90},{},[91,93,96,97,100,102,105,107,111,113,117,119,122,124,128,130,133,135,138,140,144,146,149],{"type":24,"value":92},"由于是二维向量，向量",{"type":18,"tag":46,"props":94,"children":95},{"alt":7,"src":57},[],{"type":24,"value":60},{"type":18,"tag":46,"props":98,"children":99},{"alt":7,"src":63},[],{"type":24,"value":101},"均可以用图像表示出来。将向量",{"type":18,"tag":46,"props":103,"children":104},{"alt":7,"src":57},[],{"type":24,"value":106},"分解为横轴上的",{"type":18,"tag":46,"props":108,"children":110},{"alt":7,"src":109},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/144208tcdx8uldnujo8b3f.png",[],{"type":24,"value":112},"和纵轴上的",{"type":18,"tag":46,"props":114,"children":116},{"alt":7,"src":115},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/1442330kvufcfg4iewhpwt.png",[],{"type":24,"value":118},"。现在，很容易计算的一个量就是向量",{"type":18,"tag":46,"props":120,"children":121},{"alt":7,"src":57},[],{"type":24,"value":123},"的范数。",{"type":18,"tag":46,"props":125,"children":127},{"alt":7,"src":126},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/1443067bwbr1ar7cz1saml.png",[],{"type":24,"value":129},"表示",{"type":18,"tag":46,"props":131,"children":132},{"alt":7,"src":57},[],{"type":24,"value":134},"的范数，即",{"type":18,"tag":46,"props":136,"children":137},{"alt":7,"src":57},[],{"type":24,"value":139},"的长度（这里的长度是指欧几里得长度—在m维空间中两个点之间的真实距离或者向量的自然长度，即该点到原点的距离）。根据毕达哥拉斯定理（这里小Mi故意说得气派了点，听不懂了吧，其实就是勾股定理，哈哈），",{"type":18,"tag":46,"props":141,"children":143},{"alt":7,"src":142},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/144357i1hrywkff1z4p2dn.png",[],{"type":24,"value":145},"，这是向量",{"type":18,"tag":46,"props":147,"children":148},{"alt":7,"src":57},[],{"type":24,"value":150},"的长度，它是一个实数。",{"type":18,"tag":26,"props":152,"children":153},{},[154,156,159,161,164,166,170,171,175,177,180,181,184,186,189,191,194,196,199,200,203,205,209,210,214,216,219,220,223,225,229],{"type":24,"value":155},"现在让我们来看看向量",{"type":18,"tag":46,"props":157,"children":158},{"alt":7,"src":63},[],{"type":24,"value":160},"，",{"type":18,"tag":46,"props":162,"children":163},{"alt":7,"src":63},[],{"type":24,"value":165},"是另一个向量，它的两个分量",{"type":18,"tag":46,"props":167,"children":169},{"alt":7,"src":168},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/144455ik4hckc78llfbye7.png",[],{"type":24,"value":60},{"type":18,"tag":46,"props":172,"children":174},{"alt":7,"src":173},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/1445102cvesxupbfqzhzqa.png",[],{"type":24,"value":176},"是已知的，那么",{"type":18,"tag":46,"props":178,"children":179},{"alt":7,"src":57},[],{"type":24,"value":60},{"type":18,"tag":46,"props":182,"children":183},{"alt":7,"src":63},[],{"type":24,"value":185},"之间的内积又是如何计算的呢？我们将向量",{"type":18,"tag":46,"props":187,"children":188},{"alt":7,"src":63},[],{"type":24,"value":190},"投影到向量",{"type":18,"tag":46,"props":192,"children":193},{"alt":7,"src":57},[],{"type":24,"value":195},"上，这是一个直角投影，上图中红线的长度就是向量",{"type":18,"tag":46,"props":197,"children":198},{"alt":7,"src":63},[],{"type":24,"value":190},{"type":18,"tag":46,"props":201,"children":202},{"alt":7,"src":57},[],{"type":24,"value":204},"上的量，我们设定其为",{"type":18,"tag":46,"props":206,"children":208},{"alt":7,"src":207},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