带有 Gram 矩阵的预计算 RBF 内核的 Python 实现?
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【中文标题】带有 Gram 矩阵的预计算 RBF 内核的 Python 实现?【英文标题】:Python implementation of precomputed RBF kernel with Gram matrix? 【发布时间】:2018-02-17 18:36:10 【问题描述】:Python 的信息和precomputed kernels
示例非常有限。 sklearn
仅提供linear kernel
的一个简单示例:http://scikit-learn.org/stable/modules/svm.html
这是线性内核的代码:
import numpy as np
from scipy.spatial.distance import cdist
from sklearn.datasets import load_iris
# import data
iris = datasets.load_iris()
X = iris.data
Y = iris.target
X_train, X_test, y_train, y_test = train_test_split(X, Y)
clf = svm.SVC(kernel='precomputed')
# Linear kernel
G_train = np.dot(X_train, X_train.T)
clf.fit(G_train, y_train)
G_test = np.dot(X_test, X_train.T)
y_pred = clf.predict(G_test)
这对于进一步理解其他重要内核的实现不是很有帮助,例如RBF kernel
,它将是:
K(X, X') = np.exp(divide(-cdist(X, X, 'euclidean), 2*np.std(X**2)))
如何对train
和test
进行同样的拆分并为precomputed kernel
实现RBF
?
如果内核变得更加复杂,这取决于需要在单独的函数中计算的其他参数,例如参数alpha >= 0
:
K(X, X') = alpha('some function depending on X_train, X_test')*np.exp(divide(-cdist(X, X, 'euclidean), 2*np.std(X**2)))
我们需要此类非平凡内核的示例。如有任何建议,我将不胜感激。
【问题讨论】:
【参考方案1】:我们可以手动编写内核 pca。让我们从多项式内核开始。
from sklearn.datasets import make_circles
from scipy.spatial.distance import pdist, squareform
from scipy.linalg import eigh
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
X_c, y_c = make_circles(n_samples=100, random_state=654)
plt.figure(figsize=(8,6))
plt.scatter(X_c[y_c==0, 0], X_c[y_c==0, 1], color='red')
plt.scatter(X_c[y_c==1, 0], X_c[y_c==1, 1], color='blue')
plt.ylabel('y coordinate')
plt.xlabel('x coordinate')
plt.show()
数据:
def degree_pca(X, gamma, degree, n_components):
# Calculating kernel
K = gamma*(X@X.T+1)**degree
# Obtaining eigenvalues in descending order with corresponding
# eigenvectors from the symmetric matrix.
eigvals, eigvecs = eigh(K)
# Obtaining the i eigenvectors that corresponds to the i highest eigenvalues.
X_pc = np.column_stack((eigvecs[:,-i] for i in range(1,n_components+1)))
return X_pc
现在转换数据并绘制它
X_c1 = degree_pca(X_c, gamma=5, degree=2, n_components=2)
plt.figure(figsize=(8,6))
plt.scatter(X_c1[y_c==0, 0], X_c1[y_c==0, 1], color='red')
plt.scatter(X_c1[y_c==1, 0], X_c1[y_c==1, 1], color='blue')
plt.ylabel('y coordinate')
plt.xlabel('x coordinate')
plt.show()
线性可分:
现在点可以线性分开。
接下来让我们编写 RBF 内核。为了演示,让我们以卫星为例。
from sklearn.datasets import make_moons
X, y = make_moons(n_samples=100, random_state=654)
plt.figure(figsize=(8,6))
plt.scatter(X[y==0, 0], X[y==0, 1], color='red')
plt.scatter(X[y==1, 0], X[y==1, 1], color='blue')
plt.ylabel('y coordinate')
plt.xlabel('x coordinate')
plt.show()
月亮:
内核 pca 转换:
def stepwise_kpca(X, gamma, n_components):
"""
X: A MxN dataset as NumPy array where the samples are stored as rows (M), features as columns (N).
gamma: coefficient for the RBF kernel.
n_components: number of components to be returned.
"""
# Calculating the squared Euclidean distances for every pair of points
# in the MxN dimensional dataset.
sq_dists = pdist(X, 'sqeuclidean')
# Converting the pairwise distances into a symmetric MxM matrix.
mat_sq_dists = squareform(sq_dists)
K=np.exp(-gamma*mat_sq_dists)
# Centering the symmetric NxN kernel matrix.
N = K.shape[0]
one_n = np.ones((N,N)) / N
K = K - one_n.dot(K) - K.dot(one_n) + one_n.dot(K).dot(one_n)
# Obtaining eigenvalues in descending order with corresponding
# eigenvectors from the symmetric matrix.
eigvals, eigvecs = eigh(K)
# Obtaining the i eigenvectors that corresponds to the i highest eigenvalues.
X_pc = np.column_stack((eigvecs[:,-i] for i in range(1,n_components+1)))
return X_pc
让我们开始吧
X_4 = stepwise_kpca(X, gamma=15, n_components=2)
plt.scatter(X_4[y==0, 0], X_4[y==0, 1], color='red')
plt.scatter(X_4[y==1, 0], X_4[y==1, 1], color='blue')
plt.ylabel('y coordinate')
plt.xlabel('x coordinate')
plt.show()
结果:
【讨论】:
感谢您在 Kernel PCA 上所做的努力,但这与我的问题无关。以上是关于带有 Gram 矩阵的预计算 RBF 内核的 Python 实现?的主要内容,如果未能解决你的问题,请参考以下文章
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