SVM - 如何对核化的克数矩阵进行矢量化？

Question

我使用cvxopt qp求解器在python中实现了一个支持向量机，我需要在每个元素上计算一个带有核函数的两个向量的克矩阵。我使用for循环正确实现了它，但这种策略是计算密集型的。我想对代码进行矢量化。

示例：

这是我写的：

K = np.array( [kernel(X[i], X[j],poly=poly_kernel) 
     for j in range(m)
     for i in range(m)]).reshape((m, m))

如何在没有for循环的情况下对上述代码进行矢量化以更快地获得相同的结果？

内核函数计算高斯内核。

Here is a quick explanation of an svm with kernel trick.第二页解释了这个问题。

以下是我的完整code上下文。

编辑：这是一个快速的代码片段，运行我需要以非矢量化形式进行矢量化的内容

from sklearn.datasets import make_gaussian_quantiles;
import numpy as np;


X,y = make_gaussian_quantiles(mean=None, cov=1.0, n_samples=100, n_features=2, n_classes=2, shuffle=True, random_state=5);

m = X.shape[0];


def kernel(a,b,d=20,poly=True,sigma=0.5):
    if (poly):
        return np.inner(a,b) ** d;
    else:
        return np.exp(-np.linalg.norm((a - b) ** 2)/sigma**2)

# Need to vectorize these loops

K = np.array([kernel(X[i], X[j],poly=False) 
    for j in range(m)
    for i in range(m)]).reshape((m, m))

谢谢！

Answer 1

这是一个矢量化版本。非poly分支有两个变体，一个是直接的，一个是内存，在一个特征数量很大的情况下保存一个：

from sklearn.datasets import make_gaussian_quantiles;
import numpy as np;


X,y = make_gaussian_quantiles(mean=None, cov=1.0, n_samples=100, n_features=2, n_classes=2, shuffle=True, random_state=5);
Y,_ = make_gaussian_quantiles(mean=None, cov=1.0, n_samples=200, n_features=2, n_classes=2, shuffle=True, random_state=2);

m = X.shape[0];
n = Y.shape[0]

def kernel(a,b,d=20,poly=True,sigma=0.5):
    if (poly):
        return np.inner(a,b) ** d;
    else:
        return np.exp(-np.linalg.norm((a - b) ** 2)/sigma**2)

# Need to vectorize these loops

POLY = False
LOW_MEM = 0

K = np.array([kernel(X[i], Y[j], poly=POLY) 
              for i in range(m)
              for j in range(n)]).reshape((m, n))

def kernel_v(X, Y=None, d=20, poly=True, sigma=0.5):
    Z = X if Y is None else Y
    if poly:
        return np.einsum('ik,jk', X, Z)**d
    elif X.shape[1] < LOW_MEM:
        return np.exp(-np.sqrt(((X[:, None, :] - Z[None, :, :])**4).sum(axis=-1)) / sigma**2)
    elif Y is None or Y is X:
        X2 = X*X
        H = np.einsum('ij,ij->i', X2, X2) + np.einsum('ik,jk', X2, 3*X2) - np.einsum('ik,jk', X2*X, 4*X)
        return np.exp(-np.sqrt(np.maximum(0, H+H.T)) / sigma**2)
    else:
        X2, Y2 = X*X, Y*Y
        E = np.einsum('ik,jk', X2, 6*Y2) - np.einsum('ik,jk', X2*X, 4*Y) - np.einsum('ik,jk', X, 4*Y2*Y)
        E += np.add.outer(np.einsum('ij,ij->i', X2, X2), np.einsum('ij,ij->i', Y2, Y2))
        return np.exp(-np.sqrt(np.maximum(0, E)) / sigma**2)

print(np.allclose(K, kernel_v(X, Y, poly=POLY)))