/144714klfjzjyfs2piqtwl.png",[],{"type":24,"value":160},{"type":18,"tag":46,"props":211,"children":213},{"alt":7,"src":212},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/144716yrhwfh7t89inties.png",[],{"type":24,"value":215},"是",{"type":18,"tag":46,"props":217,"children":218},{"alt":7,"src":63},[],{"type":24,"value":190},{"type":18,"tag":46,"props":221,"children":222},{"alt":7,"src":57},[],{"type":24,"value":224},"上的长度，因此可以得出",{"type":18,"tag":46,"props":226,"children":228},{"alt":7,"src":227},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/144746ikjsdtkxpupxfmtv.png",[],{"type":24,"value":230},"，这是计算内积的一种方法。",{"type":18,"tag":26,"props":232,"children":233},{},[234,236,239,241,244,246,250,252,256,258,262,264,268],{"type":24,"value":235},"如果我们从几何上画出",{"type":18,"tag":46,"props":237,"children":238},{"alt":7,"src":207},[],{"type":24,"value":240},"的值，同时画出",{"type":18,"tag":46,"props":242,"children":243},{"alt":7,"src":57},[],{"type":24,"value":245},"的范数，同样也可以计算出内积，答案是一样的。另一个计算公式是：",{"type":18,"tag":46,"props":247,"children":249},{"alt":7,"src":248},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/144912lfhp9dio1upyeqvm.png",[],{"type":24,"value":251},"就是",{"type":18,"tag":46,"props":253,"children":255},{"alt":7,"src":254},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/144937nr8h21fbbap1fppa.png",[],{"type":24,"value":257},"这个一行两列的矩阵乘以",{"type":18,"tag":46,"props":259,"children":261},{"alt":7,"src":260},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/144953dnwtpho0iwckzjj2.png",[],{"type":24,"value":263},"，结果为",{"type":18,"tag":46,"props":265,"children":267},{"alt":7,"src":266},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/145008vbwvsvagpbfkegl8.png",[],{"type":24,"value":269},"。根据线性代数的知识，这两个公式会给出同样的结果。",{"type":18,"tag":26,"props":271,"children":272},{},[273,275,279,281,284,285,288,290,293,295,298,300,303,305,308,310,313],{"type":24,"value":274},"BTW，根据计算其实我们可以证明得出",{"type":18,"tag":46,"props":276,"children":278},{"alt":7,"src":277},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/145112zv7ae7hcxsfbaw6i.png",[],{"type":24,"value":280},"。那么如果计算的时候将",{"type":18,"tag":46,"props":282,"children":283},{"alt":7,"src":57},[],{"type":24,"value":60},{"type":18,"tag":46,"props":286,"children":287},{"alt":7,"src":260},[],{"type":24,"value":289},"交换位置，将",{"type":18,"tag":46,"props":291,"children":292},{"alt":7,"src":57},[],{"type":24,"value":294},"投影到",{"type":18,"tag":46,"props":296,"children":297},{"alt":7,"src":260},[],{"type":24,"value":299},"上，实际上最终可以得到同样的结果。（等式中",{"type":18,"tag":46,"props":301,"children":302},{"alt":7,"src":57},[],{"type":24,"value":304},"的范数是一个实数，",{"type":18,"tag":46,"props":306,"children":307},{"alt":7,"src":207},[],{"type":24,"value":309},"也是一个实数，因此",{"type":18,"tag":46,"props":311,"children":312},{"alt":7,"src":248},[],{"type":24,"value":314},"就是两个实数正常相乘。）",{"type":18,"tag":26,"props":316,"children":317},{},[318],{"type":18,"tag":46,"props":319,"children":321},{"alt":7,"src":320},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/145259queypsf71ynef873.png",[],{"type":18,"tag":26,"props":323,"children":324},{},[325,327,330,332,335,337,340,341,344,346,349,350,353,355,358,360,363,365,368,370,373,375,378],{"type":24,"value":326},"Last but not least，需要注意的是",{"type":18,"tag":46,"props":328,"children":329},{"alt":7,"src":207},[],{"type":24,"value":331},"值，",{"type":18,"tag":46,"props":333,"children":334},{"alt":7,"src":207},[],{"type":24,"value":336},"实际上是有符号的，即它可能是正值，也可能是负值。