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<div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/6250abd774bc5c01056fdec4/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">Sequen<span class="_ _0"></span>tial<span class="_ _1"> </span>Minimal<span class="_ _1"> </span>Optimization<span class="_ _1"> </span>for<span class="_ _1"> </span>SVM</div><div class="t m0 x2 h3 y2 ff2 fs1 fc0 sc0 ls0 ws0">Con<span class="_ _0"></span>ten<span class="_ _0"></span>ts</div><div class="t m0 x2 h4 y3 ff3 fs2 fc0 sc0 ls0 ws0">1<span class="_ _2"> </span>In<span class="_ _0"></span>tro<span class="_ _3"></span>duction<span class="_ _4"> </span>to<span class="_ _4"> </span>Supp<span class="_ _3"></span>ort<span class="_ _4"> </span>V<span class="_ _5"></span>ector<span class="_ _4"> </span>Mac<span class="_ _0"></span>hine<span class="_ _4"> </span>(SVM)<span class="_ _6"> </span>2</div><div class="t m0 x3 h4 y4 ff4 fs2 fc0 sc0 ls0 ws0">1.1<span class="_ _7"> </span>Linear<span class="_ _8"> </span>SVM<span class="_ _9"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _b"> </span>2</div><div class="t m0 x3 h4 y5 ff4 fs2 fc0 sc0 ls0 ws0">1.2<span class="_ _7"> </span>The<span class="_ _8"> </span>dual<span class="_ _8"> </span>problem<span class="_"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _b"> </span>3</div><div class="t m0 x3 h4 y6 ff4 fs2 fc0 sc0 ls0 ws0">1.3<span class="_ _7"> </span>Non-linear<span class="_ _8"> </span>SVM<span class="_ _c"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _b"> </span>4</div><div class="t m0 x3 h4 y7 ff4 fs2 fc0 sc0 ls0 ws0">1.4<span class="_ _7"> </span>Imp<span class="_ _3"></span>erfect<span class="_ _8"> </span>separation<span class="_ _d"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _b"> </span>5</div><div class="t m0 x3 h4 y8 ff4 fs2 fc0 sc0 ls0 ws0">1.5<span class="_ _7"> </span>The<span class="_ _8"> </span>KKT<span class="_ _8"> </span>conditions<span class="_ _e"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _b"> </span>6</div><div class="t m0 x3 h4 y9 ff4 fs2 fc0 sc0 ls0 ws0">1.6<span class="_ _7"> </span>Chec<span class="_ _0"></span>king<span class="_ _8"> </span>KKT<span class="_ _8"> </span>condition<span class="_ _8"> </span>without<span class="_ _8"> </span>using<span class="_ _8"> </span>threshold<span class="_ _8"> </span><span class="ff5">b<span class="_ _9"> </span></span>.<span class="_ _a"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _b"> </span>7</div><div class="t m0 x2 h4 ya ff3 fs2 fc0 sc0 ls0 ws0">2<span class="_ _2"> </span>SMO<span class="_ _4"> </span>Algorithm<span class="_ _f"> </span>9</div><div class="t m0 x3 h4 yb ff4 fs2 fc0 sc0 ls0 ws0">2.1<span class="_ _7"> </span>Optimize<span class="_ _8"> </span>t<span class="_ _0"></span>wo<span class="_ _8"> </span><span class="ff5">α</span></div><div class="t m0 x4 h5 yc ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x5 h4 yb ff4 fs2 fc0 sc0 ls0 ws0">’s<span class="_ _7"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _b"> </span>9</div><div class="t m0 x3 h4 yd ff4 fs2 fc0 sc0 ls0 ws0">2.2<span class="_ _7"> </span>SMO<span class="_ _8"> </span>Algorithm:<span class="_ _9"> </span>Up<span class="_ _3"></span>dating<span class="_ _8"> </span>after<span class="_ _8"> </span>a<span class="_ _8"> </span>successful<span class="_ _8"> </span>optimization<span class="_ _8"> </span>step<span class="_ _10"> </span>.<span class="_ _11"> </span>13</div><div class="t m0 x3 h4 ye ff4 fs2 fc0 sc0 ls0 ws0">2.3<span class="_ _7"> </span>SMO<span class="_ _8"> </span>Algorithm:<span class="_ _9"> </span>Pick<span class="_ _8"> </span>t<span class="_ _0"></span>w<span class="_ _0"></span>o<span class="_ _8"> </span><span class="ff5">α</span></div><div class="t m0 x6 h5 yf ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x7 h4 ye ff4 fs2 fc0 sc0 ls0 ws0">’s<span class="_ _8"> </span>for<span class="_ _8"> </span>optimization<span class="_ _9"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _11"> </span>14</div><div class="t m0 x2 h4 y10 ff3 fs2 fc0 sc0 ls0 ws0">3<span class="_ _2"> </span>C</div><div class="t m0 x8 h6 y11 ff7 fs4 fc0 sc0 ls0 ws0">+<span class="_ _5"></span>+</div><div class="t m0 x9 h4 y10 ff3 fs2 fc0 sc0 ls0 ws0">Implemen<span class="_ _0"></span>tation<span class="_ _12"> </span>15</div><div class="t m0 x3 h4 y12 ff4 fs2 fc0 sc0 ls0 ws0">3.1<span class="_ _7"> </span>The<span class="_ _8"> </span><span class="ff8">main<span class="_ _8"> </span></span>routine<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _11"> </span>15</div><div class="t m0 x3 h4 y13 ff4 fs2 fc0 sc0 ls0 ws0">3.2<span class="_ _7"> </span>The<span class="_ _8"> </span><span class="ff8">examineExample<span class="_ _8"> </span></span>routine<span class="_ _13"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _1"> </span>.<span class="_ _11"> </span>18</div><div class="t m0 x3 h4 y14 ff4 fs2 fc0 sc0 ls0 ws0">3.3<span class="_ _7"> </span>The<span class="_ _8"> </span><span class="ff8">takeStep<span class="_ _8"> </span></span>routine<span class="_ _10"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _1"> </span>.<span class="_ _11"> </span>20</div><div class="t m0 x3 h4 y15 ff4 fs2 fc0 sc0 ls0 ws0">3.4<span class="_ _7"> </span>Ev<span class="_ _0"></span>aluating<span class="_ _8"> </span>classification<span class="_ _8"> </span>function<span class="_ _d"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _11"> </span>24</div><div class="t m0 x3 h4 y16 ff4 fs2 fc0 sc0 ls0 ws0">3.5<span class="_ _7"> </span>F<span class="_ _5"></span>unctions<span class="_ _8"> </span>to<span class="_ _8"> </span>compute<span class="_ _8"> </span>dot<span class="_ _8"> </span>pro<span class="_ _3"></span>duct<span class="_ _14"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _11"> </span>26</div><div class="t m0 x3 h4 y17 ff4 fs2 fc0 sc0 ls0 ws0">3.6<span class="_ _7"> </span>Kernel<span class="_ _8"> </span>functions<span class="_ _2"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _11"> </span>28</div><div class="t m0 x3 h4 y18 ff4 fs2 fc0 sc0 ls0 ws0">3.7<span class="_ _7"> </span>Input<span class="_ _8"> </span>and<span class="_ _8"> </span>output<span class="_ _a"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _11"> </span>29</div><div class="t m0 xa h4 y19 ff4 fs2 fc0 sc0 ls0 ws0">3.7.1<span class="_ _15"> </span>Get<span class="_ _8"> </span>parameters<span class="_ _8"> </span>b<span class="_ _0"></span>y<span class="_ _8"> </span>command<span class="_ _8"> </span>line<span class="_ _9"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _11"> </span>29</div><div class="t m0 xa h4 y1a ff4 fs2 fc0 sc0 ls0 ws0">3.7.2<span class="_ _15"> </span>Read<span class="_ _8"> </span>in<span class="_ _8"> </span>data<span class="_ _16"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _11"> </span>31</div><div class="t m0 xa h4 y1b ff4 fs2 fc0 sc0 ls0 ws0">3.7.3<span class="_ _15"> </span>Sa<span class="_ _0"></span>ving<span class="_ _8"> </span>and<span class="_ _8"> </span>loading<span class="_ _8"> </span>mo<span class="_ _3"></span>del<span class="_ _8"> </span>parameters<span class="_ _17"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _11"> </span>34</div><div class="t m0 x3 h4 y1c ff4 fs2 fc0 sc0 ls0 ws0">3.8<span class="_ _7"> </span>Compute<span class="_ _8"> </span>error<span class="_ _8"> </span>rate<span class="_ _4"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _11"> </span>36</div><div class="t m0 x3 h4 y1d ff4 fs2 fc0 sc0 ls0 ws0">3.9<span class="_ _7"> </span>Multiclass<span class="_ _e"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _11"> </span>37</div><div class="t m0 x3 h4 y1e ff4 fs2 fc0 sc0 ls0 ws0">3.10<span class="_ _1"> </span>Mak<span class="_ _0"></span>efiles<span class="_ _c"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _1"> </span>.<span class="_ _a"> </span>.<span class="_ _11"> </span>40</div><div class="t m0 x2 h4 y1f ff3 fs2 fc0 sc0 ls0 ws0">A<span class="_ _d"> </span>The<span class="_ _4"> </span>weigh<span class="_ _0"></span>t<span class="_ _4"> </span>v<span class="_ _0"></span>ectors<span class="_ _4"> </span>of<span class="_ _4"> </span>the<span class="_ _4"> </span>parallel<span class="_ _4"> </span>supp<span class="_ _3"></span>orting<span class="_ _4"> </span>planes<span class="_ _18"> </span>41</div><div class="t m0 x2 h4 y20 ff3 fs2 fc0 sc0 ls0 ws0">B<span class="_ _19"> </span>The<span class="_ _4"> </span>ob<span class="_ _3"></span>jective<span class="_ _4"> </span>function<span class="_ _4"> </span>of<span class="_ _4"> </span>the<span class="_ _4"> </span>dual<span class="_ _4"> </span>problem<span class="_ _1a"> </span>42</div><div class="t m0 xb h4 y21 ff4 fs2 fc0 sc0 ls0 ws0">1</div><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a 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<div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/6250abd774bc5c01056fdec4/bg2.jpg"><div class="c xc y22 w2 h7"><div class="t m2 xd h8 y23 ff9 fs5 fc0 sc0 ls0 ws0"><span class="fc1 sc0"></span></div><div class="t m2 x8 h8 y24 ff9 fs5 fc0 sc0 ls0 ws0"><span class="fc1 sc0"></span></div><div class="t m2 xe h8 y25 ff9 fs5 fc0 sc0 ls0 ws0"><span class="fc1 sc0"></span></div><div class="t m2 xf h8 y26 ff9 fs5 fc0 sc0 ls0 ws0"><span class="fc1 sc0"></span></div><div class="t m2 x10 h8 y27 ff9 fs5 fc0 sc0 ls0 ws0"><span class="fc1 sc0"></span></div><div class="t m2 x11 h8 y28 ff9 fs5 fc0 sc0 ls0 ws0"><span class="fc1 sc0"></span></div><div class="t m2 x12 h8 y29 ff9 fs5 fc0 sc0 ls0 ws0"><span class="fc1 sc0"></span></div><div class="t m2 x13 h8 y2a ff9 fs5 fc0 sc0 ls0 ws0"><span class="fc1 sc0"></span></div><div class="t m2 x14 h8 y2b ff9 fs5 fc0 sc0 ls0 ws0"><span class="fc1 sc0"></span></div><div class="t m2 x15 h8 y2c ff9 fs5 fc0 sc0 ls0 ws0"><span class="fc1 sc0"></span></div><div class="t m2 x16 h8 y2d ff9 fs5 fc0 sc0 ls0 ws0"><span class="fc1 sc0"></span></div><div class="t m2 x17 h8 y2e ff9 fs5 fc0 sc0 ls0 ws0"><span class="fc1 sc0"></span></div><div class="t m2 x18 h8 y2f ff9 fs5 fc0 sc0 ls0 ws0"><span class="fc1 sc0"></span></div></div><div class="t m0 x19 h9 y30 ffa fs6 fc0 sc0 ls0 ws0">Abstract</div><div class="t m0 x1a ha y31 ffb fs6 fc0 sc0 ls0 ws0">This<span class="_ _4"> </span>is<span class="_ _8"> </span>a<span class="_ _4"> </span>C</div><div class="t m0 x1b h6 y32 ff7 fs4 fc0 sc0 ls0 ws0">+<span class="_ _5"></span>+</div><div class="t m0 x1c ha y31 ffb fs6 fc0 sc0 ls0 ws0">implemen<span class="_ _0"></span>tation<span class="_ _4"> </span>of<span class="_ _8"> </span>John<span class="_ _4"> </span>C.