Smooth Support Vector Machine

算法特征:
①. 所有点尽可能正确分开; ②. 极大化margin; ③. 极小化非线性可分之误差.
算法推导:
Part Ⅰ
线性可分之含义:
包含同类型所有数据点的最小凸集合彼此不存在交集.
引入光滑化手段:
plus function: egin{equation*}(x)_{+} = max { x, 0 }end{equation*}
p-function: egin{equation*}p(x, eta) = frac{1}{eta}ln(mathrm{e}^{eta x} + 1)end{equation*}
可采用p-function近似plus function. 相关函数图像如下:
下面给出p-function的1阶及2阶导数:
1st order (sigmoid function): egin{equation*} s(x, eta) = frac{1}{mathrm{e}^{-eta x} + 1} end{equation*}
2nd order (delta function): egin{equation*} delta (x, eta) = frac{eta mathrm{e}^{eta x}}{(mathrm{e}^{eta x} + 1)^2} end{equation*}
相关函数图像如下:
Part Ⅱ
SVM决策超平面方程如下:
egin{equation}label{eq_1}
h(x) = W^{mathrm{T}}x + b
end{equation}
其中, $W$是超平面法向量, $b$是超平面偏置参数.
本文拟采用$L_2$范数衡量非线性可分之误差, 则SVM原始优化问题如下:
egin{equation}
egin{split}
min &quadfrac{1}{2}left|W ight|_2^2 + frac{c}{2}left|xi ight|_2^2 \
mathrm{s.t.} &quad D(A^{mathrm{T}}W + mathbf{1}b) geq mathbf{1} - xi
end{split}label{eq_2}
end{equation}
其中, $A = [x^{(1)}, x^{(2)}, cdots , x^{(n)}]$, $D = diag([ar{y}^{(1)}, ar{y}^{(2)}, cdots, ar{y}^{(n)}])$, $xi$是由所有样本非线性可分之误差组成的列矢量.
根据上述优化问题( ef{eq_2}), 误差$xi$可采用plus function等效表示:
egin{equation}label{eq_3}
xi = (mathbf{1} - D(A^{mathrm{T}}W + mathbf{1}b))_{+}
end{equation}
由此可将该有约束最优化问题转化为如下无约束最优化问题:
egin{equation}label{eq_4}
minquad frac{1}{2}left|W ight|_2^2 + frac{c}{2}left| left(mathbf{1} - D(A^{mathrm{T}}W + mathbf{1}b) ight)_{+} ight|_2^2
end{equation}
同样, 根据优化问题( ef{eq_2})KKT条件中稳定性条件有:
egin{equation}label{eq_5}
W = AD^{mathrm{T}}alpha
end{equation}
其中, $alpha$为对偶变量. 代入式( ef{eq_4})变数变换可得:
egin{equation}label{eq_6}
minquad frac{1}{2}alpha^{mathrm{T}}D^{mathrm{T}}A^{mathrm{T}}ADalpha + frac{c}{2}left| left(mathbf{1} - D(A^{mathrm{T}}AD^{mathrm{T}}alpha + mathbf{1}b) ight)_{+} ight|_2^2
end{equation}
由于权重参数$c$的存在, 上述最优化问题等效如下多目标最优化问题:
egin{equation}
egin{cases}
min& frac{1}{2}alpha^{mathrm{T}}D^{mathrm{T}}A^{mathrm{T}}ADalpha \
min& frac{1}{2}left| left(mathbf{1} - D(A^{mathrm{T}}AD^{mathrm{T}}alpha + mathbf{1}b) ight)_{+} ight|_2^2
end{cases}label{eq_7}
end{equation}
由于$D^TA^TADsucceq 0$, 有如下等效:
egin{equation}label{eq_8}
minquad frac{1}{2}alpha^{mathrm{T}}D^{mathrm{T}}A^{mathrm{T}}ADalpha quadRightarrowquad minquad frac{1}{2}alpha^{mathrm{T}}alpha
end{equation}
根据Part Ⅰ中光滑化手段, 有如下等效:
egin{equation}label{eq_9}
minquad frac{1}{2}left| left(mathbf{1} - D(A^{mathrm{T}}AD^{mathrm{T}}alpha + mathbf{1}b) ight)_{+} ight|_2^2 quadRightarrowquad minquad frac{1}{2}left| p(mathbf{1} - DA^{mathrm{T}}ADalpha - Dmathbf{1}b, eta) ight|_2^2, quad ext{where $eta oinfty$}
end{equation}
结合式( ef{eq_8})、式( ef{eq_9}), 优化问题( ef{eq_6})等效如下:
egin{equation}label{eq_10}
minquad frac{1}{2}alpha^{mathrm{T}}alpha + frac{c}{2}left| p(mathbf{1} - DA^{mathrm{T}}ADalpha - Dmathbf{1}b, eta) ight|_2^2, quad ext{where $eta oinfty$}
end{equation}
为增强模型的分类能力, 引入kernel trick, 得最终优化问题:
egin{equation}label{eq_11}