如果向量",{"type":18,"tag":46,"props":338,"children":339},{"alt":7,"src":57},[],{"type":24,"value":60},{"type":18,"tag":46,"props":342,"children":343},{"alt":7,"src":260},[],{"type":24,"value":345},"之间的夹角大于90度，则如果将",{"type":18,"tag":46,"props":347,"children":348},{"alt":7,"src":260},[],{"type":24,"value":294},{"type":18,"tag":46,"props":351,"children":352},{"alt":7,"src":57},[],{"type":24,"value":354},"上，会得到上图红线表示的一个投影，这就是",{"type":18,"tag":46,"props":356,"children":357},{"alt":7,"src":207},[],{"type":24,"value":359},"的长度，在这个情形下",{"type":18,"tag":46,"props":361,"children":362},{"alt":7,"src":248},[],{"type":24,"value":364},"等于",{"type":18,"tag":46,"props":366,"children":367},{"alt":7,"src":207},[],{"type":24,"value":369},"乘以",{"type":18,"tag":46,"props":371,"children":372},{"alt":7,"src":57},[],{"type":24,"value":374},"的范数成立，唯一一点不同的是",{"type":18,"tag":46,"props":376,"children":377},{"alt":7,"src":207},[],{"type":24,"value":379},"在这里是负的。",{"type":18,"tag":26,"props":381,"children":382},{},[383,385,388,389,392,394,397,399,402],{"type":24,"value":384},"因此，在内积计算中，如果",{"type":18,"tag":46,"props":386,"children":387},{"alt":7,"src":57},[],{"type":24,"value":60},{"type":18,"tag":46,"props":390,"children":391},{"alt":7,"src":260},[],{"type":24,"value":393},"之间的夹角小于90度，那么长度",{"type":18,"tag":46,"props":395,"children":396},{"alt":7,"src":207},[],{"type":24,"value":398},"是正值；然而如果这个夹角大于90度，则",{"type":18,"tag":46,"props":400,"children":401},{"alt":7,"src":207},[],{"type":24,"value":403},"将会是负的，从而两个向量之间的内积也是负的。",{"type":18,"tag":26,"props":405,"children":406},{},[407],{"type":24,"value":408},"以上就是关于向量内积的知识啦，接下来小Mi将会使用这些关于向量内积的性质来帮助大家理解支持向量机中的目标函数。",{"type":18,"tag":26,"props":410,"children":411},{},[412],{"type":18,"tag":46,"props":413,"children":415},{"alt":7,"src":414},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/145529apgkcec1pgrylofa.png",[],{"type":18,"tag":26,"props":417,"children":418},{},[419],{"type":24,"value":420},"先前给出的支持向量机模型中的目标函数如上图。为了更容易分析，这里做一点简化。",{"type":18,"tag":26,"props":422,"children":423},{},[424],{"type":18,"tag":46,"props":425,"children":427},{"alt":7,"src":426},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/145621qvbpqfqqpp0wsvmd.png",[],{"type":18,"tag":26,"props":429,"children":430},{},[431,433,437,439,443,445,449,451,455],{"type":24,"value":432},"首先忽略截距，令",{"type":18,"tag":46,"props":434,"children":436},{"alt":7,"src":435},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/145644ei5clipymori7w8s.png",[],{"type":24,"value":438},"，这样更容易画示意图。