<span class="_ _4"> </span>Platt’s<span class="_ _8"> </span>sequential<span class="_ _8"> </span>minimal</div><div class="t m0 x1 ha y33 ffb fs6 fc0 sc0 ls0 ws0">optimization<span class="_ _8"> </span>(SMO)<span class="_ _8"> </span>for<span class="_ _4"> </span>training<span class="_ _8"> </span>a<span class="_ _8"> </span>supp<span class="_ _3"></span>ort<span class="_ _8"> </span>vector<span class="_ _8"> </span>mac<span class="_ _0"></span>hine<span class="_ _4"> </span>(SVM).<span class="_ _8"> </span>This</div><div class="t m0 x1 ha y34 ffb fs6 fc0 sc0 ls0 ws0">program<span class="_ _1b"> </span>is<span class="_ _1b"> </span>based<span class="_ _1b"> </span>on<span class="_ _1b"> </span>the<span class="_ _1b"> </span>pseudo<span class="_ _3"></span>code<span class="_ _1b"> </span>in<span class="_ _1b"> </span>Platt<span class="_ _1b"> </span>(1998).</div><div class="t m0 x1a ha y35 ffb fs6 fc0 sc0 ls0 ws0">This<span class="_ _1b"> </span>is<span class="_ _1b"> </span>b<span class="_ _3"></span>oth<span class="_ _1b"> </span>the<span class="_ _1b"> </span>documentation<span class="_ _1b"> </span>and<span class="_ _1b"> </span>the<span class="_ _1b"> </span>C</div><div class="t m0 x1d h6 y36 ff7 fs4 fc0 sc0 ls0 ws0">+<span class="_ _5"></span>+</div><div class="t m0 x1e ha y35 ffb fs6 fc0 sc0 ls0 ws0">co<span class="_ _3"></span>de.<span class="_ _4"> </span>It<span class="_ _1b"> </span>is<span class="_ _1b"> </span>a<span class="_ _1b"> </span><span class="ffc">NUWEB<span class="_ _1b"> </span></span>do<span class="_ _3"></span>c-</div><div class="t m0 x1 ha y37 ffb fs6 fc0 sc0 ls0 ws0">umen<span class="_ _0"></span>t<span class="_ _1b"> </span>from<span class="_ _1b"> </span>whic<span class="_ _0"></span>h<span class="_ _1b"> </span>b<span class="_ _3"></span>oth<span class="_ _1b"> </span>the<span class="_"> </span>L</div><div class="t m0 x1f hb y38 ffd fs7 fc0 sc0 ls0 ws0">A</div><div class="t m0 x20 ha y37 ffb fs6 fc0 sc0 ls0 ws0">T</div><div class="t m0 x21 ha y39 ffb fs6 fc0 sc0 ls0 ws0">E</div><div class="t m0 x22 ha y37 ffb fs6 fc0 sc0 ls0 ws0">X<span class="_ _1b"> </span>file<span class="_ _1b"> </span>and<span class="_"> </span>the<span class="_ _1b"> </span>C</div><div class="t m0 x23 h6 y3a ff7 fs4 fc0 sc0 ls0 ws0">+<span class="_ _5"></span>+</div><div class="t m0 x24 ha y37 ffb fs6 fc0 sc0 ls0 ws0">file<span class="_ _1b"> </span>can<span class="_ _1b"> </span>be<span class="_ _1b"> </span>generated.</div><div class="t m0 x1 ha y3b ffb fs6 fc0 sc0 ls0 ws0">The<span class="_"> </span>documentation<span class="_"> </span>is<span class="_ _1c"> </span>essen<span class="_ _0"></span>tially<span class="_"> </span>my<span class="_ _1c"> </span>notes<span class="_"> </span>when<span class="_"> </span>reading<span class="_ _1c"> </span>the<span class="_"> </span>papers<span class="_"> </span>(most</div><div class="t m0 x1 ha y3c ffb fs6 fc0 sc0 ls0 ws0">of<span class="_ _1b"> </span>them<span class="_ _1b"> </span>b<span class="_ _3"></span>eing<span class="_ _1b"> </span><span class="ffe">cut-and-p<span class="_ _0"></span>aste<span class="_ _1b"> </span><span class="ffb">from<span class="_ _1b"> </span>the<span class="_ _1b"> </span>papers).</span></span></div><div class="t m0 x2 h3 y3d ff2 fs1 fc0 sc0 ls0 ws0">1<span class="_ _1d"> </span>In<span class="_ _0"></span>tro<span class="_ _3"></span>duction<span class="_ _4"> </span>to<span class="_ _8"> </span>Supp<span class="_ _3"></span>ort<span class="_ _4"> </span>V<span class="_ _1e"></span>ector<span class="_ _4"> </span>Mac<span class="_ _0"></span>hine<span class="_ _8"> </span>(SVM)</div><div class="t m0 x2 h4 y3e ff4 fs2 fc0 sc0 ls0 ws0">This<span class="_"> </span>introductio<span class="_"> </span>to<span class="_"> </span>Supp<span class="_ _3"></span>ort<span class="_"> </span>V<span class="_ _5"></span>ector<span class="_"> </span>Machine<span class="_"> </span>for<span class="_"> </span>binary<span class="_"> </span>classification<span class="_"> </span>is<span class="_"> </span>based<span class="_"> </span>on</div><div class="t m0 x2 h4 y3f ff4 fs2 fc0 sc0 ls0 ws0">Burges<span class="_ _8"> </span>(1998).</div><div class="t m0 x2 hc y40 ff2 fs8 fc0 sc0 ls0 ws0">1.1<span class="_ _1f"> </span>Linear<span class="_ _9"> </span>SVM</div><div class="t m0 x2 h4 y41 ff4 fs2 fc0 sc0 ls0 ws0">First<span class="_ _4"> </span>let<span class="_ _4"> </span>us<span class="_ _8"> </span>lo<span class="_ _3"></span>ok<span class="_ _4"> </span>at<span class="_ _4"> </span>the<span class="_ _8"> </span>linear<span class="_ _4"> </span>supp<span class="_ _3"></span>ort<span class="_ _4"> </span>v<span class="_ _0"></span>ector<span class="_ _4"> </span>mac<span class="_ _0"></span>hine.<span class="_ _e"> </span>It<span class="_ _4"> </span>is<span class="_ _4"> </span>based<span class="_ _4"> </span>on<span class="_ _8"> </span>the<span class="_ _4"> </span>idea</div><div class="t m0 x2 h4 y42 ff4 fs2 fc0 sc0 ls0 ws0">of<span class="_ _8"> </span>h<span class="_ _0"></span>yp<span class="_ _3"></span>erplane<span class="_ _1b"> </span>classifier,<span class="_ _8"> </span>or<span class="_ _8"> </span>linearly<span class="_ _8"> </span>separability<span class="_ _5"></span>.</div><div class="t m0 x3 hd y43 ff4 fs2 fc0 sc0 ls0 ws0">Supp<span class="_ _3"></span>ose<span class="_ _a"> </span>w<span class="_ _0"></span>e<span class="_ _a"> </span>hav<span class="_ _0"></span>e<span class="_ _a"> </span><span class="ff5">N<span class="_ _d"> </span></span>training<span class="_ _a"> </span>data<span class="_ _a"> </span>points<span class="_ _a"> </span><span class="fff">{</span>(<span class="ff3">x</span></div><div class="t m0 x23 he y44 ff10 fs3 fc0 sc0 ls0 ws0">1</div><div class="t m0 x25 hf y43 ff5 fs2 fc0 sc0 ls0 ws0">,<span class="_ _20"> </span>y</div><div class="t m0 x24 he y44 ff10 fs3 fc0 sc0 ls0 ws0">1</div><div class="t m0 x26 h4 y43 ff4 fs2 fc0 sc0 ls0 ws0">)<span class="ff5">,<span class="_ _20"> </span></span>(<span class="ff3">x</span></div><div class="t m0 x27 he y44 ff10 fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x28 hf y43 ff5 fs2 fc0 sc0 ls0 ws0">,<span class="_ _20"> </span>y</div><div class="t m0 x29 he y44 ff10 fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x2a h4 y43 ff4 fs2 fc0 sc0 ls0 ws0">)<span class="ff5">,<span class="_ _20"> </span>.<span class="_ _20"> </span>.<span class="_ _20"> </span>.<span class="_ _20"> </span>,<span class="_ _20"> </span></span>(<span class="ff3">x</span></div><div class="t m0 x2b h5 y44 ff6 fs3 fc0 sc0 ls0 ws0">N</div><div class="t m0 x2c hf y43 ff5 fs2 fc0 sc0 ls0 ws0">,<span class="_ _20"> </span>y</div><div class="t m0 x2d h5 y44 ff6 fs3 fc0 sc0 ls0 ws0">N</div><div class="t m0 x2e hd y43 ff4 fs2 fc0 sc0 ls0 ws0">)<span class="fff">}</span></div><div class="t m0 x2 h4 y45 ff4 fs2 fc0 sc0 ls0 ws0">where<span class="_ _a"> </span><span class="ff3">x</span></div><div class="t m0 x2f h5 y46 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x30 hd y45 fff fs2 fc0 sc0 ls0 ws0">∈<span class="_ _a"> </span>R</div><div class="t m0 x31 h5 y47 ff6 fs3 fc0 sc0 ls0 ws0">d</div><div class="t m0 x32 h4 y45 ff4 fs2 fc0 sc0 ls0 ws0">and<span class="_ _a"> </span><span class="ff5">y</span></div><div class="t m0 x1c h5 y46 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x33 hd y45 fff fs2 fc0 sc0 ls0 ws0">∈<span class="_ _a"> </span>{±<span class="ff4">1</span>}<span class="ff4">.<span class="_ _14"> </span>W<span class="_ _5"></span>e<span class="_ _a"> </span>w<span class="_ _0"></span>ould<span class="_ _a"> </span>lik<span class="_ _0"></span>e<span class="_ _a"> </span>to<span class="_ _9"> </span>learn<span class="_ _a"> </span>a<span class="_ _9"> </span>linear<span class="_ _a"> </span>separating</span></div><div class="t m0 x2 h4 y48 ff4 fs2 fc0 sc0 ls0 ws0">h<span class="_ _0"></span>yp<span class="_ _3"></span>erplane<span class="_ _1b"> </span>classifier:</div><div class="t m0 x1 hd y49 ff5 fs2 fc0 sc0 ls0 ws0">f<span class="_ _21"></span><span class="ff4">(<span class="ff3">x</span>)<span class="_"> </span>=<span class="_"> </span></span>sg<span class="_ _3"></span>n<span class="ff4">(<span class="ff3">w<span class="_ _1c"> </span><span class="fff">·<span class="_ _1c"> </span></span>x<span class="_ _1c"> </span><span class="fff">−<span class="_ _1c"> </span></span></span></span>b<span class="ff4">)</span>.</div><div class="t m0 x3 h4 y4a ff4 fs2 fc0 sc0 ls0 ws0">F<span class="_ _5"></span>urthermore,<span class="_"> </span>we<span class="_ _1c"> </span>w<span class="_ _0"></span>ant<span class="_ _1c"> </span>this<span class="_ _22"> </span>h<span class="_ _0"></span>yp<span class="_ _3"></span>erplane<span class="_ _1c"> </span>to<span class="_"> </span>ha<span class="_ _0"></span>v<span class="_ _0"></span>e<span class="_"> </span>the<span class="_ _1c"> </span>maxim<span class="_ _0"></span>um<span class="_"> </span>separating<span class="_ _1c"> </span>mar-</div><div class="t m0 x2 h4 y4b ff4 fs2 fc0 sc0 ls0 ws0">gin<span class="_"> </span>with<span class="_"> </span>resp<span class="_ _3"></span>ect<span class="_"> </span>to<span class="_"> </span>the<span class="_"> </span>tw<span class="_ _0"></span>o<span class="_"> </span>classes.