在解决SVM双重问题时，CVXPY错误“无法乘以两个非常数”

Question

我正在尝试使用CVXPY解决SVM双重问题。

以下是Python代码：

import numpy as np
import cvxpy as cvx

# note: X and Y are numpy arrays generated for testing purpose
# calculating guassian kernal
def kg(a, b, theta=1):
    sim = np.exp( -0.5 * np.dot(a, b) / (theta ** 2))
    return sim

# generating kernal matrix
def k_mat(x, k_func=kg):
    m = x.shape[0]
    mat = np.zeros((m, m))
    for i in range(m):
        for j in range(m):
            mat[i, j] = k_func(x[i, :], x[j, :])
    return mat

k=k_mat(X)

# setup parameters
a = cvx.Variable(m)
C = cvx.Parameter(sign="positive")
C.value = 0.01

# start convex optimization
obj = cvx.Maximize(cvx.sum_entries(a) - \
      0.5 * cvx.mul_elemwise(Y, a).T * k * cvx.mul_elemwise(Y, a))
constraints = [a>=0, a<=C, cvx.sum_entries(Y, a)==0]
prob = cvx.Problem(obj, constraints)

prob.solve()
print(a.value)

我收到了错误：

Traceback (most recent call last):
  File "/Library/Frameworks/Python.framework/Versions/3.5/lib/python3.5/site-packages/IPython/core/interactiveshell.py", line 3066, in run_code
    exec(code_obj, self.user_global_ns, self.user_ns)
  File "<ipython-input-44-d3b220364629>", line 4, in <module>
    obj = cvx.Maximize(cvx.sum_entries(a) - 0.5 * cvx.mul_elemwise(Y, a).T * k * cvx.mul_elemwise(Y, a))
  File "/Library/Frameworks/Python.framework/Versions/3.5/lib/python3.5/site-packages/cvxpy/expressions/expression.py", line 43, in cast_op
    return binary_op(self, other)
  File "/Library/Frameworks/Python.framework/Versions/3.5/lib/python3.5/site-packages/cvxpy/expressions/expression.py", line 224, in __mul__
    raise DCPError("Cannot multiply two non-constants.")
cvxpy.error.DCPError: Cannot multiply two non-constants.

似乎cvxpy不支持内核矩阵的二次形式优化。但是我看到人们在Matlab中使用cvx解决了本演示文稿第13页（35）中的相同问题：

http://users.isy.liu.se/en/rt/schon/CourseMLlund/le5.pdf

我对cvx很新。请帮我改正一下。感谢。

Answer 1

找到了解决方案。

问题1

我犯了一个愚蠢的错误，错误的高斯内核定义应该是：

def kg(a, b, theta=1):
    sim = np.exp(np.dot((a - b), (a - b)) / (2 * theta ** 2))
    return sim

问题2

应该是正确的二次形式。 （我真的希望cvx公式能与numpy惯例兼容）

cvx.quad_form(cvx.mul_elemwise(Y, a), k)