将特征数n设置为2，因此我们仅有两个特征",{"type":18,"tag":46,"props":440,"children":442},{"alt":7,"src":441},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/145708sppv184pwoinhjpz.png",[],{"type":24,"value":444},"，即",{"type":18,"tag":46,"props":446,"children":448},{"alt":7,"src":447},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/145724aweaw5hvdvkebfzz.png",[],{"type":24,"value":450},"时，支持向量机的优化目标函数可以写为：",{"type":18,"tag":46,"props":452,"children":454},{"alt":7,"src":453},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/145748aiujrze1pvxuiqbp.png",[],{"type":24,"value":456},"。",{"type":18,"tag":26,"props":458,"children":459},{},[460,462,466,468,472,474,478,480,483,485,489,491,494],{"type":24,"value":461},"这里忽略",{"type":18,"tag":46,"props":463,"children":465},{"alt":7,"src":464},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/145806psq482jcrwksqfu4.png",[],{"type":24,"value":467},"的话，",{"type":18,"tag":46,"props":469,"children":471},{"alt":7,"src":470},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/145821xixtprjnkaec0ytf.png",[],{"type":24,"value":473},"的范数仅仅和",{"type":18,"tag":46,"props":475,"children":477},{"alt":7,"src":476},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/14583594ga8uupjagmihku.png",[],{"type":24,"value":479},"有关。事实上，数学里不管是否包含",{"type":18,"tag":46,"props":481,"children":482},{"alt":7,"src":464},[],{"type":24,"value":484},"，其实后面的推导并没有太大差别，最终也就意味着我们的目标函数是等于",{"type":18,"tag":46,"props":486,"children":488},{"alt":7,"src":487},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/145916qo7ulo9srjufygjm.png",[],{"type":24,"value":490},"。因此支持向量机做的全部事情，就是极小化参数向量",{"type":18,"tag":46,"props":492,"children":493},{"alt":7,"src":470},[],{"type":24,"value":495},"范数的平方，或者说长度的平方。",{"type":18,"tag":26,"props":497,"children":498},{},[499,501,505,507,510,511,515,517,520,521,524],{"type":24,"value":500},"而",{"type":18,"tag":46,"props":502,"children":504},{"alt":7,"src":503},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/145953n7gysgizczuyojao.png",[],{"type":24,"value":506},"中",{"type":18,"tag":46,"props":508,"children":509},{"alt":7,"src":470},[],{"type":24,"value":60},{"type":18,"tag":46,"props":512,"children":514},{"alt":7,"src":513},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/150023quuum4wuiuerscjx.png",[],{"type":24,"value":516},"就类似于之前提到的向量",{"type":18,"tag":46,"props":518,"children":519},{"alt":7,"src":57},[],{"type":24,"value":60},{"type":18,"tag":46,"props":522,"children":523},{"alt":7,"src":260},[],{"type":24,"value":456},{"type":18,"tag":26,"props":526,"children":527},{},[528],{"type":18,"tag":46,"props":529,"children":531},{"alt":7,"src":530},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/150119gnoq55ettdu665np.png",[],{"type":18,"tag":26,"props":533,"children":534},{},[535,537,541,543,547,549,553,555,559,561,565],{"type":24,"value":536},"如上图所示，我们考察一个单一的训练样本，有一个正样本",{"type":18,"tag":46,"props":538,"children":540},{"alt":7,"src":539},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/150202tqvbmsngpinq0hqk.png",[],{"type":24,"value":542},"水平轴上取值为",{"type":18,"tag":46,"props":544,"children":546},{"alt":7,"src":545},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/150246n9upyknngncoscxt.png",[],{"type":24,"value":548},"，竖直轴上取值为",{"type":18,"tag":46,"props":550,"children":552},{"alt":7,"src":551},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/1503020fuwtl6alu9b4ysp.png",[],{"type":24,"value":554},"。事实上，它就是一个始于原点，终点位置在训练样本点（红色“×”处）的向量。