<span class="_ _9"> </span>Sp<span class="_ _3"></span>ecifically<span class="_ _5"></span>,<span class="_ _1b"> </span>we<span class="_"> </span>w<span class="_ _0"></span>an<span class="_ _0"></span>t<span class="_"> </span>to<span class="_ _1b"> </span>find<span class="_"> </span>this<span class="_"> </span>hyperplane</div><div class="t m0 x2 hd y4c ff5 fs2 fc0 sc0 ls0 ws0">H<span class="_ _4"> </span><span class="ff4">:<span class="_ _1"> </span></span>y<span class="_ _1b"> </span><span class="ff4">=<span class="_"> </span><span class="ff3">w<span class="_ _21"></span><span class="fff">·<span class="_ _3"></span></span>x<span class="_ _21"></span><span class="fff">−<span class="_ _3"></span></span></span></span>b<span class="_ _1b"> </span><span class="ff4">=<span class="_"> </span>0<span class="_ _1c"> </span>and<span class="_"> </span>t<span class="_ _0"></span>wo<span class="_ _22"> </span>h<span class="_ _0"></span>yp<span class="_ _3"></span>erplanes<span class="_"> </span>parallel<span class="_ _1c"> </span>to<span class="_"> </span>it<span class="_"> </span>and<span class="_ _1c"> </span>with<span class="_"> </span>equal<span class="_"> </span>distances</span></div><div class="t m0 x2 h4 y4d ff4 fs2 fc0 sc0 ls0 ws0">to<span class="_ _8"> </span>it,</div><div class="t m0 x1 hf y4e ff5 fs2 fc0 sc0 ls0 ws0">H</div><div class="t m0 x34 he y4f ff10 fs3 fc0 sc0 ls0 ws0">1</div><div class="t m0 x35 hd y4e ff4 fs2 fc0 sc0 ls0 ws0">:<span class="_ _d"> </span><span class="ff5">y<span class="_ _1b"> </span></span>=<span class="_"> </span><span class="ff3">w<span class="_ _1c"> </span><span class="fff">·<span class="_ _1c"> </span></span>x<span class="_ _1c"> </span><span class="fff">−<span class="_ _1c"> </span><span class="ff5">b<span class="_ _1b"> </span></span></span></span>=<span class="_"> </span>+1<span class="ff5">,</span></div><div class="t m0 xb h4 y21 ff4 fs2 fc0 sc0 ls0 ws0">2</div><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a></div><div class="pi" data-data='{"ctm":[1.568633,0.000000,0.000000,1.568633,0.000000,0.000000]}'></div></div>
<div id="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/6250abd774bc5c01056fdec4/bg3.jpg"><div class="t m0 x1 hf y50 ff5 fs2 fc0 sc0 ls0 ws0">H</div><div class="t m0 x34 he y51 ff10 fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x35 hd y50 ff4 fs2 fc0 sc0 ls0 ws0">:<span class="_ _d"> </span><span class="ff5">y<span class="_ _1b"> </span></span>=<span class="_"> </span><span class="ff3">w<span class="_ _1c"> </span><span class="fff">·<span class="_ _1c"> </span></span>x<span class="_ _1c"> </span><span class="fff">−<span class="_ _1c"> </span><span class="ff5">b<span class="_ _1b"> </span></span></span></span>=<span class="_"> </span><span class="fff">−</span>1<span class="ff5">,</span></div><div class="t m0 x2 h4 y52 ff4 fs2 fc0 sc0 ls0 ws0">with<span class="_ _4"> </span>the<span class="_ _8"> </span>condition<span class="_ _4"> </span>that<span class="_ _8"> </span>there<span class="_ _4"> </span>are<span class="_ _8"> </span>no<span class="_ _4"> </span>data<span class="_ _8"> </span>p<span class="_ _3"></span>oints<span class="_ _8"> </span>b<span class="_ _3"></span>et<span class="_ _0"></span>ween<span class="_ _8"> </span><span class="ff5">H</span></div><div class="t m0 x36 he y53 ff10 fs3 fc0 sc0 ls0 ws0">1</div><div class="t m0 x37 h4 y52 ff4 fs2 fc0 sc0 ls0 ws0">and<span class="_ _4"> </span><span class="ff5">H</span></div><div class="t m0 x38 he y53 ff10 fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x39 h4 y52 ff4 fs2 fc0 sc0 ls0 ws0">,<span class="_ _4"> </span>and<span class="_ _8"> </span>the</div><div class="t m0 x2 h4 y54 ff4 fs2 fc0 sc0 ls0 ws0">distance<span class="_ _8"> </span>b<span class="_ _3"></span>et<span class="_ _0"></span>w<span class="_ _0"></span>een<span class="_ _8"> </span><span class="ff5">H</span></div><div class="t m0 x3a he y55 ff10 fs3 fc0 sc0 ls0 ws0">1</div><div class="t m0 x3b h4 y54 ff4 fs2 fc0 sc0 ls0 ws0">and<span class="_ _1b"> </span><span class="ff5">H</span></div><div class="t m0 x3c he y55 ff10 fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x3d h4 y54 ff4 fs2 fc0 sc0 ls0 ws0">is<span class="_ _8"> </span>maximized.</div><div class="t m0 x3 h4 y56 ff4 fs2 fc0 sc0 ls0 ws0">F<span class="_ _5"></span>or<span class="_"> </span>an<span class="_ _0"></span>y<span class="_ _22"> </span>separating<span class="_ _1c"> </span>plane<span class="_"> </span><span class="ff5">H<span class="_ _1b"> </span></span>and<span class="_ _1c"> </span>the<span class="_"> </span>corresponding<span class="_"> </span><span class="ff5">H</span></div><div class="t m0 x3e he y57 ff10 fs3 fc0 sc0 ls0 ws0">1</div><div class="t m0 x3f h4 y56 ff4 fs2 fc0 sc0 ls0 ws0">and<span class="_ _1c"> </span><span class="ff5">H</span></div><div class="t m0 x2a he y57 ff10 fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x40 h4 y56 ff4 fs2 fc0 sc0 ls0 ws0">,<span class="_"> </span>w<span class="_ _0"></span>e<span class="_ _1c"> </span>can<span class="_"> </span>alw<span class="_ _0"></span>ays</div><div class="t m0 x2 h4 y58 ff4 fs2 fc0 sc0 ls0 ws0">“normalize”<span class="_ _4"> </span>the<span class="_ _4"> </span>co<span class="_ _3"></span>efficients<span class="_ _4"> </span>v<span class="_ _0"></span>ector<span class="_ _4"> </span><span class="ff3">w<span class="_ _9"> </span></span>so<span class="_ _4"> </span>that<span class="_ _4"> </span><span class="ff5">H</span></div><div class="t m0 x41 he y59 ff10 fs3 fc0 sc0 ls0 ws0">1</div><div class="t m0 x23 hd y58 ff4 fs2 fc0 sc0 ls0 ws0">will<span class="_ _4"> </span>b<span class="_ _3"></span>e<span class="_ _4"> </span><span class="ff5">y<span class="_ _9"> </span></span>=<span class="_ _8"> </span><span class="ff3">w<span class="_"> </span><span class="fff">·<span class="_ _1c"> </span></span>x<span class="_ _1b"> </span><span class="fff">−<span class="_ _1c"> </span><span class="ff5">b<span class="_ _4"> </span></span></span></span>=<span class="_ _4"> </span>+1,</div><div class="t m0 x2 h4 y5a ff4 fs2 fc0 sc0 ls0 ws0">and<span class="_ _1b"> </span><span class="ff5">H</span></div><div class="t m0 x42 he y5b ff10 fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x43 hd y5a ff4 fs2 fc0 sc0 ls0 ws0">will<span class="_ _8"> </span>b<span class="_ _3"></span>e<span class="_ _1b"> </span><span class="ff5">y<span class="_ _1b"> </span></span>=<span class="_"> </span><span class="ff3">w<span class="_ _22"> </span><span class="fff">·<span class="_ _1c"> </span></span>x<span class="_ _1c"> </span><span class="fff">−<span class="_ _1c"> </span><span class="ff5">b<span class="_ _1b"> </span></span></span></span>=<span class="_"> </span><span class="fff">−</span>1.<span class="_ _9"> </span>See<span class="_ _1b"> </span>App<span class="_ _3"></span>endix<span class="_ _1b"> </span>A<span class="_ _8"> </span>for<span class="_ _8"> </span>details.</div><div class="t m0 x3 h4 y5c ff4 fs2 fc0 sc0 ls0 ws0">W<span class="_ _5"></span>e<span class="_"> </span>w<span class="_ _0"></span>an<span class="_ _0"></span>t<span class="_"> </span>to<span class="_ _1c"> </span>maximize<span class="_ _1c"> </span>the<span class="_ _22"> </span>distance<span class="_ _1c"> </span>b<span class="_ _3"></span>etw<span class="_ _0"></span>een<span class="_ _1c"> </span><span class="ff5">H</span></div><div class="t m0 x1d he y5d ff10 fs3 fc0 sc0 ls0 ws0">1</div><div class="t m0 x44 h4 y5c ff4 fs2 fc0 sc0 ls0 ws0">and<span class="_ _1c"> </span><span class="ff5">H</span></div><div class="t m0 x45 he y5d ff10 fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x3f h4 y5c ff4 fs2 fc0 sc0 ls0 ws0">.<span class="_ _9"> </span>So<span class="_ _1c"> </span>there<span class="_ _1c"> </span>will<span class="_ _22"> </span>b<span class="_ _3"></span>e<span class="_ _1c"> </span>some</div><div class="t m0 x2 h4 y5e ff4 fs2 fc0 sc0 ls0 ws0">p<span class="_ _3"></span>ositiv<span class="_ _0"></span>e<span class="_ _4"> </span>examples<span class="_ _4"> </span>on<span class="_ _8"> </span><span class="ff5">H</span></div><div class="t m0 x46 he y5f ff10 fs3 fc0 sc0 ls0 ws0">1</div><div class="t m0 x47 h4 y5e ff4 fs2 fc0 sc0 ls0 ws0">and<span class="_ _4"> </span>some<span class="_ _4"> </span>negativ<span class="_ _0"></span>e<span class="_ _4"> </span>examples<span class="_ _4"> </span>on<span class="_ _8"> </span><span class="ff5">H</span></div><div class="t m0 x48 he y5f ff10 fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x49 h4 y5e ff4 fs2 fc0 sc0 ls0 ws0">.<span class="_ _e"> </span>These<span class="_ _4"> </span>examples</div><div class="t m0 x2 h4 y60 ff4 fs2 fc0 sc0 ls0 ws0">are<span class="_ _a"> </span>called<span class="_ _1"> </span><span class="ff11">supp<span class="_ _0"></span>ort<span class="_ _a"> </span>ve<span class="_ _0"></span>ctors<span class="_ _1"> </span><span class="ff4">because<span class="_ _1"> </span>only<span class="_ _a"> </span>they<span class="_ _a"> </span>participate<span class="_ _1"> </span>in<span class="_ _a"> </span>the<span class="_ _1"> </span>definition<span class="_ _a"> </span>of</span></span></div><div class="t m0 x2 h4 y61 ff4 fs2 fc0 sc0 ls0 ws0">the<span class="_ _4"> </span>separating<span class="_ _8"> </span>hyperplane,<span class="_ _4"> </span>and<span class="_ _4"> </span>other<span class="_ _8"> </span>examples<span class="_ _4"> </span>can<span class="_ _4"> </span>be<span class="_ _4"> </span>remo<span class="_ _0"></span>ved<span class="_ _8"> </span>and/or<span class="_ _4"> </span>mo<span class="_ _0"></span>ved</div><div class="t m0 x2 h4 y62 ff4 fs2 fc0 sc0 ls0 ws0">around<span class="_ _1b"> </span>as<span class="_ _8"> </span>long<span class="_ _8"> </span>as<span class="_ _4"> </span>they<span class="_ _1b"> </span>do<span class="_ _1b"> </span>not<span class="_ _8"> </span>cross<span class="_ _4"> </span>the<span class="_ _1b"> </span>planes<span class="_ _8"> </span><span class="ff5">H</span></div><div class="t m0 x4a he y63 ff10 fs3 fc0 sc0 ls0 ws0">1</div><div class="t m0 x4b h4 y62 ff4 fs2 fc0 sc0 ls0 ws0">and<span class="_ _1b"> </span><span class="ff5">H</span></div><div class="t m0 x4c he y63 ff10 fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x4d h4 y62 ff4 fs2 fc0 sc0 ls0 ws0">.