minquad frac{1}{2}alpha^{mathrm{T}}alpha + frac{c}{2}left| p(mathbf{1} - DK(A^{mathrm{T}}, A)Dalpha - Dmathbf{1}b, eta) ight|_2^2, quad ext{where $eta oinfty$}
end{equation}
其中, $K$代表kernel矩阵, 内部元素$K_{ij}=k(x^{(i)}, x^{(j)})$由kernel函数决定. 常见的kernel函数如下:
①. 多项式核: $k(x^{(i)}, x^{(j)}) = ({x^{(i)}}cdot x^{(j)})^n$
②. 高斯核: $k(x^{(i)}, x^{(j)}) = mathrm{e}^{-mu| x^{(i)} - x^{(j)}|_2^2}$
结合式( ef{eq_1})、式( ef{eq_5})及kernel trick可得最终决策超平面方程:
egin{equation}
h(x) = alpha^{mathrm{T}}DK(A^{mathrm{T}}, x) + b
end{equation}

Part Ⅲ
以下给出优化相关协助信息. 拟设式( ef{eq_11})之目标函数符号为$J$, 并令:
egin{equation}
J = J_1 + J_2quadquad ext{其中,}
egin{cases}
J_1 = frac{1}{2}alpha^{mathrm{T}}alpha \
J_2 = frac{c}{2}left| p(mathbf{1} - DK(A^{mathrm{T}}, A)Dalpha - Dmathbf{1}b, eta) ight|_2^2
end{cases}
end{equation}
$J_1$相关:
egin{gather*}
frac{partial J_1}{partial alpha} = alpha quad frac{partial J_1}{partial b} = 0 \
frac{partial^2J_1}{partialalpha^2} = I quad frac{partial^2J_1}{partialalphapartial b} = 0 quad frac{partial^2J_1}{partial bpartialalpha} = 0 quad frac{partial^2J_1}{partial b^2} = 0
end{gather*}
egin{gather*}
abla J_1 = egin{bmatrix} frac{partial J_1}{partial alpha} \ frac{partial J_1}{partial b}end{bmatrix} \ \
abla^2 J_1 = egin{bmatrix}frac{partial^2J_1}{partialalpha^2} & frac{partial^2J_1}{partialalphapartial b} \ frac{partial^2J_1}{partial bpartialalpha} & frac{partial^2J_1}{partial b^2}end{bmatrix}
end{gather*}
$J_2$相关:
egin{gather*}
z_i = 1 - sum_{j}K_{ij}ar{y}^{(i)}ar{y}^{(j)}alpha_j - ar{y}^{(i)}b \
frac{partial J_2}{partial alpha_k} = -csum_{i}ar{y}^{(i)}ar{y}^{(k)}K_{ik}p(z_i, eta)s(z_i, eta) \
frac{partial J_2}{partial b} = -csum_{i}ar{y}^{(i)}p(z_i, eta)s(z_i, eta) \
frac{partial^2 J_2}{partial alpha_kpartial alpha_l} = csum_{i}ar{y}^{(k)}ar{y}^{(l)}K_{ik}K_{il}[s^2(z_i, eta) + p(z_i, eta)delta(z_i, eta)] \
frac{partial^2 J_2}{partial alpha_kpartial b} = csum_{i}ar{y}^{(k)}K_{ik}[s^2(z_i, eta) + p(z_i, eta)delta(z_i, eta)] \
frac{partial^2 J_2}{partial bpartialalpha_l} = csum_{i}ar{y}^{(l)}K_{il}[s^2(z_i, eta) + p(z_i, eta)delta(z_i, eta)] \
frac{partial^2 J_2}{partial b^2} = csum_{i}[s^2(z_i, eta) + p(z_i, eta)delta(z_i, eta)]
end{gather*}
egin{gather*}
abla J_2 = egin{bmatrix} frac{partial J_2}{partialalpha_k} \ frac{partial J_2}{partial b} end{bmatrix} \ \
abla^2 J_2 = egin{bmatrix} frac{partial^2 J_2}{partialalpha_kpartialalpha_l} & frac{partial^2 J_2}{partialalpha_kpartial b} \ frac{partial^2 J_2}{partial bpartialalpha_l} & frac{partial^2 J_2}{partial b^2} end{bmatrix}
end{gather*}
目标函数$J$之gradient及Hessian如下:
egin{gather}
abla J = abla J_1 + abla J_2 \
H = abla^2 J = abla^2 J_1 + abla^2 J_2
end{gather}
Part Ⅳ
现以二分类为例进行算法实施:
egin{equation*}
egin{cases}
h(x) geq 0 & y = 1 & ext{正例} \
h(x) < 0 & y = -1 & ext{负例}
end{cases}
end{equation*}

代码实现:

  1 # Smooth Support Vector Machine之实现
  2 
  3 import numpy
  4 from matplotlib import pyplot as plt
  5 
  6 