同样，",{"type":18,"tag":46,"props":556,"children":558},{"alt":7,"src":557},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/150324e3mavs4ez9beqzqo.png",[],{"type":24,"value":560},"也类似表示，那么内积",{"type":18,"tag":46,"props":562,"children":564},{"alt":7,"src":563},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/150353tbtkqxmogilzjim0.png",[],{"type":24,"value":566},"将会是什么样的呢？",{"type":18,"tag":26,"props":568,"children":569},{},[570,572,575,577,581,583,587,589,592,594,597,599,603,605,608,610,614],{"type":24,"value":571},"使用之前的方法，计算方式就是将训练样本投影到参数向量",{"type":18,"tag":46,"props":573,"children":574},{"alt":7,"src":557},[],{"type":24,"value":576},"，将这个线段画成红色。这里将其称为",{"type":18,"tag":46,"props":578,"children":580},{"alt":7,"src":579},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/150442tmjhwyni9tonbu8z.png",[],{"type":24,"value":582},"，表示这是第",{"type":18,"tag":46,"props":584,"children":586},{"alt":7,"src":585},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/150651zt7ywrzzqeohqjxu.png",[],{"type":24,"value":588},"个训练样本在参数向量",{"type":18,"tag":46,"props":590,"children":591},{"alt":7,"src":557},[],{"type":24,"value":593},"上的投影。根据我们之前求证出的结果，我们知道",{"type":18,"tag":46,"props":595,"children":596},{"alt":7,"src":563},[],{"type":24,"value":598},"将会等于",{"type":18,"tag":46,"props":600,"children":602},{"alt":7,"src":601},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/150827buxmkvxmnqhzqpfh.png",[],{"type":24,"value":604},"乘以向量",{"type":18,"tag":46,"props":606,"children":607},{"alt":7,"src":557},[],{"type":24,"value":609},"的长度或范数，也等于",{"type":18,"tag":46,"props":611,"children":613},{"alt":7,"src":612},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/150904umzplnaooql9j9cw.png",[],{"type":24,"value":456},{"type":18,"tag":26,"props":616,"children":617},{},[618,620,624,626,630,632,636,638,642],{"type":24,"value":619},"这里小Mi有了一个新发现！说白了，约束条件",{"type":18,"tag":46,"props":621,"children":623},{"alt":7,"src":622},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/150919i5zclfzo1wcuhbym.png",[],{"type":24,"value":625},"或者",{"type":18,"tag":46,"props":627,"children":629},{"alt":7,"src":628},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/150942cfnvnmjcs2bgfr6e.png",[],{"type":24,"value":631},"是可以被",{"type":18,"tag":46,"props":633,"children":635},{"alt":7,"src":634},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/150957qggpengwdrhci3gv.png",[],{"type":24,"value":637},"这个约束所代替的。",{"type":18,"tag":46,"props":639,"children":641},{"alt":7,"src":640},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/151015zu344i4fqupb7wkc.png",[],{"type":24,"value":643},"，将其写入优化目标：",{"type":18,"tag":26,"props":645,"children":646},{},[647],{"type":18,"tag":46,"props":648,"children":650},{"alt":7,"src":649},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/151043vjidwofypjae7pjn.png",[],{"type":18,"tag":26,"props":652,"children":653},{},[654],{"type":18,"tag":46,"props":655,"children":657},{"alt":7,"src":656},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/151109xu8mmaafmgzb1xyk.png",[],{"type":18,"tag":26,"props":659,"children":660},{},[661,663,667,668,672],{"type":24,"value":662},"值得注意的是，优化目标函数可以被写成等于",{"type":18,"tag":46,"props":664,"children":666},{"alt":7,"src":665},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/151127gxwbusdx1kfw6n9y.png",[],{"type":24,"value":160},{"type":18,"tag":46,"props":669,"children":671},{"alt":7,"src":670},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/1511491cvufcvcsaitdai8.png",[],{"type":24,"value":673},"。