</div><div class="t m0 x3 h4 y64 ff4 fs2 fc0 sc0 ls0 ws0">Recall<span class="_ _1c"> </span>that<span class="_ _1c"> </span>in<span class="_ _22"> </span>2-D,<span class="_ _1c"> </span>the<span class="_ _1c"> </span>distance<span class="_ _22"> </span>from<span class="_ _1c"> </span>a<span class="_ _1c"> </span>p<span class="_ _3"></span>oin<span class="_ _0"></span>t<span class="_ _22"> </span>(<span class="ff5">x</span></div><div class="t m0 x4a he y65 ff10 fs3 fc0 sc0 ls0 ws0">0</div><div class="t m0 x4e hf y64 ff5 fs2 fc0 sc0 ls0 ws0">,<span class="_ _20"> </span>y</div><div class="t m0 x4f he y65 ff10 fs3 fc0 sc0 ls0 ws0">0</div><div class="t m0 x50 h4 y64 ff4 fs2 fc0 sc0 ls0 ws0">)<span class="_ _1c"> </span>to<span class="_ _1c"> </span>a<span class="_ _22"> </span>line<span class="_ _1c"> </span><span class="ff5">Ax<span class="_ _3"></span></span>+<span class="ff5">B<span class="_ _3"></span>y<span class="_ _3"></span></span>+<span class="_ _3"></span><span class="ff5">C<span class="_ _1b"> </span></span>=<span class="_"> </span>0</div><div class="t m0 x2 h4 y66 ff4 fs2 fc0 sc0 ls0 ws0">is</div><div class="t m0 x51 h5 y67 ff12 fs3 fc0 sc0 ls0 ws0">|<span class="ff6">Ax</span></div><div class="t m0 x1 h10 y68 ff13 fs4 fc0 sc0 ls0 ws0">0</div><div class="t m0 x42 h5 y67 ff10 fs3 fc0 sc0 ls0 ws0">+<span class="ff6">B<span class="_ _3"></span>y</span></div><div class="t m0 x30 h10 y68 ff13 fs4 fc0 sc0 ls0 ws0">0</div><div class="t m0 x52 h5 y67 ff10 fs3 fc0 sc0 ls0 ws0">+<span class="ff6">C<span class="_ _3"></span><span class="ff12">|</span></span></div><div class="t m0 x53 h5 y69 ff12 fs3 fc0 sc0 ls0 ws0">√</div><div class="t m0 x42 h5 y6a ff6 fs3 fc0 sc0 ls0 ws0">A</div><div class="t m0 x34 h10 y6b ff13 fs4 fc0 sc0 ls0 ws0">2</div><div class="t m0 x9 h5 y6a ff10 fs3 fc0 sc0 ls0 ws0">+<span class="ff6">B</span></div><div class="t m0 x54 h10 y6b ff13 fs4 fc0 sc0 ls0 ws0">2</div><div class="t m0 x31 h4 y66 ff4 fs2 fc0 sc0 ls0 ws0">.<span class="_ _1"> </span>Similarly<span class="_ _23"></span>,<span class="_ _4"> </span>the<span class="_ _8"> </span>distance<span class="_ _4"> </span>of<span class="_ _4"> </span>a<span class="_ _8"> </span>p<span class="_ _3"></span>oin<span class="_ _0"></span>t<span class="_ _4"> </span>on<span class="_ _8"> </span><span class="ff5">H</span></div><div class="t m0 x45 he y6c ff10 fs3 fc0 sc0 ls0 ws0">1</div><div class="t m0 x55 hd y66 ff4 fs2 fc0 sc0 ls0 ws0">to<span class="_ _4"> </span><span class="ff5">H<span class="_ _4"> </span></span>:<span class="_ _19"> </span><span class="ff3">w<span class="_ _24"> </span><span class="fff">·<span class="_ _1c"> </span></span>x<span class="_ _22"> </span><span class="fff">−<span class="_ _22"> </span><span class="ff5">b<span class="_ _8"> </span></span></span></span>=<span class="_ _8"> </span>0</div><div class="t m0 x2 h4 y6d ff4 fs2 fc0 sc0 ls0 ws0">is</div><div class="t m0 x51 h5 y6e ff12 fs3 fc0 sc0 ls0 ws0">|<span class="ff14">w</span>·<span class="ff14">x</span>−<span class="ff6">b</span>|</div><div class="t m0 x56 h5 y6f ff12 fs3 fc0 sc0 ls0 ws0">k<span class="ff14">w</span>k</div><div class="t m0 x57 h4 y6d ff4 fs2 fc0 sc0 ls0 ws0">=</div><div class="t m0 x31 he y70 ff10 fs3 fc0 sc0 ls0 ws0">1</div><div class="t m0 x58 h5 y6f ff12 fs3 fc0 sc0 ls0 ws0">k<span class="ff14">w</span>k</div><div class="t m0 x59 h4 y6d ff4 fs2 fc0 sc0 ls0 ws0">,<span class="_ _9"> </span>and<span class="_ _9"> </span>the<span class="_ _9"> </span>distance<span class="_ _4"> </span>b<span class="_ _3"></span>et<span class="_ _0"></span>ween<span class="_ _4"> </span><span class="ff5">H</span></div><div class="t m0 x5a he y71 ff10 fs3 fc0 sc0 ls0 ws0">1</div><div class="t m0 x44 h4 y6d ff4 fs2 fc0 sc0 ls0 ws0">and<span class="_ _9"> </span><span class="ff5">H</span></div><div class="t m0 x5b he y71 ff10 fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x5c h4 y6d ff4 fs2 fc0 sc0 ls0 ws0">is</div><div class="t m0 x29 he y70 ff10 fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x5d h5 y6f ff12 fs3 fc0 sc0 ls0 ws0">k<span class="ff14">w</span>k</div><div class="t m0 x5e h4 y6d ff4 fs2 fc0 sc0 ls0 ws0">.<span class="_ _19"> </span>So,<span class="_ _9"> </span>in<span class="_ _9"> </span>order</div><div class="t m0 x2 hd y72 ff4 fs2 fc0 sc0 ls0 ws0">to<span class="_ _8"> </span>maximize<span class="_ _8"> </span>the<span class="_ _8"> </span>distance,<span class="_ _4"> </span>w<span class="_ _23"></span>e<span class="_ _4"> </span>should<span class="_ _1b"> </span>minimize<span class="_ _8"> </span><span class="fff">k<span class="ff3">w<span class="_ _3"></span></span>k<span class="_ _24"> </span></span>=<span class="_"> </span><span class="ff3">w</span></div><div class="t m0 x5f h5 y73 ff6 fs3 fc0 sc0 ls0 ws0">T</div><div class="t m0 x55 h4 y72 ff3 fs2 fc0 sc0 ls0 ws0">w<span class="_ _8"> </span><span class="ff4">with<span class="_ _4"> </span>the<span class="_ _1b"> </span>condition</span></div><div class="t m0 x2 h4 y74 ff4 fs2 fc0 sc0 ls0 ws0">that<span class="_ _8"> </span>there<span class="_ _8"> </span>are<span class="_ _8"> </span>no<span class="_ _8"> </span>data<span class="_ _8"> </span>p<span class="_ _3"></span>oin<span class="_ _23"></span>ts<span class="_ _8"> </span>b<span class="_ _3"></span>etw<span class="_ _23"></span>een<span class="_ _8"> </span><span class="ff5">H</span></div><div class="t m0 x60 he y75 ff10 fs3 fc0 sc0 ls0 ws0">1</div><div class="t m0 x61 h4 y74 ff4 fs2 fc0 sc0 ls0 ws0">and<span class="_ _1b"> </span><span class="ff5">H</span></div><div class="t m0 x62 he y75 ff10 fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x63 h4 y74 ff4 fs2 fc0 sc0 ls0 ws0">:</div><div class="t m0 x1 hd y76 ff3 fs2 fc0 sc0 ls0 ws0">w<span class="_ _1c"> </span><span class="fff">·<span class="_ _1c"> </span></span>x<span class="_ _1c"> </span><span class="fff">−<span class="_ _1c"> </span><span class="ff5">b<span class="_ _24"> </span></span>≥<span class="_ _24"> </span><span class="ff4">+1<span class="ff5">,<span class="_ _25"> </span></span>for<span class="_ _1b"> </span>p<span class="_ _3"></span>ositive<span class="_ _1b"> </span>examples<span class="_ _8"> </span><span class="ff5">y</span></span></span></div><div class="t m0 x64 h5 y77 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x1d h4 y76 ff4 fs2 fc0 sc0 ls0 ws0">=<span class="_"> </span>+1<span class="ff5">,</span></div><div class="t m0 x1 hd y78 ff3 fs2 fc0 sc0 ls0 ws0">w<span class="_ _1c"> </span><span class="fff">·<span class="_ _1c"> </span></span>x<span class="_ _1c"> </span><span class="fff">−<span class="_ _1c"> </span><span class="ff5">b<span class="_ _24"> </span></span>≤<span class="_ _24"> </span>−<span class="ff4">1<span class="ff5">,<span class="_ _25"> </span></span>for<span class="_ _1b"> </span>negative<span class="_ _1b"> </span>examples<span class="_ _8"> </span><span class="ff5">y</span></span></span></div><div class="t m0 x65 h5 y79 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x62 hd y78 ff4 fs2 fc0 sc0 ls0 ws0">=<span class="_"> </span><span class="fff">−</span>1<span class="ff5">.</span></div><div class="t m0 x2 h4 y7a ff4 fs2 fc0 sc0 ls0 ws0">These<span class="_ _8"> </span>t<span class="_ _0"></span>wo<span class="_ _1b"> </span>conditions<span class="_ _8"> </span>can<span class="_ _8"> </span>b<span class="_ _3"></span>e<span class="_ _1b"> </span>combined<span class="_ _1b"> </span>into</div><div class="t m0 x1 hf y7b ff5 fs2 fc0 sc0 ls0 ws0">y</div><div class="t m0 x66 h5 y7c ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x34 hd y7b ff4 fs2 fc0 sc0 ls0 ws0">(<span class="ff3">w<span class="_ _1c"> </span><span class="fff">·<span class="_ _1c"> </span></span>x</span></div><div class="t m0 x58 h5 y7c ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x31 hd y7b fff fs2 fc0 sc0 ls0 ws0">−<span class="_ _1c"> </span><span class="ff5">b<span class="ff4">)<span class="_"> </span></span></span>≥<span class="_ _24"> </span><span class="ff4">1<span class="ff5">.</span></span></div><div class="t m0 x3 h4 y7d ff4 fs2 fc0 sc0 ls0 ws0">So<span class="_ _8"> </span>our<span class="_ _1b"> </span>problem<span class="_ _8"> </span>can<span class="_ _8"> </span>b<span class="_ _3"></span>e<span class="_ _8"> </span>formulated<span class="_ _1b"> </span>as</div><div class="t m0 x1 h4 y7e ff4 fs2 fc0 sc0 ls0 ws0">min</div><div class="t m0 x67 h5 y7f ff14 fs3 fc0 sc0 ls0 ws0">w<span class="ff6">,b</span></div><div class="t m0 x30 h4 y80 ff4 fs2 fc0 sc0 ls0 ws0">1</div><div class="t m0 x30 h4 y81 ff4 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x54 h4 y7e ff3 fs2 fc0 sc0 ls0 ws0">w</div><div class="t m0 x58 h5 y82 ff6 fs3 fc0 sc0 ls0 ws0">T</div><div class="t m0 xc h4 y7e ff3 fs2 fc0 sc0 ls0 ws0">w<span class="_ _1f"> </span><span class="ff4">sub<span class="_ _3"></span>ject<span class="_ _8"> </span>to<span class="_ _8"> </span><span class="ff5">y</span></span></div><div class="t m0 x68 h5 y83 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x69 hd y7e ff4 fs2 fc0 sc0 ls0 ws0">(<span class="ff3">w<span class="_ _1c"> </span><span class="fff">·<span class="_ _1c"> </span></span>x</span></div><div class="t m0 x6a h5 y83 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x6b hd y7e fff fs2 fc0 sc0 ls0 ws0">−<span class="_ _1c"> </span><span class="ff5">b<span class="ff4">)<span class="_"> </span></span></span>≥<span class="_ _24"> </span><span class="ff4">1<span class="ff5">.