  7 def spiral_point(val, center=(0, 0)):
  8     rn = 0.4 * (105 - val) / 104
  9     an = numpy.pi * (val - 1) / 25
 10 
 11     x0 = center[0] + rn * numpy.sin(an)
 12     y0 = center[1] + rn * numpy.cos(an)
 13     z0 = -1
 14     x1 = center[0] - rn * numpy.sin(an)
 15     y1 = center[1] - rn * numpy.cos(an)
 16     z1 = 1
 17 
 18     return (x0, y0, z0), (x1, y1, z1)
 19 
 20 
 21 def spiral_data(valList):
 22     dataList = list(spiral_point(val) for val in valList)
 23     data0 = numpy.array(list(item[0] for item in dataList))
 24     data1 = numpy.array(list(item[1] for item in dataList))
 25     return data0, data1
 26 
 27 
 28 # 生成训练数据集
 29 trainingValList = numpy.arange(1, 101, 1)
 30 trainingData0, trainingData1 = spiral_data(trainingValList)
 31 trainingSet = numpy.vstack((trainingData0, trainingData1))
 32 # 生成测试数据集
 33 testValList = numpy.arange(1.5, 101.5, 1)
 34 testData0, testData1 = spiral_data(testValList)
 35 testSet = numpy.vstack((testData0, testData1))
 36 
 37 
 38 class SSVM(object):
 39 
 40     def __init__(self, trainingSet, c=1, mu=1, beta=100):
 41         self.__trainingSet = trainingSet                     # 训练集数据
 42         self.__c = c                                         # 误差项权重
 43         self.__mu = mu                                       # gaussian kernel参数
 44         self.__beta = beta                                   # 光滑化参数
 45 
 46         self.__A, self.__D = self.__get_AD()
 47 
 48 
 49     def get_cls(self, x, alpha, b):
 50         A, D = self.__A, self.__D
 51         mu = self.__mu
 52 
 53         x = numpy.array(x).reshape((-1, 1))
 54         KAx = self.__get_KAx(A, x, mu)
 55         clsVal = self.__calc_hVal(KAx, D, alpha, b)
 56         if clsVal >= 0:
 57             return 1
 58         else:
 59             return -1
 60 
 61 
 62     def get_accuracy(self, dataSet, alpha, b):
 63         '''
 64         正确率计算
 65         '''
 66         rightCnt = 0
 67         for row in dataSet:
 68             clsVal = self.get_cls(row[:2], alpha, b)
 69             if clsVal == row[2]:
 70                 rightCnt += 1
 71         accuracy = rightCnt / dataSet.shape[0]
 72         return accuracy
 73 
 74 
 75     def optimize(self, maxIter=100, epsilon=1.e-9):
 76         '''
 77         maxIter: 最大迭代次数
 78         epsilon: 收敛判据, 梯度趋于0则收敛
 79         '''
 80         A, D = self.__A, self.__D
 81         c = self.__c
 82         mu = self.__mu
 83         beta = self.__beta
 84 
 85         alpha, b = self.__init_alpha_b((A.shape[1], 1))
 86         KAA = self.__get_KAA(A, mu)
 87 
 88         JVal = self.__calc_JVal(KAA, D, c, beta, alpha, b)
 89         grad = self.__calc_grad(KAA, D, c, beta, alpha, b)
 90         Hess = self.__calc_Hess(KAA, D, c, beta, alpha, b)
 91 
 92         for i in range(maxIter):
 93             print("iterCnt: {:3d},   JVal: {}".format(i, JVal))
