核脊和具有多项式特征的简单脊

Question

Kernel Ridge（来自 sklearn.kernel_ridge）与多项式内核和使用 PolynomialFeatures + Ridge（来自 sklearn.linear_model）有什么区别？

Answer 1

区别在于特征计算。PolynomialFeatures显式计算输入特征之间达到所需程度的多项式组合，同时仅考虑以原始特征表示的KernelRidge(kernel='poly')多项式内核（特征点积的多项式表示）。本文档总体上提供了很好的概述。

关于计算我们可以从源码中查看相关部分：

• 岭回归
  • 实际计算从这里开始（对于默认设置）；您可以与上面链接文档中的方程（5）进行比较。计算过程包括计算特征向量（内核）之间的点积，然后计算对偶系数 ( alpha )，最后计算与特征向量的点积以获得权重。
• 内核岭
  • 类似地计算对偶系数并存储它们（而不是计算一些权重）。这是因为在进行预测时，会再次计算训练样本和预测样本之间的内核。然后将结果用对偶系数点缀。

（训练）内核的计算遵循类似的过程：比较Ridge和KernelRidge。主要区别在于，Ridge显式考虑其收到的任何（多项式）特征之间的点积，而KernelRidge这些多项式特征是在计算过程中隐式生成的。例如考虑单个特征x；与gamma = coef0 = 1计算。KernelRidge​ (x**2 + 1)**2 == (x**4 + 2*x**2 + 1)如果您现在考虑PolynomialFeatures这将提供功能x**2, x, 1，并且相应的点积为x**4 + x**2 + 1。因此点积因项而异x**2。当然，我们可以重新调整多特征的大小，x**2, sqrt(2)*x, 1但KernelRidge(kernel='poly')我们没有这种灵活性。另一方面，差异可能并不重要（在大多数情况下）。

请注意，对偶系数的计算也以类似的方式执行：Ridge和KernelRidge。最后KernelRidge保留对偶系数，同时Ridge直接计算权重。

让我们看一个小例子：

import matplotlib.pyplot as plt
import numpy as np
from sklearn.linear_model import Ridge
from sklearn.kernel_ridge import KernelRidge
from sklearn.preprocessing import PolynomialFeatures
from sklearn.utils.extmath import safe_sparse_dot

np.random.seed(20181001)

a, b = 1, 4
x = np.linspace(0, 2, 100).reshape(-1, 1)
y = a*x**2 + b*x + np.random.normal(scale=0.2, size=(100,1))

poly = PolynomialFeatures(degree=2, include_bias=True)
xp = poly.fit_transform(x)
print('We can see that the new features are now [1, x, x**2]:')
print(f'xp.shape: {xp.shape}')
print(f'xp[-5:]:\n{xp[-5:]}', end='\n\n')
# Scale the `x` columns so we obtain similar results.
xp[:, 1] *= np.sqrt(2)

ridge = Ridge(alpha=0, fit_intercept=False, solver='cholesky')
ridge.fit(xp, y)

krr = KernelRidge(alpha=0, kernel='poly', degree=2, gamma=1, coef0=1)
krr.fit(x, y)

# Let's try to reproduce some of the involved steps for the different models.
ridge_K = safe_sparse_dot(xp, xp.T)
krr_K = krr._get_kernel(x)
print('The computed kernels are (alomst) similar:')
print(f'Max. kernel difference: {np.abs(ridge_K - krr_K).max()}', end='\n\n')
print('Predictions slightly differ though:')
print(f'Max. difference: {np.abs(krr.predict(x) - ridge.predict(xp)).max()}', end='\n\n')

# Let's see if the fit changes if we provide `x**2, x, 1` instead of `x**2, sqrt(2)*x, 1`.
xp_2 = xp.copy()
xp_2[:, 1] /= np.sqrt(2)
ridge_2 = Ridge(alpha=0, fit_intercept=False, solver='cholesky')
ridge_2.fit(xp_2, y)
print('Using features "[x**2, x, 1]" instead of "[x**2, sqrt(2)*x, 1]" predictions are (almost) the same:')
print(f'Max. difference: {np.abs(ridge_2.predict(xp_2) - ridge.predict(xp)).max()}', end='\n\n')
print('Interpretability of the coefficients changes though:')
print(f'ridge.coef_[1:]: {ridge.coef_[0, 1:]}, ridge_2.coef_[1:]: {ridge_2.coef_[0, 1:]}')
print(f'ridge.coef_[1]*sqrt(2): {ridge.coef_[0, 1]*np.sqrt(2)}')
print(f'Compare with: a, b = ({a}, {b})')

plt.plot(x.ravel(), y.ravel(), 'o', color='skyblue', label='Data')
plt.plot(x.ravel(), ridge.predict(xp).ravel(), '-', label='Ridge', lw=3)
plt.plot(x.ravel(), krr.predict(x).ravel(), '--', label='KRR', lw=3)
plt.grid()
plt.legend()
plt.show()

从中我们得到：

We can see that the new features are now [x, x**2]:
xp.shape: (100, 3)
xp[-5:]:
[[1.         1.91919192 3.68329762]
 [1.         1.93939394 3.76124885]
 [1.         1.95959596 3.84001632]
 [1.         1.97979798 3.91960004]
 [1.         2.         4.        ]]

The computed kernels are (alomst) similar:
Max. kernel difference: 1.0658141036401503e-14

Predictions slightly differ though:
Max. difference: 0.04244651134471766

Using features "[x**2, x, 1]" instead of "[x**2, sqrt(2)*x, 1]" predictions are (almost) the same:
Max. difference: 7.15642822779472e-14

Interpretability of the coefficients changes though:
ridge.coef_[1:]: [2.73232239 1.08868872], ridge_2.coef_[1:]: [3.86408737 1.08868872]
ridge.coef_[1]*sqrt(2): 3.86408737392841
Compare with: a, b = (1, 4)