</span></span></div><div class="t m0 x2 h4 y84 ff4 fs2 fc0 sc0 ls0 ws0">This<span class="_ _8"> </span>is<span class="_ _8"> </span>a<span class="_ _8"> </span>con<span class="_ _0"></span>vex,<span class="_ _1b"> </span>quadratic<span class="_ _8"> </span>programming<span class="_ _8"> </span>problem<span class="_ _8"> </span>(in<span class="_ _8"> </span><span class="ff3">w<span class="_ _3"></span><span class="ff5">,<span class="_ _20"> </span>b</span></span>),<span class="_ _1b"> </span>in<span class="_ _8"> </span>a<span class="_ _8"> </span>conv<span class="_ _23"></span>ex<span class="_ _8"> </span>set.</div><div class="t m0 x3 h4 y85 ff4 fs2 fc0 sc0 ls0 ws0">In<span class="_ _0"></span>tro<span class="_ _3"></span>ducing<span class="_ _8"> </span>Lagrange<span class="_ _4"> </span>m<span class="_ _23"></span>ultipliers<span class="_ _4"> </span><span class="ff5">α</span></div><div class="t m0 xb he y86 ff10 fs3 fc0 sc0 ls0 ws0">1</div><div class="t m0 x6b hf y85 ff5 fs2 fc0 sc0 ls0 ws0">,<span class="_ _20"> </span>α</div><div class="t m0 x6c he y86 ff10 fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x6d hf y85 ff5 fs2 fc0 sc0 ls0 ws0">,<span class="_ _20"> </span>.<span class="_ _20"> </span>.<span class="_ _20"> </span>.<span class="_ _20"> </span>,<span class="_ _20"> </span>α</div><div class="t m0 x63 h5 y86 ff6 fs3 fc0 sc0 ls0 ws0">N</div><div class="t m0 x4f hd y85 fff fs2 fc0 sc0 ls0 ws0">≥<span class="_ _1b"> </span><span class="ff4">0,<span class="_ _8"> </span>we<span class="_ _8"> </span>hav<span class="_ _23"></span>e<span class="_ _4"> </span>the<span class="_ _8"> </span>following</span></div><div class="t m0 x2 h4 y87 ff4 fs2 fc0 sc0 ls0 ws0">Lagrangian:</div><div class="t m0 x1 hd y88 ff15 fs2 fc0 sc0 ls0 ws0">L<span class="_ _26"> </span><span class="ff4">(<span class="ff3">w<span class="ff5">,<span class="_ _20"> </span>b,<span class="_ _20"> </span><span class="ff16">α</span></span></span>)<span class="_"> </span><span class="fff">≡</span></span></div><div class="t m0 x6e h4 y89 ff4 fs2 fc0 sc0 ls0 ws0">1</div><div class="t m0 x6e h4 y8a ff4 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x3b h4 y88 ff3 fs2 fc0 sc0 ls0 ws0">w</div><div class="t m0 x6f h5 y8b ff6 fs3 fc0 sc0 ls0 ws0">T</div><div class="t m0 x33 hd y88 ff3 fs2 fc0 sc0 ls0 ws0">w<span class="_ _1c"> </span><span class="fff">−</span></div><div class="t m0 x70 h5 y8c ff6 fs3 fc0 sc0 ls0 ws0">N</div><div class="t m0 x3d h11 y8d ff17 fs2 fc0 sc0 ls0 ws0">X</div><div class="t m0 x3d h5 y8e ff6 fs3 fc0 sc0 ls0 ws0">i<span class="ff10">=1</span></div><div class="t m0 x20 hf y88 ff5 fs2 fc0 sc0 ls0 ws0">α</div><div class="t m0 x71 h5 y8f ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x72 hf y88 ff5 fs2 fc0 sc0 ls0 ws0">y</div><div class="t m0 x73 h5 y8f ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x6 hd y88 ff4 fs2 fc0 sc0 ls0 ws0">(<span class="ff3">w<span class="_ _1c"> </span><span class="fff">·<span class="_ _1c"> </span></span>x</span></div><div class="t m0 x6d h5 y8f ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x74 hd y88 fff fs2 fc0 sc0 ls0 ws0">−<span class="_ _1c"> </span><span class="ff5">b<span class="ff4">)<span class="_ _1c"> </span>+</span></span></div><div class="t m0 x75 h5 y8c ff6 fs3 fc0 sc0 ls0 ws0">N</div><div class="t m0 x1e h11 y8d ff17 fs2 fc0 sc0 ls0 ws0">X</div><div class="t m0 x25 h5 y8e ff6 fs3 fc0 sc0 ls0 ws0">i<span class="ff10">=1</span></div><div class="t m0 x76 hf y88 ff5 fs2 fc0 sc0 ls0 ws0">α</div><div class="t m0 x77 h5 y8f ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x4c hf y88 ff5 fs2 fc0 sc0 ls0 ws0">.</div><div class="t m0 x2 hc y90 ff2 fs8 fc0 sc0 ls0 ws0">1.2<span class="_ _1f"> </span>The<span class="_ _9"> </span>dual<span class="_ _a"> </span>problem</div><div class="t m0 x2 h4 y91 ff4 fs2 fc0 sc0 ls0 ws0">W<span class="_ _5"></span>e<span class="_ _4"> </span>can<span class="_ _4"> </span>solv<span class="_ _0"></span>e<span class="_ _4"> </span>the<span class="_ _8"> </span>W<span class="_ _23"></span>olfe<span class="_ _4"> </span>dual<span class="_ _8"> </span>instead:<span class="_ _1"> </span><span class="ff11">maximize<span class="_ _4"> </span><span class="ff15">L<span class="_ _20"> </span></span></span>(<span class="ff3">w<span class="_ _3"></span><span class="ff5">,<span class="_ _20"> </span>b,<span class="_ _20"> </span><span class="ff16">α</span></span></span>)<span class="_ _8"> </span>with<span class="_ _4"> </span>resp<span class="_ _3"></span>ect<span class="_ _8"> </span>to<span class="_ _4"> </span><span class="ff16">α</span>,</div><div class="t m0 x2 h4 y92 ff4 fs2 fc0 sc0 ls0 ws0">sub<span class="_ _3"></span>ject<span class="_ _4"> </span>to<span class="_ _4"> </span>the<span class="_ _4"> </span>constraints<span class="_ _4"> </span>that<span class="_ _4"> </span>the<span class="_ _4"> </span>gradien<span class="_ _0"></span>t<span class="_ _4"> </span>of<span class="_ _4"> </span><span class="ff15">L<span class="_ _26"> </span></span>(<span class="ff3">w<span class="_ _3"></span><span class="ff5">,<span class="_ _20"> </span>b,<span class="_ _20"> </span><span class="ff16">α</span></span></span>)<span class="_ _4"> </span>with<span class="_ _4"> </span>respect<span class="_ _4"> </span>to<span class="_ _4"> </span>the</div><div class="t m0 x2 h4 y93 ff4 fs2 fc0 sc0 ls0 ws0">primal<span class="_ _1b"> </span>v<span class="_ _0"></span>ariables<span class="_ _8"> </span><span class="ff3">w<span class="_ _4"> </span></span>and<span class="_ _1b"> </span><span class="ff5">b<span class="_ _8"> </span></span>v<span class="_ _23"></span>anish:</div><div class="t m0 x78 h12 y94 ff5 fs2 fc0 sc0 ls0 ws0">∂<span class="_ _3"></span><span class="ff15">L</span></div><div class="t m0 x67 h4 y95 ff5 fs2 fc0 sc0 ls0 ws0">∂<span class="_ _3"></span><span class="ff3">w</span></div><div class="t m0 x79 h4 y96 ff4 fs2 fc0 sc0 ls0 ws0">=<span class="_"> </span><span class="ff3">0<span class="ff5">,<span class="_ _27"> </span></span></span>(1)</div><div class="t m0 x78 h12 y97 ff5 fs2 fc0 sc0 ls0 ws0">∂<span class="_ _3"></span><span class="ff15">L</span></div><div class="t m0 x66 hf y98 ff5 fs2 fc0 sc0 ls0 ws0">∂<span class="_ _3"></span>b</div><div class="t m0 x79 h4 y99 ff4 fs2 fc0 sc0 ls0 ws0">=<span class="_"> </span>0<span class="ff5">,<span class="_ _28"> </span></span>(2)</div><div class="t m0 xb h4 y9a ff4 fs2 fc0 sc0 ls0 ws0">3</div><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a></div><div class="pi" data-data='{"ctm":[1.568633,0.000000,0.000000,1.568633,0.000000,0.000000]}'></div></div>
<div id="pf4" class="pf w0 h0" data-page-no="4"><div class="pc pc4 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/6250abd774bc5c01056fdec4/bg4.jpg"><div class="t m0 x2 h4 y50 ff4 fs2 fc0 sc0 ls0 ws0">and<span class="_ _1b"> </span>that</div><div class="t m0 x1 hd y9b ff16 fs2 fc0 sc0 ls0 ws0">α<span class="_ _24"> </span><span class="fff">≥<span class="_ _24"> </span><span class="ff3">0<span class="ff5">.</span></span></span></div><div class="t m0 x2 h4 y9c ff4 fs2 fc0 sc0 ls0 ws0">F<span class="_ _5"></span>rom<span class="_ _8"> </span>Equations<span class="_ _8"> </span>1<span class="_ _8"> </span>and<span class="_ _8"> </span><span class="fc1 sc0">2</span>,<span class="_ _8"> </span>we<span class="_ _1b"> </span>hav<span class="_ _0"></span>e</div><div class="t m0 x1 h4 y9d ff3 fs2 fc0 sc0 ls0 ws0">w<span class="_ _24"> </span><span class="ff4">=</span></div><div class="t m0 x54 h5 y9e ff6 fs3 fc0 sc0 ls0 ws0">N</div><div class="t m0 x79 h11 y9f ff17 fs2 fc0 sc0 ls0 ws0">X</div><div class="t m0 x7a h5 ya0 ff6 fs3 fc0 sc0 ls0 ws0">i<span class="ff10">=1</span></div><div class="t m0 x7b hf y9d ff5 fs2 fc0 sc0 ls0 ws0">α</div><div class="t m0 x7c h5 ya1 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x7d hf y9d ff5 fs2 fc0 sc0 ls0 ws0">y</div><div class="t m0 x7e h5 ya1 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x7f h4 y9d ff3 fs2 fc0 sc0 ls0 ws0">x</div><div class="t m0 x6e h5 ya1 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x80 hf y9d ff5 fs2 fc0 sc0 ls0 ws0">,</div><div class="t m0 x81 h5 ya2 ff6 fs3 fc0 sc0 ls0 ws0">N</div><div class="t m0 x1 h11 ya3 ff17 fs2 fc0 sc0 ls0 ws0">X</div><div class="t m0 x78 h5 ya4 ff6 fs3 fc0 sc0 ls0 ws0">i<span class="ff10">=1</span></div><div class="t m0 x35 hf ya5 ff5 fs2 fc0 sc0 ls0 ws0">α</div><div class="t m0 x7a h5 ya6 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x82 hf ya5 ff5 fs2 fc0 sc0 ls0 ws0">y</div><div class="t m0 x83 h5 ya6 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x84 h4 ya5 ff4 fs2 fc0 sc0 ls0 ws0">=<span class="_"> </span>0<span class="ff5">.