 94             if self.__converged1(grad, epsilon):
 95                 return alpha, b, True
 96 
 97             dCurr = -numpy.matmul(numpy.linalg.inv(Hess), grad)
 98             ALPHA = self.__calc_ALPHA_by_ArmijoRule(alpha, b, JVal, grad, dCurr, KAA, D, c, beta)
 99 
100             delta = ALPHA * dCurr
101             alphaNew = alpha + delta[:-1, :]
102             bNew = b + delta[-1, -1]
103             JValNew = self.__calc_JVal(KAA, D, c, beta, alphaNew, bNew)
104             if self.__converged2(delta, JValNew - JVal, epsilon ** 2):
105                 return alphaNew, bNew, True
106 
107             alpha, b, JVal = alphaNew, bNew, JValNew
108             grad = self.__calc_grad(KAA, D, c, beta, alpha, b)
109             Hess = self.__calc_Hess(KAA, D, c, beta, alpha, b)
110         else:
111             if self.__converged1(grad, epsilon):
112                 return alpha, b, True
113         return alpha, b, False
114 
115 
116     def __converged2(self, delta, JValDelta, epsilon):
117         val1 = numpy.linalg.norm(delta, ord=numpy.inf)
118         val2 = numpy.abs(JValDelta)
119         if val1 <= epsilon or val2 <= epsilon:
120             return True
121         return False
122 
123 
124     def __calc_ALPHA_by_ArmijoRule(self, alphaCurr, bCurr, JCurr, gCurr, dCurr, KAA, D, c, beta, C=1.e-4, v=0.5):
125         i = 0
126         ALPHA = v ** i
127         delta = ALPHA * dCurr
128         alphaNext = alphaCurr + delta[:-1, :]
129         bNext = bCurr + delta[-1, -1]
130         JNext = self.__calc_JVal(KAA, D, c, beta, alphaNext, bNext)
131         while True:
132             if JNext <= JCurr + C * ALPHA * numpy.matmul(dCurr.T, gCurr)[0, 0]: break
133             i += 1
134             ALPHA = v ** i
135             delta = ALPHA * dCurr
136             alphaNext = alphaCurr + delta[:-1, :]
137             bNext = bCurr + delta[-1, -1]
138             JNext = self.__calc_JVal(KAA, D, c, beta, alphaNext, bNext)
139         return ALPHA
140 
141 
142     def __converged1(self, grad, epsilon):
143         if numpy.linalg.norm(grad, ord=numpy.inf) <= epsilon:
144             return True
145         return False
146 
147 
148     def __p(self, x, beta):
149         term = x * beta
150         if term > 10:
151             val = x + numpy.math.log(1 + numpy.math.exp(-term)) / beta
152         else:
153             val = numpy.math.log(numpy.math.exp(term) + 1) / beta
154         return val
155 
156 
157     def __s(self, x, beta):
158         term = x * beta
159         if term > 10:
160             val = 1 / (numpy.math.exp(-beta * x) + 1)
161         else:
162             term1 = numpy.math.exp(term)
163             val = term1 / (1 + term1)
164         return val
165 
166 
167     def __d(self, x, beta):
168         term = x * beta
169         if term > 10:
170             term1 = numpy.math.exp(-term)