以上图的这些训练样本为例，支持向量机又会选择什么样的决策界呢？",{"type":18,"tag":26,"props":675,"children":676},{},[677],{"type":18,"tag":46,"props":678,"children":680},{"alt":7,"src":679},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/151217bxdqp9kusgva2p7z.png",[],{"type":18,"tag":26,"props":682,"children":683},{},[684],{"type":24,"value":685},"我们假设支持向量机会选择绿线这个决策边界，但是这并不是一个非常好的选择，因为它的间距很小，决策界离训练样本的距离很近。",{"type":18,"tag":26,"props":687,"children":688},{},[689],{"type":18,"tag":46,"props":690,"children":692},{"alt":7,"src":691},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/151241tywawcfvntz7kkre.png",[],{"type":18,"tag":26,"props":694,"children":695},{},[696,698,702,704,708,710,714,716,720],{"type":24,"value":697},"对于参数选择，可以看到参数向量",{"type":18,"tag":46,"props":699,"children":701},{"alt":7,"src":700},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/151304dz80ndryqa6ea5cr.png",[],{"type":24,"value":703},"事实上是和决策界90度正交的，因此这个绿色的决策界对应着一个参数向量",{"type":18,"tag":46,"props":705,"children":707},{"alt":7,"src":706},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/151313h9geclawnhhz1c1c.png",[],{"type":24,"value":709},"，而",{"type":18,"tag":46,"props":711,"children":713},{"alt":7,"src":712},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/1513285ylzb9zwx1ikhvgc.png",[],{"type":24,"value":715},"的简化意味着决策界必须通过原点",{"type":18,"tag":46,"props":717,"children":719},{"alt":7,"src":718},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/151339j2dzwgxufmhtgjnz.png",[],{"type":24,"value":721},"。现在让我们看一下这对于优化目标函数意味着什么。",{"type":18,"tag":26,"props":723,"children":724},{},[725],{"type":18,"tag":46,"props":726,"children":728},{"alt":7,"src":727},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/151413pawspvoo4jatdt7i.png",[],{"type":18,"tag":26,"props":730,"children":731},{},[732,734,738,740,744,746,750,752,756,758,761,762,766,768,771,773,776,778,781,783,786],{"type":24,"value":733},"比如第一个样本",{"type":18,"tag":46,"props":735,"children":737},{"alt":7,"src":736},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/151433envyjvtcqyxyixcq.png",[],{"type":24,"value":739},"，该样本到",{"type":18,"tag":46,"props":741,"children":743},{"alt":7,"src":742},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/151446gmkvqybmzltmp5z1.png",[],{"type":24,"value":745},"的投影",{"type":18,"tag":46,"props":747,"children":749},{"alt":7,"src":748},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/151512nhbjuklvj6xbffjw.png",[],{"type":24,"value":751},"，是上图中这个短短的红线段。类似地，样本",{"type":18,"tag":46,"props":753,"children":755},{"alt":7,"src":754},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/151527q2vsfc7ruwhtxjod.png",[],{"type":24,"value":757},"到",{"type":18,"tag":46,"props":759,"children":760},{"alt":7,"src":742},[],{"type":24,"value":745},{"type":18,"tag":46,"props":763,"children":765},{"alt":7,"src":764},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/151548g4wuuhjomuag0zyf.png",[],{"type":24,"value":767},"，是上图中的粉色线段。