To determine which of the given options is not a kernel, we need to analyze each of the suggested kernels to verify their validity under the mathematical framework of kernel functions.

1. **Polynomial Kernel:** This kernel is defined as $ K(x, y) = (x \cdot y + c)^d $, where $ c $ is a constant and $ d $ is the degree of the polynomial. It is a valid kernel since it satisfies Mercer's theorem.

2. **Gaussian Kernel:** Also known as the Radial Basis Function (RBF) kernel, it is defined as $ K(x, y) = e^{-\frac{\|x - y\|^2}{2\sigma^2}} $, where $ \sigma $ is a parameter. This kernel is proven to be valid as it meets the requirements of a kernel function.

3. **Sigmoid Kernel:** This kernel is defined as $ K(x, y) = \tanh(\alpha (x \cdot y) + c) $, where $ \alpha $ and $ c $ are constants. However, this kernel does not always satisfy the conditions of Mercer's theorem for certain parameters, which means it is not a general valid kernel.

4. **None of these:** This option suggests that all the listed kernels are valid.

### Final Answer
The option that is **not a kernel** is the **Sigmoid Kernel**.

Question

To determine which of the given options is not a kernel, we need to analyze each of the suggested kernels to verify their validity under the mathematical framework of kernel functions.

1. **Polynomial Kernel:** This kernel is defined as $ K(x, y) = (x \cdot y + c)^d $, where $ c $ is a constant and $ d $ is the degree of the polynomial. It is a valid kernel since it satisfies Mercer's theorem.

2. **Gaussian Kernel:** Also known as the Radial Basis Function (RBF) kernel, it is defined as $ K(x, y) = e^{-\frac{\|x - y\|^2}{2\sigma^2}} $, where $ \sigma $ is a parameter. This kernel is proven to be valid as it meets the requirements of a kernel function.

3. **Sigmoid Kernel:** This kernel is defined as $ K(x, y) = \tanh(\alpha (x \cdot y) + c) $, where $ \alpha $ and $ c $ are constants. However, this kernel does not always satisfy the conditions of Mercer's theorem for certain parameters, which means it is not a general valid kernel.

4. **None of these:** This option suggests that all the listed kernels are valid.

### Final Answer
The option that is **not a kernel** is the **Sigmoid Kernel**.

Knowee AI · Accepted Answer

To determine which of the given options is not a kernel, we need to analyze each of the suggested kernels to verify their validity under the mathematical framework of kernel functions.

1. **Polynomial Kernel:** This kernel is defined as $ K(x, y) = (x \cdot y + c)^d $, where $ c $ is a constant and $ d $ is the degree of the polynomial. It is a valid kernel since it satisfies Mercer's theorem.

2. **Gaussian Kernel:** Also known as the Radial Basis Function (RBF) kernel, it is defined as $ K(x, y) = e^{-\frac{\|x - y\|^2}{2\sigma^2}} $, where $ \sigma $ is a parameter. This kernel is proven to be valid as it meets the requirements of a kernel function.

3. **Sigmoid Kernel:** This kernel is defined as $ K(x, y) = \tanh(\alpha (x \cdot y) + c) $, where $ \alpha $ and $ c $ are constants. However, this kernel does not always satisfy the conditions of Mercer's theorem for certain parameters, which means it is not a general valid kernel.

4. **None of these:** This option suggests that all the listed kernels are valid.

### Final Answer
The option that is **not a kernel** is the **Sigmoid Kernel**.

Which of the following is not a kernel?(1 Point)Polynomial KernelGaussian KernelSigmoid KernelNone of these

Question

Which of the following is not a kernel?

Solution

Final Answer

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