171             val = beta * term1 / (1 + term1) ** 2
172         else:
173             term1 = numpy.math.exp(term)
174             val = beta * term1 / (term1 + 1) ** 2
175         return val
176 
177 
178     def __calc_Hess(self, KAA, D, c, beta, alpha, b):
179         Hess_J1 = self.__calc_Hess_J1(alpha)
180         Hess_J2 = self.__calc_Hess_J2(KAA, D, c, beta, alpha, b)
181         Hess = Hess_J1 + Hess_J2
182         return Hess
183 
184 
185     def __calc_Hess_J2(self, KAA, D, c, beta, alpha, b):
186         Hess_J2 = numpy.zeros((KAA.shape[0] + 1, KAA.shape[0] + 1))
187         Y = numpy.matmul(D, numpy.ones((D.shape[0], 1)))
188         YY = numpy.matmul(Y, Y.T)
189         KAAYY = KAA * YY
190 
191         z = 1 - numpy.matmul(KAAYY, alpha) - Y * b
192         p = numpy.array(list(self.__p(z[i, 0], beta) for i in range(z.shape[0]))).reshape((-1, 1))
193         s = numpy.array(list(self.__s(z[i, 0], beta) for i in range(z.shape[0]))).reshape((-1, 1))
194         d = numpy.array(list(self.__d(z[i, 0], beta) for i in range(z.shape[0]))).reshape((-1, 1))
195         term = s * s + p * d
196 
197         for k in range(Hess_J2.shape[0] - 1):
198             for l in range(k + 1):
199                 val = c * Y[k, 0] * Y[l, 0] * numpy.sum(KAA[:, k:k+1] * KAA[:, l:l+1] * term)
200                 Hess_J2[k, l] = Hess_J2[l, k] = val
201             val = c * Y[k, 0] * numpy.sum(KAA[:, k:k+1] * term)
202             Hess_J2[k, -1] = Hess_J2[-1, k] = val
203         val = c * numpy.sum(term)
204         Hess_J2[-1, -1] = val
205         return Hess_J2
206 
207 
208     def __calc_Hess_J1(self, alpha):
209         I = numpy.identity(alpha.shape[0])
210         term = numpy.hstack((I, numpy.zeros((I.shape[0], 1))))
211         Hess_J1 = numpy.vstack((term, numpy.zeros((1, term.shape[1]))))
212         return Hess_J1
213 
214 
215     def __calc_grad(self, KAA, D, c, beta, alpha, b):
216         grad_J1 = self.__calc_grad_J1(alpha)
217         grad_J2 = self.__calc_grad_J2(KAA, D, c, beta, alpha, b)
218         grad = grad_J1 + grad_J2
219         return grad
220 
221 
222     def __calc_grad_J2(self, KAA, D, c, beta, alpha, b):
223         grad_J2 = numpy.zeros((KAA.shape[0] + 1, 1))
224         Y = numpy.matmul(D, numpy.ones((D.shape[0], 1)))
225         YY = numpy.matmul(Y, Y.T)
226         KAAYY = KAA * YY
227 
228         z = 1 - numpy.matmul(KAAYY, alpha) - Y * b
229         p = numpy.array(list(self.__p(z[i, 0], beta) for i in range(z.shape[0]))).reshape((-1, 1))
230         s = numpy.array(list(self.__s(z[i, 0], beta) for i in range(z.shape[0]))).reshape((-1, 1))
231         term = p * s
232 
233         for k in range(grad_J2.shape[0] - 1):
234             val = -c * Y[k, 0] * numpy.sum(Y * KAA[:, k:k+1] * term)
235             grad_J2[k, 0] = val
236         grad_J2[-1, 0] = -c * numpy.sum(Y * term)
237         return grad_J2
238 
239 