</span></div><div class="t m0 x2 h4 ya7 ff4 fs2 fc0 sc0 ls0 ws0">Substitute<span class="_ _1b"> </span>them<span class="_ _8"> </span>into<span class="_ _1b"> </span><span class="ff15">L<span class="_ _26"> </span></span>(<span class="ff3">w<span class="_ _3"></span><span class="ff5">,<span class="_ _20"> </span>b,<span class="_ _20"> </span><span class="ff16">α</span></span></span>),<span class="_ _1b"> </span>we<span class="_ _1b"> </span>hav<span class="_ _0"></span>e</div><div class="t m0 x1 h12 ya8 ff15 fs2 fc0 sc0 ls0 ws0">L</div><div class="t m0 x34 h5 ya9 ff6 fs3 fc0 sc0 ls0 ws0">D</div><div class="t m0 x85 hd ya8 fff fs2 fc0 sc0 ls0 ws0">≡</div><div class="t m0 x86 h5 yaa ff6 fs3 fc0 sc0 ls0 ws0">N</div><div class="t m0 x87 h11 yab ff17 fs2 fc0 sc0 ls0 ws0">X</div><div class="t m0 x88 h5 yac ff6 fs3 fc0 sc0 ls0 ws0">i<span class="ff10">=1</span></div><div class="t m0 x89 hf ya8 ff5 fs2 fc0 sc0 ls0 ws0">α</div><div class="t m0 x8a h5 ya9 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x8b hd ya8 fff fs2 fc0 sc0 ls0 ws0">−</div><div class="t m0 x3b h4 yad ff4 fs2 fc0 sc0 ls0 ws0">1</div><div class="t m0 x3b h4 yae ff4 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x6f h11 yab ff17 fs2 fc0 sc0 ls0 ws0">X</div><div class="t m0 x8c h5 yac ff6 fs3 fc0 sc0 ls0 ws0">i,j</div><div class="t m0 x8d hf ya8 ff5 fs2 fc0 sc0 ls0 ws0">α</div><div class="t m0 x8e h5 ya9 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x8f hf ya8 ff5 fs2 fc0 sc0 ls0 ws0">α</div><div class="t m0 x90 h5 ya9 ff6 fs3 fc0 sc0 ls0 ws0">j</div><div class="t m0 x91 hf ya8 ff5 fs2 fc0 sc0 ls0 ws0">y</div><div class="t m0 x69 h5 ya9 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x20 hf ya8 ff5 fs2 fc0 sc0 ls0 ws0">y</div><div class="t m0 x21 h5 ya9 ff6 fs3 fc0 sc0 ls0 ws0">j</div><div class="t m0 x72 h4 ya8 ff3 fs2 fc0 sc0 ls0 ws0">x</div><div class="t m0 x92 h5 ya9 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x7 hd ya8 fff fs2 fc0 sc0 ls0 ws0">·<span class="_ _1c"> </span><span class="ff3">x</span></div><div class="t m0 x60 h5 ya9 ff6 fs3 fc0 sc0 ls0 ws0">j</div><div class="t m0 x93 hf ya8 ff5 fs2 fc0 sc0 ls0 ws0">,</div><div class="t m0 x2 h4 yaf ff4 fs2 fc0 sc0 ls0 ws0">in<span class="_ _8"> </span>whic<span class="_ _0"></span>h<span class="_ _8"> </span>the<span class="_ _8"> </span>primal<span class="_ _8"> </span>v<span class="_ _0"></span>ariables<span class="_ _8"> </span>are<span class="_ _8"> </span>eliminated.</div><div class="t m0 x3 h4 yb0 ff4 fs2 fc0 sc0 ls0 ws0">When<span class="_ _1b"> </span>we<span class="_ _1b"> </span>solve<span class="_ _1b"> </span><span class="ff5">α</span></div><div class="t m0 x80 h5 yb1 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x3b h4 yb0 ff4 fs2 fc0 sc0 ls0 ws0">,<span class="_ _1b"> </span>we<span class="_ _1b"> </span>can<span class="_ _8"> </span>get<span class="_ _8"> </span><span class="ff3">w<span class="_"> </span></span>=</div><div class="t m0 xb h11 yb2 ff17 fs2 fc0 sc0 ls0 ws0">P</div><div class="t m0 x94 h5 yb3 ff6 fs3 fc0 sc0 ls0 ws0">N</div><div class="t m0 x94 h5 yb4 ff6 fs3 fc0 sc0 ls0 ws0">i<span class="ff10">=1</span></div><div class="t m0 x95 hf yb0 ff5 fs2 fc0 sc0 ls0 ws0">α</div><div class="t m0 x96 h5 yb1 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x65 hf yb0 ff5 fs2 fc0 sc0 ls0 ws0">y</div><div class="t m0 x1d h5 yb1 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x97 h4 yb0 ff3 fs2 fc0 sc0 ls0 ws0">x</div><div class="t m0 x23 h5 yb1 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x1e h4 yb0 ff4 fs2 fc0 sc0 ls0 ws0">,<span class="_ _1b"> </span>(we<span class="_ _1b"> </span>will<span class="_ _8"> </span>later<span class="_ _8"> </span>show<span class="_ _1b"> </span>how<span class="_ _1b"> </span>to</div><div class="t m0 x2 h4 yb5 ff4 fs2 fc0 sc0 ls0 ws0">compute<span class="_ _8"> </span>the<span class="_ _8"> </span>threshold<span class="_ _1b"> </span><span class="ff5">b</span>),<span class="_ _8"> </span>and<span class="_ _8"> </span>we<span class="_ _1b"> </span>can<span class="_ _8"> </span>classify<span class="_ _8"> </span>a<span class="_ _8"> </span>new<span class="_ _8"> </span>ob<span class="_ _29"></span>ject<span class="_ _8"> </span><span class="ff3">x<span class="_"> </span></span>with</div><div class="t m0 x1 hd yb6 ff5 fs2 fc0 sc0 ls0 ws0">f<span class="_ _21"></span><span class="ff4">(<span class="ff3">x</span>)<span class="_ _7"> </span>=<span class="_ _c"> </span></span>sg<span class="_ _3"></span>n<span class="ff4">(<span class="ff3">w<span class="_ _1c"> </span><span class="fff">·<span class="_ _1c"> </span></span>x<span class="_ _1c"> </span></span>+<span class="_ _1c"> </span></span>b<span class="ff4">)</span></div><div class="t m0 x87 h4 yb7 ff4 fs2 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff5">sgn</span>((</div><div class="t m0 x98 h5 yb8 ff6 fs3 fc0 sc0 ls0 ws0">N</div><div class="t m0 x99 h11 yb9 ff17 fs2 fc0 sc0 ls0 ws0">X</div><div class="t m0 x9a h5 yba ff6 fs3 fc0 sc0 ls0 ws0">i<span class="ff10">=1</span></div><div class="t m0 x47 hf yb7 ff5 fs2 fc0 sc0 ls0 ws0">α</div><div class="t m0 x9b h5 ybb ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x3c hf yb7 ff5 fs2 fc0 sc0 ls0 ws0">y</div><div class="t m0 x8f h5 ybb ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x9c h4 yb7 ff3 fs2 fc0 sc0 ls0 ws0">x</div><div class="t m0 x9d h5 ybb ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x68 hd yb7 ff4 fs2 fc0 sc0 ls0 ws0">)<span class="_ _1c"> </span><span class="fff">·<span class="_ _1c"> </span><span class="ff3">x<span class="_ _1c"> </span></span></span>+<span class="_ _1c"> </span><span class="ff5">b</span>)</div><div class="t m0 x87 h4 ybc ff4 fs2 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff5">sgn</span>(</div><div class="t m0 x99 h5 ybd ff6 fs3 fc0 sc0 ls0 ws0">N</div><div class="t m0 x9e h11 ybe ff17 fs2 fc0 sc0 ls0 ws0">X</div><div class="t m0 x3b h5 ybf ff6 fs3 fc0 sc0 ls0 ws0">i<span class="ff10">=1</span></div><div class="t m0 x9f hf ybc ff5 fs2 fc0 sc0 ls0 ws0">α</div><div class="t m0 xa0 h5 yc0 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x8d hf ybc ff5 fs2 fc0 sc0 ls0 ws0">y</div><div class="t m0 xa1 h5 yc0 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 xa2 h4 ybc ff4 fs2 fc0 sc0 ls0 ws0">(<span class="ff3">x</span></div><div class="t m0 x9d h5 yc0 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x1f hd ybc fff fs2 fc0 sc0 ls0 ws0">·<span class="_ _1c"> </span><span class="ff3">x<span class="ff4">)<span class="_ _1c"> </span>+<span class="_ _1c"> </span><span class="ff5">b</span>)<span class="ff5">.</span></span></span></div><div class="t m0 x3 h4 yc1 ff4 fs2 fc0 sc0 ls0 ws0">Please<span class="_ _a"> </span>note<span class="_ _a"> </span>that<span class="_ _a"> </span>in<span class="_ _a"> </span>the<span class="_ _9"> </span>ob<span class="_ _29"></span>jectiv<span class="_ _0"></span>e<span class="_ _a"> </span>function<span class="_ _a"> </span>and<span class="_ _a"> </span>the<span class="_ _a"> </span>solution,<span class="_ _a"> </span>the<span class="_ _a"> </span>training</div><div class="t m0 x2 h4 yc2 ff4 fs2 fc0 sc0 ls0 ws0">v<span class="_ _0"></span>ectors<span class="_ _8"> </span><span class="ff3">x</span></div><div class="t m0 x35 h5 yc3 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x79 h4 yc2 ff4 fs2 fc0 sc0 ls0 ws0">o<span class="_ _3"></span>ccur<span class="_ _1b"> </span>only<span class="_ _8"> </span>in<span class="_ _8"> </span>the<span class="_ _8"> </span>form<span class="_ _8"> </span>of<span class="_ _8"> </span>dot<span class="_ _8"> </span>pro<span class="_ _3"></span>duct.</div><div class="t m0 x3 h4 yc4 ff4 fs2 fc0 sc0 ls0 ws0">Before<span class="_ _9"> </span>going<span class="_ _9"> </span>in<span class="_ _0"></span>to<span class="_ _9"> </span>the<span class="_ _9"> </span>details<span class="_ _9"> </span>to<span class="_ _9"> </span>ho<span class="_ _0"></span>w<span class="_ _9"> </span>to<span class="_ _9"> </span>solv<span class="_ _0"></span>e<span class="_ _9"> </span>this<span class="_ _9"> </span>quadratic<span class="_ _9"> </span>programming</div><div class="t m0 x2 h4 yc5 ff4 fs2 fc0 sc0 ls0 ws0">problem,<span class="_ _1b"> </span>let’s<span class="_ _4"> </span>extend<span class="_ _1b"> </span>it<span class="_ _8"> </span>in<span class="_ _1b"> </span>tw<span class="_ _0"></span>o<span class="_ _8"> </span>directions.</div><div class="t m0 x2 hc yc6 ff2 fs8 fc0 sc0 ls0 ws0">1.3<span class="_ _1f"> </span>Non-linear<span class="_ _9"> </span>SVM</div><div class="t m0 x2 h4 yc7 ff4 fs2 fc0 sc0 ls0 ws0">What<span class="_ _9"> </span>if<span class="_ _9"> </span>the<span class="_ _a"> </span>surface<span class="_ _9"> </span>separating<span class="_ _9"> </span>the<span class="_ _a"> </span>t<span class="_ _0"></span>w<span class="_ _0"></span>o<span class="_ _a"> </span>classes<span class="_ _9"> </span>are<span class="_ _9"> </span>not<span class="_ _a"> </span>linear?<span class="_ _10"> </span>W<span class="_ _23"></span>ell,<span class="_ _a"> </span>w<span class="_ _0"></span>e<span class="_ _a"> </span>can</div><div class="t m0 x2 h4 yc8 ff4 fs2 fc0 sc0 ls0 ws0">transform<span class="_"> </span>the<span class="_ _1b"> </span>data<span class="_"> </span>p<span class="_ _3"></span>oin<span class="_ _0"></span>ts<span class="_ _1b"> </span>to<span class="_"> </span>another<span class="_ _1b"> </span>high<span class="_"> </span>dimensional<span class="_ _1b"> </span>space<span class="_"> </span>such<span class="_"> </span>that<span class="_"> </span>the<span class="_ _1b"> </span>data</div><div class="t m0 x2 hd yc9 ff4 fs2 fc0 sc0 ls0 ws0">p<span class="_ _3"></span>oin<span class="_ _23"></span>ts<span class="_ _9"> </span>will<span class="_ _4"> </span>be<span class="_ _4"> </span>linearly<span class="_ _4"> </span>separable.<span class="_ _d"> </span>Let<span class="_ _4"> </span>the<span class="_ _4"> </span>transformation<span class="_ _4"> </span>be<span class="_ _4"> </span>Φ(<span class="fff">·</span>).