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### Feature map for the Gaussian kernel

In SVM, the Gaussian kernel is defined as: $$K(x,y)=\exp\left({-\frac{\|x-y\|_2^2}{2\sigma^2}}\right)=\phi(x)^T\phi(y)$$ where $x, y\in \mathbb{R^n}$. I do not know the explicit equation of $\phi$. I ...
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### Linear combination of two kernel functions

How can I prove that linear combination of two kernel functions is also a kernel function? \begin{align} k_{p}( x, y) = a_1k_1( x, y) + a_2k_2(x,y) \end{align} given $k_1(,)$ and $k_2(,)$ are valid ...
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In support vector machines (SVMs) and other Kernel based methods, like Gaussian processes, the Kernel replaces the inner product of two feature vectors $k(x_n,x_m)=x_n^Tx_m$. The Gaussian kernel $$k(... 1answer 10k views ### Use Gaussian RBF kernel for mapping of 2D data to 3D I am working on SVMs and try to get all the concepts involved. For instance, the kernel mapping. I would like to construct some parts of the algorithm by myself, to understand what is happening. My ... 1answer 4k views ### Proof that exponential of a kernel is a kernel How can I prove that the exponential \exp(K) of a kernel function K is again a kernel? I think it can be proved using Taylor expansion but I am not sure how. 2answers 938 views ### Prove that the squared exponential covariance is non-negative definite Consider a covariance function of the form$$K_{i,j}=\alpha\times exp(-0.5 (x_i-x_j)^2/l^2)$$This is a very common function used in Gaussian processes. How to show that this covariance is non-... 1answer 2k views ### Combination of two SVM Kernels According to the book "Support Vector Machines" from Cristianini and Shawe-Taylor, it is feasible to make kernels from kernels. My question is now more in application of this methods with tools like ... 1answer 1k views ### Given a kernel, how to find mapping phi? I'm not clear about kernel. How could I construct my own kernel that is valid? Is the only method the Mercer Theorem (positive semi-definite)? I mean if I know K is a valid kernel, do I know that ... 1answer 727 views ### Validating Kernel Functions I'd appreciate help in clarifying my understanding of how to valid kernel functions, using the following two examples: K(x, t) = x^Tt - (x^Tt)^2 K(x, t) = e^{(x_1t_1)} where x_1\ and\ t_1 are ... 1answer 1k views ### Prove that this kernel is a valid kernel How would you argument or prove that this is a valid kernel:$$ K_a(x, t) = \prod_{i=1}^{n} (1 + x_it_i + (1-x_i)(1-t_i)).  I know that there are two conditions that a kernel must satisfy to be a ...
I'm trying to implement a paper that used SVM and an improve of it with Bayesian decision theory. How do I do the mapping feature $\phi(x)$ that appears in the decision function? The paper used an ...