关于SVM核函数参数及高斯/RBF核中向量a、b含义的技术问询

Hey there! Let's break down your questions about SVM kernel functions clearly, since you already have a grasp on the basics of the RBF kernel.

SVM中核函数的常见参数 #

Different kernel types in SVM come with their own tunable parameters—here are the most common ones you’ll encounter:

• RBF/Gaussian Kernel (K(a,b) = exp(-gamma||a-b||²)):
  • The key parameter here is gamma: It controls the "width" of the Gaussian curve. A large gamma means the curve is narrow, so only samples very close to each other are considered similar; a small gamma makes the curve wider, allowing more distant samples to influence the decision boundary.
• Polynomial Kernel (K(a,b) = (gamma*a·b + coef0)^degree):
  • degree: The degree of the polynomial (e.g., 2 for quadratic, 3 for cubic). Higher degrees can model more complex boundaries but risk overfitting.
  • gamma: Scales the dot product of the vectors, similar to its role in RBF.
  • coef0: Shifts the polynomial, affecting how much lower-degree terms contribute to the kernel value.
• Sigmoid Kernel (K(a,b) = tanh(gamma*a·b + coef0)):
  • gamma: Scales the dot product.
  • coef0: Shifts the input to the tanh function, which can change the shape of the decision boundary.
• Linear Kernel (K(a,b) = a·b):
  • No tunable parameters—this is the simplest kernel, equivalent to using a linear SVM without the kernel trick.

RBF核中向量a和b的含义 #

The meaning of a and b depends on whether we’re in the training phase or prediction phase of the SVM:

• Training Phase:
  Both a and b are individual samples from your training dataset. The kernel calculates the similarity between every pair of training samples to build the decision boundary in the high-dimensional space. For example, when computing the kernel matrix during training, each entry K(i,j) is the similarity between the i-th training sample (a) and j-th training sample (b).
• Prediction Phase:
  Here, a is the new test sample you want to classify, and b refers to each of the support vectors (the critical training samples that define the decision boundary). The SVM computes the similarity between the test sample and each support vector, combines those values with the learned weights, and outputs the classification result.

内容的提问来源于stack exchange，提问作者Justin