<span class="_ _d"> </span>In<span class="_ _4"> </span>the<span class="_ _4"> </span>high</div><div class="t m0 x2 h4 yca ff4 fs2 fc0 sc0 ls0 ws0">dimensional<span class="_ _1b"> </span>space,<span class="_ _4"> </span>w<span class="_ _23"></span>e<span class="_ _8"> </span>solve</div><div class="t m0 x1 h12 ycb ff15 fs2 fc0 sc0 ls0 ws0">L</div><div class="t m0 x34 h5 ycc ff6 fs3 fc0 sc0 ls0 ws0">D</div><div class="t m0 x85 hd ycb fff fs2 fc0 sc0 ls0 ws0">≡</div><div class="t m0 x86 h5 ycd ff6 fs3 fc0 sc0 ls0 ws0">N</div><div class="t m0 x87 h11 yce ff17 fs2 fc0 sc0 ls0 ws0">X</div><div class="t m0 x88 h5 ycf ff6 fs3 fc0 sc0 ls0 ws0">i<span class="ff10">=1</span></div><div class="t m0 x89 hf ycb ff5 fs2 fc0 sc0 ls0 ws0">α</div><div class="t m0 x8a h5 ycc ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x8b hd ycb fff fs2 fc0 sc0 ls0 ws0">−</div><div class="t m0 x3b h4 yd0 ff4 fs2 fc0 sc0 ls0 ws0">1</div><div class="t m0 x3b h4 yd1 ff4 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x6f h11 yce ff17 fs2 fc0 sc0 ls0 ws0">X</div><div class="t m0 x8c h5 ycf ff6 fs3 fc0 sc0 ls0 ws0">i,j</div><div class="t m0 x8d hf ycb ff5 fs2 fc0 sc0 ls0 ws0">α</div><div class="t m0 x8e h5 ycc ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x8f hf ycb ff5 fs2 fc0 sc0 ls0 ws0">α</div><div class="t m0 x90 h5 ycc ff6 fs3 fc0 sc0 ls0 ws0">j</div><div class="t m0 x91 hf ycb ff5 fs2 fc0 sc0 ls0 ws0">y</div><div class="t m0 x69 h5 ycc ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x20 hf ycb ff5 fs2 fc0 sc0 ls0 ws0">y</div><div class="t m0 x21 h5 ycc ff6 fs3 fc0 sc0 ls0 ws0">j</div><div class="t m0 x72 h4 ycb ff4 fs2 fc0 sc0 ls0 ws0">Φ(<span class="ff3">x</span></div><div class="t m0 xa3 h5 ycc ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 xa4 hd ycb ff4 fs2 fc0 sc0 ls0 ws0">)<span class="_ _1c"> </span><span class="fff">·<span class="_ _1c"> </span></span>Φ(<span class="ff3">x</span></div><div class="t m0 x64 h5 ycc ff6 fs3 fc0 sc0 ls0 ws0">j</div><div class="t m0 x5a h4 ycb ff4 fs2 fc0 sc0 ls0 ws0">)</div><div class="t m0 x3 h4 yd2 ff4 fs2 fc0 sc0 ls0 ws0">Supp<span class="_ _3"></span>ose,<span class="_ _1b"> </span>in<span class="_ _8"> </span>addition,<span class="_ _8"> </span>Φ(<span class="ff3">x</span></div><div class="t m0 xa5 h5 yd3 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 xa6 hd yd2 ff4 fs2 fc0 sc0 ls0 ws0">)<span class="_ _1c"> </span><span class="fff">·<span class="_ _1c"> </span></span>Φ(<span class="ff3">x</span></div><div class="t m0 x73 h5 yd3 ff6 fs3 fc0 sc0 ls0 ws0">j</div><div class="t m0 x6 h4 yd2 ff4 fs2 fc0 sc0 ls0 ws0">)<span class="_"> </span>=<span class="_"> </span><span class="ff5">k<span class="_ _3"></span></span>(<span class="ff3">x</span></div><div class="t m0 x95 h5 yd3 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 xa7 h4 yd2 ff5 fs2 fc0 sc0 ls0 ws0">,<span class="_ _20"> </span><span class="ff3">x</span></div><div class="t m0 x1d h5 yd3 ff6 fs3 fc0 sc0 ls0 ws0">j</div><div class="t m0 x4a h4 yd2 ff4 fs2 fc0 sc0 ls0 ws0">).<span class="_ _9"> </span>That<span class="_ _8"> </span>is,<span class="_ _4"> </span>the<span class="_ _1b"> </span>dot<span class="_ _8"> </span>pro<span class="_ _3"></span>duct<span class="_ _1b"> </span>in</div><div class="t m0 x2 h4 yd4 ff4 fs2 fc0 sc0 ls0 ws0">that<span class="_"> </span>high<span class="_ _22"> </span>dimensional<span class="_"> </span>space<span class="_"> </span>is<span class="_ _1c"> </span>equiv<span class="_ _0"></span>alent<span class="_ _1c"> </span>to<span class="_"> </span>a<span class="_"> </span><span class="ff11">kernel<span class="_"> </span></span>function<span class="_"> </span>of<span class="_"> </span>the<span class="_ _1c"> </span>input<span class="_"> </span>space.</div><div class="t m0 x2 hd y4e ff4 fs2 fc0 sc0 ls0 ws0">So<span class="_ _4"> </span>we<span class="_ _4"> </span>need<span class="_ _4"> </span>not<span class="_ _4"> </span>be<span class="_ _4"> </span>explicit<span class="_ _4"> </span>ab<span class="_ _3"></span>out<span class="_ _4"> </span>the<span class="_ _4"> </span>transformation<span class="_ _4"> </span>Φ(<span class="fff">·</span>)<span class="_ _4"> </span>as<span class="_ _9"> </span>long<span class="_ _4"> </span>as<span class="_ _4"> </span>w<span class="_ _0"></span>e<span class="_ _4"> </span>know</div><div class="t m0 xb h4 y9a ff4 fs2 fc0 sc0 ls0 ws0">4</div><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a></div><div class="pi" data-data='{"ctm":[1.568633,0.000000,0.000000,1.568633,0.000000,0.000000]}'></div></div>
<div id="pf5" class="pf w0 h0" data-page-no="5"><div class="pc pc5 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/6250abd774bc5c01056fdec4/bg5.jpg"><div class="t m0 x2 h4 y50 ff4 fs2 fc0 sc0 ls0 ws0">that<span class="_"> </span>the<span class="_ _1b"> </span>kernel<span class="_"> </span>function<span class="_"> </span><span class="ff5">k<span class="_ _3"></span></span>(<span class="ff3">x</span></div><div class="t m0 x3c h5 y51 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 xa8 h4 y50 ff5 fs2 fc0 sc0 ls0 ws0">,<span class="_ _20"> </span><span class="ff3">x</span></div><div class="t m0 xa9 h5 y51 ff6 fs3 fc0 sc0 ls0 ws0">j</div><div class="t m0 x68 h4 y50 ff4 fs2 fc0 sc0 ls0 ws0">)<span class="_"> </span>is<span class="_ _1b"> </span>equiv<span class="_ _23"></span>alent<span class="_"> </span>to<span class="_ _1b"> </span>the<span class="_"> </span>dot<span class="_ _1b"> </span>pro<span class="_ _3"></span>duct<span class="_"> </span>of<span class="_"> </span>some<span class="_ _1b"> </span>other</div><div class="t m0 x2 h4 yd5 ff4 fs2 fc0 sc0 ls0 ws0">high<span class="_ _1b"> </span>dimensional<span class="_ _8"> </span>space.<span class="_ _9"> </span>There<span class="_ _8"> </span>are<span class="_ _8"> </span>man<span class="_ _0"></span>y<span class="_ _8"> </span>kernel<span class="_ _1b"> </span>functions<span class="_ _1b"> </span>that<span class="_ _8"> </span>can<span class="_ _8"> </span>b<span class="_ _3"></span>e<span class="_ _1b"> </span>used<span class="_ _1b"> </span>this</div><div class="t m0 x2 h4 yd6 ff4 fs2 fc0 sc0 ls0 ws0">w<span class="_ _0"></span>ay<span class="_ _5"></span>,<span class="_ _8"> </span>for<span class="_ _8"> </span>example,<span class="_ _8"> </span>the<span class="_ _8"> </span>radial<span class="_ _8"> </span>basis<span class="_ _8"> </span>function<span class="_ _8"> </span>(Gaussian<span class="_ _8"> </span>kernel)</div><div class="t m0 x1 h4 yd7 ff5 fs2 fc0 sc0 ls0 ws0">K<span class="_ _29"></span><span class="ff4">(<span class="ff3">x</span></span></div><div class="t m0 x85 h5 yd8 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x7a h4 yd7 ff5 fs2 fc0 sc0 ls0 ws0">,<span class="_ _20"> </span><span class="ff3">x</span></div><div class="t m0 x86 h5 yd8 ff6 fs3 fc0 sc0 ls0 ws0">j</div><div class="t m0 x84 h4 yd7 ff4 fs2 fc0 sc0 ls0 ws0">)<span class="_"> </span>=<span class="_"> </span><span class="ff5">e</span></div><div class="t m0 xaa h5 yd9 ff12 fs3 fc0 sc0 ls0 ws0">−k<span class="ff14">x</span></div><div class="t m0 x1c h13 yda ff18 fs4 fc0 sc0 ls0 ws0">i</div><div class="t m0 x6f h5 yd9 ff12 fs3 fc0 sc0 ls0 ws0">−<span class="ff14">x</span></div><div class="t m0 xab h13 yda ff18 fs4 fc0 sc0 ls0 ws0">j</div><div class="t m0 x8d h5 yd9 ff12 fs3 fc0 sc0 ls0 ws0">k</div><div class="t m0 x3c h10 ydb ff13 fs4 fc0 sc0 ls0 ws0">2</div><div class="t m0 xa8 h5 yd9 ff6 fs3 fc0 sc0 ls0 ws0">/<span class="ff10">2</span>σ</div><div class="t m0 x91 h10 ydb ff13 fs4 fc0 sc0 ls0 ws0">2</div><div class="t m0 x69 hf yd7 ff5 fs2 fc0 sc0 ls0 ws0">.</div><div class="t m0 x3 h4 ydc ff4 fs2 fc0 sc0 ls0 ws0">The<span class="_ _4"> </span>Mercer’s<span class="_ _8"> </span>condition<span class="_ _4"> </span>can<span class="_ _8"> </span>b<span class="_ _3"></span>e<span class="_ _8"> </span>used<span class="_ _4"> </span>to<span class="_ _8"> </span>determine<span class="_ _4"> </span>if<span class="_ _8"> </span>a<span class="_ _4"> </span>function<span class="_ _8"> </span>can<span class="_ _4"> </span>be<span class="_ _4"> </span>used</div><div class="t m0 x2 h4 ydd ff4 fs2 fc0 sc0 ls0 ws0">as<span class="_ _8"> </span>a<span class="_ _8"> </span>k<span class="_ _0"></span>ernel<span class="_ _8"> </span>function:</div><div class="t m0 x1 h4 yde ff4 fs2 fc0 sc0 ls0 ws0">There<span class="_ _8"> </span>exists<span class="_ _8"> </span>a<span class="_ _8"> </span>mapping<span class="_ _8"> </span>Φ<span class="_ _8"> </span>and<span class="_ _8"> </span>an<span class="_ _8"> </span>expansion</div><div class="t m0 x54 h4 ydf ff5 fs2 fc0 sc0 ls0 ws0">K<span class="_ _29"></span><span class="ff4">(<span class="ff3">x</span></span>,<span class="_ _20"> </span><span class="ff3">y<span class="ff4">)<span class="_"> </span>=</span></span></div><div class="t m0 x9a h11 ye0 ff17 fs2 fc0 sc0 ls0 ws0">X</div><div class="t m0 x46 h5 ye1 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 xab h4 ydf ff4 fs2 fc0 sc0 ls0 ws0">Φ(<span class="ff3">x</span>)</div><div class="t m0 x90 h5 ye2 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x91 h4 ydf ff4 fs2 fc0 sc0 ls0 ws0">Φ(<span class="ff3">y</span>)</div><div class="t m0 xac h5 ye2 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 xad hf ydf ff5 fs2 fc0 sc0 ls0 ws0">,</div><div class="t m0 x1 h4 ye3 ff4 fs2 fc0 sc0 ls0 ws0">if<span class="_ _8"> </span>and<span class="_ _1b"> </span>only<span class="_ _8"> </span>if,<span class="_ _8"> </span>for<span class="_ _8"> </span>any<span class="_ _1b"> </span><span class="ff5">g<span class="_ _3"></span></span>(<span class="ff3">x</span>)<span class="_ _8"> </span>such<span class="_ _1b"> </span>that</div><div class="t m0 x54 h11 ye4 ff17 fs2 fc0 sc0 ls0 ws0">Z</div><div class="t m0 x84 h4 ye5 ff5 fs2 fc0 sc0 ls0 ws0">g<span class="_ _3"></span><span class="ff4">(<span class="ff3">x</span>)</span></div><div class="t m0 x7f he ye6 ff10 fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x1b h4 ye5 ff5 fs2 fc0 sc0 ls0 ws0">d<span class="ff3">x<span class="_"> </span><span class="ff4">is<span class="_ _8"> </span>finite</span></span>,</div><div class="t m0 x1 h4 ye7 ff4 fs2 fc0 sc0 ls0 ws0">then</div><div class="t m0 x54 h11 ye8 ff17 fs2 fc0 sc0 ls0 ws0">Z</div><div class="t m0 x84 hd ye9 ff5 fs2 fc0 sc0 ls0 ws0">K<span class="_ _29"></span><span class="ff4">(<span class="ff3">x</span></span>,<span class="_ _20"> </span><span class="ff3">y<span class="ff4">)</span></span>g<span class="_ _3"></span><span class="ff4">(<span class="ff3">x</span>)</span>g<span class="_ _3"></span><span class="ff4">(<span class="ff3">y</span>)</span>d<span class="ff3">x</span>d<span class="ff3">y<span class="_"> </span><span class="fff">≥<span class="_ _22"> </span><span class="ff4">0</span></span></span>.