实际上，",{"type":18,"tag":46,"props":769,"children":770},{"alt":7,"src":764},[],{"type":24,"value":772},"是一个负值，在相反的方向，由于向量",{"type":18,"tag":46,"props":774,"children":775},{"alt":7,"src":754},[],{"type":24,"value":777},"和参数向量",{"type":18,"tag":46,"props":779,"children":780},{"alt":7,"src":742},[],{"type":24,"value":782},"的夹角大于90度，所以",{"type":18,"tag":46,"props":784,"children":785},{"alt":7,"src":764},[],{"type":24,"value":787},"的值小于0。",{"type":18,"tag":26,"props":789,"children":790},{},[791,793,797,799,803,805,809,811,814,816,819,821,824],{"type":24,"value":792},"一个个试验了之后，我们会发现这些",{"type":18,"tag":46,"props":794,"children":796},{"alt":7,"src":795},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/151641pl9somdftibvbnhp.png",[],{"type":24,"value":798},"将会是非常小的数。而优化目标函数时，对于正样本而言，需要",{"type":18,"tag":46,"props":800,"children":802},{"alt":7,"src":801},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/151656ifktdjm8ywgvetni.png",[],{"type":24,"value":804},"，如果",{"type":18,"tag":46,"props":806,"children":808},{"alt":7,"src":807},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/151714cwfeqovzkosr3gb4.png",[],{"type":24,"value":810},"非常小，那就意味着",{"type":18,"tag":46,"props":812,"children":813},{"alt":7,"src":742},[],{"type":24,"value":815},"的范数需要非常大。但是我们的目标函数是希望找到一个参数",{"type":18,"tag":46,"props":817,"children":818},{"alt":7,"src":742},[],{"type":24,"value":820},"，它的范数是小的。因此，绿色的这条线并不是一个好的参数向量",{"type":18,"tag":46,"props":822,"children":823},{"alt":7,"src":742},[],{"type":24,"value":825},"的选择。",{"type":18,"tag":26,"props":827,"children":828},{},[829],{"type":18,"tag":46,"props":830,"children":832},{"alt":7,"src":831},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/151753bapdc1bht3v7yibn.png",[],{"type":18,"tag":26,"props":834,"children":835},{},[836,838,841,843,846,848,852,854,857,859,863,865,869],{"type":24,"value":837},"来看另一条完全不同的决策边界，图像会因此有很大不同。如果上图的绿线是决策边界，那么相对应的参数",{"type":18,"tag":46,"props":839,"children":840},{"alt":7,"src":742},[],{"type":24,"value":842},"的方向与之垂直，是纵向的直线。那么样本",{"type":18,"tag":46,"props":844,"children":845},{"alt":7,"src":736},[],{"type":24,"value":847},"在横轴",{"type":18,"tag":46,"props":849,"children":851},{"alt":7,"src":850},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/151934ewwszgjfz3gsljjw.png",[],{"type":24,"value":853},"上的投影就可以得出",{"type":18,"tag":46,"props":855,"children":856},{"alt":7,"src":748},[],{"type":24,"value":858},"。样本",{"type":18,"tag":46,"props":860,"children":862},{"alt":7,"src":861},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/1520074rfhfazulfjqjzrl.png",[],{"type":24,"value":864},"的投影为",{"type":18,"tag":46,"props":866,"children":868},{"alt":7,"src":867},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/152029jklilurohqra5dre.png",[],{"type":24,"value":870},"，是负值。",{"type":18,"tag":26,"props":872,"children":873},{},[874,876,879,880,883,885,889,891,894,896,899],{"type":24,"value":875},"这时候我们会注意到现在的",{"type":18,"tag":46,"props":877,"children":878},{"alt":7,"src":748},[],{"type":24,"value":60},{"type":18,"tag":46,"props":881,"children":882},{"alt":7,"src":867},[],{"type":24,"value":884},"长度比之前长多了。