240     def __calc_grad_J1(self, alpha):
241         grad_J1 = numpy.vstack((alpha, [[0]]))
242         return grad_J1
243 
244 
245     def __calc_JVal(self, KAA, D, c, beta, alpha, b):
246         J1 = self.__calc_J1(alpha)
247         J2 = self.__calc_J2(KAA, D, c, beta, alpha, b)
248         JVal = J1 + J2
249         return JVal
250 
251 
252     def __calc_J2(self, KAA, D, c, beta, alpha, b):
253         tmpOne = numpy.ones((KAA.shape[0], 1))
254         x = tmpOne - numpy.matmul(numpy.matmul(numpy.matmul(D, KAA), D), alpha) - numpy.matmul(D, tmpOne) * b
255         p = numpy.array(list(self.__p(x[i, 0], beta) for i in range(x.shape[0])))
256         J2 = numpy.sum(p * p) * c / 2
257         return J2
258 
259 
260     def __calc_J1(self, alpha):
261         J1 = numpy.sum(alpha * alpha) / 2
262         return J1
263 
264 
265     def __get_KAA(self, A, mu):
266         KAA = numpy.zeros((A.shape[1], A.shape[1]))
267         for rowIdx in range(KAA.shape[0]):
268             for colIdx in range(rowIdx + 1):
269                 x1 = A[:, rowIdx:rowIdx+1]
270                 x2 = A[:, colIdx:colIdx+1]
271                 val = self.__calc_gaussian(x1, x2, mu)
272                 KAA[rowIdx, colIdx] = KAA[colIdx, rowIdx] = val
273         return KAA
274 
275 
276     def __init_alpha_b(self, shape):
277         '''
278         alpha, b之初始化
279         '''
280         alpha, b = numpy.zeros(shape), 0
281         return alpha, b
282 
283 
284     def __get_KAx(self, A, x, mu):
285         KAx = numpy.zeros((A.shape[1], 1))
286         for rowIdx in range(KAx.shape[0]):
287             x1 = A[:, rowIdx:rowIdx+1]
288             val = self.__calc_gaussian(x1, x, mu)
289             KAx[rowIdx, 0] = val
290         return KAx
291 
292 
293     def __calc_hVal(self, KAx, D, alpha, b):
294         hVal = numpy.matmul(numpy.matmul(alpha.T, D), KAx)[0, 0] + b
295         return hVal
296 
297 
298     def __calc_gaussian(self, x1, x2, mu):
299         val = numpy.math.exp(-mu * numpy.linalg.norm(x1 - x2) ** 2)
300         # val = numpy.sum(x1 * x2)
301         return val
302 
303 
304     def __get_AD(self):
305         A = self.__trainingSet[:, :2].T
306         D = numpy.diag(self.__trainingSet[:, 2])
307         return A, D
308 
309 
310 class SpiralPlot(object):
311 
312     def spiral_data_plot(self, trainingData0, trainingData1, testData0, testData1):
313         fig = plt.figure(figsize=(5, 5))
314         ax1 = plt.subplot()
315         ax1.scatter(trainingData1[:, 0], trainingData1[:, 1], c="red", marker="o", s=10, label="training data - Positive")
316         ax1.scatter(trainingData0[:, 0], trainingData0[:, 1], c="blue", marker="o", s=10, label="training data - Negative")
317         ax1.scatter(testData1[:, 0], testData1[:, 1], c="red", marker="x", s=10, label="test data - Positive")
