Linked Questions

2
votes
0answers
670 views

Is a covariance matrix defined through a Gaussian covariance function always positive-definite? [duplicate]

When using Gaussian processes, the covariance matrix $\mathbf{\Sigma}$ is often defined via a covariance function $K$ as follows $$ \mathbf{\Sigma}_{ij} = K(\underline{x}_i, \underline{x}_j) $$ where $...
1
vote
0answers
348 views

What is the most intuitive proof that Gaussian kernel is positive definite? [duplicate]

I have general form of Gaussian kernel $K(x,x')=\exp(-\|x-x'\|^{2})$ (just not considering $\sigma$). I tried to prove its positive definiteness via Gram matrix properties, but couldn't. Is there any ...
1
vote
0answers
182 views

Derive squared exponential covariance function [duplicate]

In Gaussian Processes, SVMs, kernels are used (as to my understanding) as similarity measure. However, they have the constraint that any kernel has to be represented as a dot product. i.e. $k(x_1,x_2)=...
31
votes
4answers
20k views

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 ...
8
votes
2answers
6k views

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 ...
9
votes
3answers
3k views

Is the Gaussian Kernel still a valid Kernel when taking the negative of the inner function?

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(...
3
votes
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 ...
3
votes
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.
6
votes
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-...
3
votes
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 ...
2
votes
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 $...
0
votes
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 ...
2
votes
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 ...
2
votes
2answers
455 views

Is alpha*RBF a valid kernel, where alpha >= 0 is a parameter?

I wonder if K = alpha*RBF can be a valid kernel satisfying Mercer's condition, where ...
0
votes
1answer
583 views

How do the mapping function phi(x) of a RBF kernel?

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 ...

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