</div><div class="t m0 x2 hc yea ff2 fs8 fc0 sc0 ls0 ws0">1.4<span class="_ _1f"> </span>Imp<span class="_ _3"></span>erfect<span class="_ _9"> </span>separation</div><div class="t m0 x2 h4 yeb ff4 fs2 fc0 sc0 ls0 ws0">The<span class="_"> </span>other<span class="_ _1c"> </span>direction<span class="_"> </span>to<span class="_"> </span>extend<span class="_ _1c"> </span>SVM<span class="_"> </span>is<span class="_"> </span>to<span class="_ _1c"> </span>allow<span class="_ _22"> </span>for<span class="_"> </span>noise,<span class="_"> </span>or<span class="_ _1c"> </span>imp<span class="_ _3"></span>erfect<span class="_"> </span>separation.</div><div class="t m0 x2 h4 yec ff4 fs2 fc0 sc0 ls0 ws0">That<span class="_"> </span>is,<span class="_ _1b"> </span>we<span class="_"> </span>do<span class="_"> </span>not<span class="_"> </span>strictly<span class="_ _1b"> </span>enforce<span class="_"> </span>that<span class="_ _1b"> </span>there<span class="_"> </span>b<span class="_ _3"></span>e<span class="_"> </span>no<span class="_ _1b"> </span>data<span class="_"> </span>p<span class="_ _3"></span>oin<span class="_ _0"></span>ts<span class="_"> </span>b<span class="_ _3"></span>etw<span class="_ _23"></span>een<span class="_ _1b"> </span><span class="ff5">H</span></div><div class="t m0 xae he yed ff10 fs3 fc0 sc0 ls0 ws0">1</div><div class="t m0 x2d h4 yec ff4 fs2 fc0 sc0 ls0 ws0">and</div><div class="t m0 x2 hf yee ff5 fs2 fc0 sc0 ls0 ws0">H</div><div class="t m0 xaf he yef ff10 fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 xe h4 yee ff4 fs2 fc0 sc0 ls0 ws0">,<span class="_"> </span>but<span class="_ _22"> </span>we<span class="_ _1c"> </span>definitely<span class="_"> </span>w<span class="_ _0"></span>ant<span class="_ _1c"> </span>to<span class="_"> </span>penalize<span class="_"> </span>the<span class="_"> </span>data<span class="_ _1c"> </span>p<span class="_ _3"></span>oin<span class="_ _0"></span>ts<span class="_"> </span>that<span class="_ _22"> </span>cross<span class="_"> </span>the<span class="_ _1c"> </span>b<span class="_ _3"></span>oundaries.</div><div class="t m0 x2 hd yf0 ff4 fs2 fc0 sc0 ls0 ws0">The<span class="_ _4"> </span>penalty<span class="_ _8"> </span><span class="ff5">C<span class="_ _9"> </span></span>will<span class="_ _4"> </span>be<span class="_ _4"> </span>finite.<span class="_ _a"> </span>(If<span class="_ _4"> </span><span class="ff5">C<span class="_ _4"> </span></span>=<span class="_ _1b"> </span><span class="fff">∞</span>,<span class="_ _4"> </span>w<span class="_ _0"></span>e<span class="_ _4"> </span>come<span class="_ _8"> </span>back<span class="_ _8"> </span>to<span class="_ _4"> </span>the<span class="_ _8"> </span>original<span class="_ _4"> </span>perfect</div><div class="t m0 x2 h4 yf1 ff4 fs2 fc0 sc0 ls0 ws0">separating<span class="_ _1b"> </span>case.)</div><div class="t m0 x3 h4 yf2 ff4 fs2 fc0 sc0 ls0 ws0">W<span class="_ _5"></span>e<span class="_ _8"> </span>introduce<span class="_ _8"> </span>non-negative<span class="_ _1b"> </span>slack<span class="_ _1b"> </span>v<span class="_ _0"></span>ariables<span class="_ _8"> </span><span class="ff5">ξ</span></div><div class="t m0 xb0 h5 yf3 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x41 hd yf2 fff fs2 fc0 sc0 ls0 ws0">≥<span class="_ _24"> </span><span class="ff4">0,<span class="_ _8"> </span>so<span class="_ _8"> </span>that</span></div><div class="t m0 x1 hd yf4 ff3 fs2 fc0 sc0 ls0 ws0">w<span class="_ _1c"> </span><span class="fff">·<span class="_ _1c"> </span></span>x</div><div class="t m0 x79 h5 yf5 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 xb1 hd yf4 fff fs2 fc0 sc0 ls0 ws0">−<span class="_ _1c"> </span><span class="ff5">b<span class="_ _24"> </span></span>≥<span class="_ _24"> </span><span class="ff4">+1<span class="_ _1c"> </span></span>−<span class="_ _1c"> </span><span class="ff5">ξ</span></div><div class="t m0 xb2 h5 yf5 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 xa1 h4 yf4 ff4 fs2 fc0 sc0 ls0 ws0">for<span class="_ _1b"> </span><span class="ff5">y</span></div><div class="t m0 x69 h5 yf5 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 xb3 h4 yf4 ff4 fs2 fc0 sc0 ls0 ws0">=<span class="_"> </span>+1<span class="ff5">,</span></div><div class="t m0 x1 hd yf6 ff3 fs2 fc0 sc0 ls0 ws0">w<span class="_ _1c"> </span><span class="fff">·<span class="_ _1c"> </span></span>x</div><div class="t m0 x79 h5 yf7 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 xb1 hd yf6 fff fs2 fc0 sc0 ls0 ws0">−<span class="_ _1c"> </span><span class="ff5">b<span class="_ _24"> </span></span>≤<span class="_ _24"> </span>−<span class="ff4">1<span class="_ _1c"> </span>+<span class="_ _1c"> </span><span class="ff5">ξ</span></span></div><div class="t m0 xb2 h5 yf7 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 xa1 h4 yf6 ff4 fs2 fc0 sc0 ls0 ws0">for<span class="_ _1b"> </span><span class="ff5">y</span></div><div class="t m0 x69 h5 yf7 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 xb3 hd yf6 ff4 fs2 fc0 sc0 ls0 ws0">=<span class="_"> </span><span class="fff">−</span>1<span class="ff5">,</span></div><div class="t m0 x1 hf yf8 ff5 fs2 fc0 sc0 ls0 ws0">ξ</div><div class="t m0 x66 h5 yf9 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x43 hd yf8 fff fs2 fc0 sc0 ls0 ws0">≥<span class="_ _24"> </span><span class="ff4">0<span class="ff5">,<span class="_ _15"> </span></span></span>∀<span class="ff5">i.</span></div><div class="t m0 x2 h4 yfa ff4 fs2 fc0 sc0 ls0 ws0">and<span class="_ _1b"> </span>we<span class="_ _1b"> </span>add<span class="_ _8"> </span>to<span class="_ _8"> </span>the<span class="_ _8"> </span>ob<span class="_ _29"></span>jectiv<span class="_ _0"></span>e<span class="_ _8"> </span>function<span class="_ _8"> </span>a<span class="_ _8"> </span>p<span class="_ _3"></span>enalizing<span class="_ _1b"> </span>term:</div><div class="t m0 x1 h4 yfb ff4 fs2 fc0 sc0 ls0 ws0">minimize</div><div class="t m0 x43 h5 yfc ff14 fs3 fc0 sc0 ls0 ws0">w<span class="ff6">,b,<span class="ff19">ξ</span></span></div><div class="t m0 xb4 h4 yfd ff4 fs2 fc0 sc0 ls0 ws0">1</div><div class="t m0 xb4 h4 yfe ff4 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x32 h4 yfb ff3 fs2 fc0 sc0 ls0 ws0">w</div><div class="t m0 x8b h5 yff ff6 fs3 fc0 sc0 ls0 ws0">T</div><div class="t m0 xb5 h4 yfb ff3 fs2 fc0 sc0 ls0 ws0">w<span class="_ _1c"> </span><span class="ff4">+<span class="_ _1c"> </span><span class="ff5">C<span class="_ _29"></span></span>(</span></div><div class="t m0 xb6 h11 y100 ff17 fs2 fc0 sc0 ls0 ws0">X</div><div class="t m0 xa2 h5 y101 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x9d hf yfb ff5 fs2 fc0 sc0 ls0 ws0">ξ</div><div class="t m0 x1f h5 y102 ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 xb7 h4 yfb ff4 fs2 fc0 sc0 ls0 ws0">)</div><div class="t m0 xb3 h5 yff ff6 fs3 fc0 sc0 ls0 ws0">m</div><div class="t m0 x2 h4 y103 ff4 fs2 fc0 sc0 ls0 ws0">where<span class="_ _8"> </span><span class="ff5">m<span class="_ _8"> </span></span>is<span class="_ _8"> </span>usually<span class="_ _8"> </span>set<span class="_ _8"> </span>to<span class="_ _8"> </span>1,<span class="_ _8"> </span>whic<span class="_ _0"></span>h<span class="_ _8"> </span>gives<span class="_ _1b"> </span>us</div><div class="t m0 xb8 h4 y104 ff4 fs2 fc0 sc0 ls0 ws0">minimize</div><div class="t m0 xb9 h5 y105 ff14 fs3 fc0 sc0 ls0 ws0">w<span class="ff6">,b,ξ</span></div><div class="t m0 xba h13 y106 ff18 fs4 fc0 sc0 ls0 ws0">i</div><div class="t m0 xbb he y107 ff10 fs3 fc0 sc0 ls0 ws0">1</div><div class="t m0 xbb he y108 ff10 fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x46 h4 y104 ff3 fs2 fc0 sc0 ls0 ws0">w</div><div class="t m0 x47 h5 y109 ff6 fs3 fc0 sc0 ls0 ws0">T</div><div class="t m0 x9b h4 y104 ff3 fs2 fc0 sc0 ls0 ws0">w<span class="_ _1c"> </span><span class="ff4">+<span class="_ _1c"> </span><span class="ff5">C</span></span></div><div class="t m0 xb3 h11 y10a ff17 fs2 fc0 sc0 ls0 ws0">P</div><div class="t m0 xac h5 y10b ff6 fs3 fc0 sc0 ls0 ws0">N</div><div class="t m0 xac h5 y10c ff6 fs3 fc0 sc0 ls0 ws0">i<span class="ff10">=1</span></div><div class="t m0 x6b hf y104 ff5 fs2 fc0 sc0 ls0 ws0">ξ</div><div class="t m0 xbc h5 y10d ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x1 h4 y10e ff4 fs2 fc0 sc0 ls0 ws0">sub<span class="_ _3"></span>ject<span class="_ _8"> </span>to<span class="_ _1f"> </span><span class="ff5">y</span></div><div class="t m0 x6e h5 y10f ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x80 h4 y10e ff4 fs2 fc0 sc0 ls0 ws0">(<span class="ff3">w</span></div><div class="t m0 xbd h5 y110 ff6 fs3 fc0 sc0 ls0 ws0">T</div><div class="t m0 x9f h4 y10e ff3 fs2 fc0 sc0 ls0 ws0">x</div><div class="t m0 xa0 h5 y10f ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 xb6 hd y10e fff fs2 fc0 sc0 ls0 ws0">−<span class="_ _1c"> </span><span class="ff5">b<span class="ff4">)<span class="_ _1c"> </span>+<span class="_ _1c"> </span></span>ξ</span></div><div class="t m0 x72 h5 y10f ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 x92 hd y10e fff fs2 fc0 sc0 ls0 ws0">−<span class="_ _1c"> </span><span class="ff4">1<span class="_"> </span></span>≥<span class="_ _24"> </span><span class="ff4">0<span class="ff5">,<span class="_ _7"> </span></span>1<span class="_"> </span></span>≤<span class="_ _22"> </span><span class="ff5">i<span class="_ _24"> </span></span>≤<span class="_ _24"> </span><span class="ff5">N</span></div><div class="t m0 xa8 hf y4e ff5 fs2 fc0 sc0 ls0 ws0">ξ</div><div class="t m0 x3d h5 y4f ff6 fs3 fc0 sc0 ls0 ws0">i</div><div class="t m0 xa9 hd y4e fff fs2 fc0 sc0 ls0 ws0">≥<span class="_ _24"> </span><span class="ff4">0<span class="ff5">,<span class="_ _2a"> </span></span>1<span class="_"> </span></span>≤<span class="_ _24"> </span><span class="ff5">i<span class="_ _24"> </span></span>≤<span class="_ _24"> </span><span class="ff5">N</span></div><div class="t m0 xb h4 y9a ff4 fs2 fc0 sc0 ls0 ws0">5</div></div><div class="pi" data-data='{"ctm":[1.568633,0.000000,0.000000,1.568633,0.000000,0.000000]}'></div></div>