如果需要满足",{"type":18,"tag":46,"props":886,"children":888},{"alt":7,"src":887},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/152118j2wpuifp9rn73uyg.png",[],{"type":24,"value":890},"这个约束条件，现在",{"type":18,"tag":46,"props":892,"children":893},{"alt":7,"src":748},[],{"type":24,"value":895},"变大了，那么",{"type":18,"tag":46,"props":897,"children":898},{"alt":7,"src":742},[],{"type":24,"value":900},"的范数就可以变小了。",{"type":18,"tag":26,"props":902,"children":903},{},[904,906,909],{"type":24,"value":905},"这时候，支持向量机就可以使参数",{"type":18,"tag":46,"props":907,"children":908},{"alt":7,"src":742},[],{"type":24,"value":910},"的范数变小很多，这也是支持向量机为什么能够有效地产生大间距分类的原因。",{"type":18,"tag":26,"props":912,"children":913},{},[914],{"type":18,"tag":46,"props":915,"children":917},{"alt":7,"src":916},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/152239cvpnitu5lvhxtvh9.png",[],{"type":18,"tag":26,"props":919,"children":920},{},[921,923,926,928,932,934,938,940,943,945,949],{"type":24,"value":922},"总结来说，我们希望正样本和负样本投影到",{"type":18,"tag":46,"props":924,"children":925},{"alt":7,"src":742},[],{"type":24,"value":927},"的值足够大，要做到这一点的唯一方式就是选择这条绿线作为决边策界，通过大间距决策界来区分开正样本和负样本。实际上这个间距的值就是",{"type":18,"tag":46,"props":929,"children":931},{"alt":7,"src":930},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/152314zktaskd6rm66r9jh.png",[],{"type":24,"value":933},"等等的值。因此通过让间距变大，即使",{"type":18,"tag":46,"props":935,"children":937},{"alt":7,"src":936},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/152322wom7bkpj8cbl6isf.png",[],{"type":24,"value":939},"的值变大，支持向量机最终可以找到一个较小的",{"type":18,"tag":46,"props":941,"children":942},{"alt":7,"src":742},[],{"type":24,"value":944},"的范数。这正是支持向量机中最小化目标函数的目的，也是支持向量机最终会找到大间距分类器的原因。因为它试图极大化这些",{"type":18,"tag":46,"props":946,"children":948},{"alt":7,"src":947},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/152351tdj6klvvhqmfsuuy.png",[],{"type":24,"value":950},"的范数，它们是训练样本到决策边界的距离。",{"type":18,"tag":26,"props":952,"children":953},{},[954],{"type":18,"tag":46,"props":955,"children":957},{"alt":7,"src":956},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/1524095xer1fiojiffgzwe.png",[],{"type":18,"tag":26,"props":959,"children":960},{},[961,963,967,969,973],{"type":24,"value":962},"最后一点需要注意的是，我们的推导自始至终使用了",{"type":18,"tag":46,"props":964,"children":966},{"alt":7,"src":965},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/152428dcxfxcenuuamtxkd.png",[],{"type":24,"value":968},"这个简化假设。作用就是：最终得到的决策界是通过原点的。当然如果不简化公式，",{"type":18,"tag":46,"props":970,"children":972},{"alt":7,"src":971},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/152502jgnngech4ypzl7wp.png",[],{"type":24,"value":974},"的话，支持向量机产生大间距分类器的结论同样会被证明同样成立。（大家可以动动手自己推导试试哦~）",{"type":18,"tag":26,"props":976,"children":977},{},[978,980,984],{"type":24,"value":979},"废话不多说，今天小Mi想告诉大家的就是，支持向量机优化目标函数对应着C值非常大的情况，即便",{"type":18,"tag":46,"props":981,"children":983},{"alt":7,"src":982},"https://bbs-img.huaweicloud.com/data/forums/attachment/forum/202108/28/152529satvbd6q3pztm96n.png",[],{"type":24,"value":985},"不等于0，支持向量机仍然会找到正样本和负样本之间的大间距分隔。",{"type":18,"tag":26,"props":987,"children":988},{},[989],{"type":24,"value":990},"下周小Mi带大家学习如何利用支持向量机的原理建立一个复杂的非线性分类器，我们下周，再见呦（挥手十分钟！）。",{"title":7,"searchDepth":992,"depth":992,"links":993},4,[994],{"id":34,"depth":995,"text":41},2,"markdown","content:technology-blogs:zh:698.md","content","technology-blogs/zh/698.md","technology-blogs/zh/698","md",1776506139987]