318         ax1.scatter(testData0[:, 0], testData0[:, 1], c="blue", marker="x", s=10, label="test data - Negative")
319         ax1.set(xlim=(-0.5, 0.5), ylim=(-0.5, 0.5), xlabel="$x_1$", ylabel="$x_2$")
320         plt.legend(fontsize="x-small")
321         fig.tight_layout()
322         fig.savefig("spiral.png", dpi=100)
323         plt.close()
324 
325 
326     def spiral_pred_plot(self, trainingData0, trainingData1, testData0, testData1, ssvmObj, alpha, b):
327         x = numpy.linspace(-0.5, 0.5, 500)
328         y = numpy.linspace(-0.5, 0.5, 500)
329         x, y = numpy.meshgrid(x, y)
330         z = numpy.zeros(shape=x.shape)
331         for rowIdx in range(x.shape[0]):
332             for colIdx in range(x.shape[1]):
333                 z[rowIdx, colIdx] = ssvmObj.get_cls((x[rowIdx, colIdx], y[rowIdx, colIdx]), alpha, b)
334         cls2color = {-1: "blue", 0: "white", 1: "red"}
335 
336         fig = plt.figure(figsize=(5, 5))
337         ax1 = plt.subplot()
338         ax1.contourf(x, y, z, levels=[-1.5, -0.5, 0.5, 1.5], colors=["blue", "white", "red"], alpha=0.3)
339         ax1.scatter(trainingData1[:, 0], trainingData1[:, 1], c="red", marker="o", s=10, label="training data - Positive")
340         ax1.scatter(trainingData0[:, 0], trainingData0[:, 1], c="blue", marker="o", s=10, label="training data - Negative")
341         ax1.scatter(testData1[:, 0], testData1[:, 1], c="red", marker="x", s=10, label="test data - Positive")
342         ax1.scatter(testData0[:, 0], testData0[:, 1], c="blue", marker="x", s=10, label="test data - Negative")
343         ax1.set(xlim=(-0.5, 0.5), ylim=(-0.5, 0.5), xlabel="$x_1$", ylabel="$x_2$")
344         plt.legend(loc="upper left", fontsize="x-small")
345         fig.tight_layout()
346         fig.savefig("pred.png", dpi=100)
347         plt.close()
348 
349 
350 
351 if __name__ == "__main__":
352     ssvmObj = SSVM(trainingSet, c=0.1, mu=250, beta=100)
353     alpha, b, tab = ssvmObj.optimize()
354     accuracy1 = ssvmObj.get_accuracy(trainingSet, alpha, b)
355     print("Accuracy on trainingSet is {}%".format(accuracy1 * 100))
356     accuracy2 = ssvmObj.get_accuracy(testSet, alpha, b)
357     print("Accuracy on testSet is {}%".format(accuracy2 * 100))
358 
359     spObj = SpiralPlot()
360     spObj.spiral_data_plot(trainingData0, trainingData1, testData0, testData1)
361     spObj.spiral_pred_plot(trainingData0, trainingData1, testData0, testData1, ssvmObj, alpha, b)

View Code

笔者所用训练集、测试集数据分布如下:

很显然, 此数据集非线性可分.

结果展示:
此模型在训练集、测试集上的准确率均达到100%.
使用建议:
①. 通过核函数映射至高维后, 支持向量可能存在很多个;
②. SSVM本质上求解的仍然是原问题, 此时$alpha$仅作为变数变换引入, 无需满足KKT条件中对偶可行性;
③. 数据集是否需要归一化, 应按实际情况予以考虑.
参考文档:
Yuh-Jye Lee and O. L. Mangasarian. "SSVM: A Smooth Support Vector Machine for Classification", Computational Optimization and Applications, 20, (2001) 5-22.

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原文地址：https://www.cnblogs.com/xxhbdk/p/12275567.html