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Questions tagged [kernel]

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Sum of two kernel is also a kernel, how can we prove that? [duplicate]

Suppose that k(·, ·) and k 0 (·, ·) are kernels. Prove that l(·, ·) where l(x, y) = k(x, y) + k0 (x, y) is also a kernel. I am having trouble proving this, can you help?
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Linear separation in higher dimension

I am having a problem comprehending with the relation of kernel, weight and linear separation. I have a case where I am given a kernel k1. that has a corresponding mapping phi1. And we know that ...
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What kind of kernel is used by statsmodels.nonparametric.kernel_regression.KernelReg?

I am doing multivariate nonparametric kernel regression using the Python function as mentioned in the title. The documentation can be found here: https://www.statsmodels.org/stable/generated/...
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Intuition behind the length-scale of the Rational Quadratic Kernel

What is the meaning of the length-scale in a rational quadratic? \begin{equation} k_{\textrm{RQ}}(t, t') = \sigma^2 \left( 1 + \frac{(t - t')^2}{2 \alpha \ell^2} \right)^{-\alpha} \label{eq:...
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Can SVM with Gaussian RBF kernel separate all kinds of data theoretically?

Gaussian is well known because its corresponding feature mapping is to infinite dimension. So with finite number of training data, is that the case that we can achieve zero training error with some ...
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Kernel Density Estimation for non-parametric

I'm writing an R function to get the fitted values of the kernel density estimate. For that I use the computational formula of summation of {n-1 h-1 K{(x - Xi)/h}} where n is the number of ...
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The role of Fisher information matrix in Fisher kernel

I read the original paper proposed the Fisher kernel. The Fisher kernel is defined as $K(X_i,X_j) \propto U_{X_i}I^{-1}U_{X_j}$, where $U_X$ is the Fisher schore and $I$ is the Fisher information ...
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Kernlab, user-defined kernel on chosen variables [closed]

I want to make a user-defined kernel with different variables in the kernel and combine them. Does anyone know if this is possible with kernlab or what is wrong with my code? I use these packages: <...
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Why does the DCGAN output degrade with an increase in the kernel size?

Thank you for the explanation on the kernel size. I have been experimenting with the sample Generative Adversarial Network (GAN) code from the book on Deep learning with Python by François Chollet, ...
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Kernel function with a feature space equipped with an inner product that is not the dot product

Premise: A function $K: \mathbb R^d \times \mathbb R^d \to \mathbb R$ is called a kernel function on $\mathbb{R}^d$ if there exists a Hilbert space $\mathcal{H}$ and a map $\phi: \mathbb R^d \...
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What does bandwidth in kernel regression mean?

here https://stat.ethz.ch/R-manual/R-devel/library/stats/html/ksmooth.html is bandwidth explained as "the bandwidth. The kernels are scaled so that their quartiles (viewed as probability densities) ...
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Integration of Kernel and density product

Im considering kernels of the form $$K_s(u) = A(s)k_s(u)I[\lvert u \lvert \leq 1]$$ and $$k_s(u) = (1-u^2)^s$$ with $r$'th derivative $$K_s^r(u) = A(s)\frac{d^r k_s(u)}{du^r}I[\lvert u \lvert \leq ...
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Expectation of derivative of kernel density estimator

I am trying to calculate the expectation of the $s$'th derivative of a kernel density estimator. This problem arises naturally when trying to estimate the derivative of a density, because one approach ...
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Is there a Gaussian Process Kernel that limits functions to sigmoids?

I am modeling a large number of Dose-response curves. I have strong reason to believe that the generating function will be sigmoidal against the concentration of the assay (Michaelis-Menten kinetics). ...
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Estimate RBF-kernel mapping function given graph/space

Problem Provide a mapping function $𝜑(x)$ that enables us to draw a linear separator between the two classes in the mapped space. Attempt I tried to use a radial basis function by finding 4 ...
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Kernel Density Estimation of Hazard rate using R

How can we estimate the hazard rate using Kernel density estimation? For the time being I am simulating 1000 observations from exponential distribution, then estimate its hazard rate using gaussian ...
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Simulating from an Epanechnikov kernel density estimate in MATLAB / exact form of the Epanechnikov kernel in MATLAB?

It's my first time posting, so apologies if I'm breaking any etiquette. I've used MATLAB's ksdensity function to estimate a density using the Epanechnikov kernel and would now like to make repeated ...
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Is there a term to refer to a weighted mean that's weighted by a function of percentile (using a kernel)?

I came-up with a type of central tendency which is a weighted mean. The weighting is based on percentile, with values closer to the median having a higher weight. It's similar to the idea of a ...
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What is the most intuitive proof that Gaussian kernel is positive definite?

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 ...
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1answer
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Is the absolute value of the difference a kernel?

In particular is $$ k(x_i,x_j)=|x_i-x_j|, \quad x_i,x_j\in \mathbb{R}$$ a valid kernel?
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Why is the convolution of two box kernels a triangle kernel? [duplicate]

Can anyone show the mathematical steps proving $K_{box}*K_{box} = K_{triangle}$
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Gaussian Processes, Prior function of constant, Linear and Polynomial Kernel

In my university course slide, regarding the exponential and Gaussian kernel there are these two relevant pictures. They show the prior function of each kernel: You can see that the resulting prior ...
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Weighted kernel density

I would like to produce a 3-d plot based on density of 2-d data. This can be achieved for example in R using the kde2d function from the ...
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Prove this Kernel 1 - 2* angle(x,y)/pi is positive semi definite [closed]

How should we prove this kernel is positive semi definite? K(x,y) = 1- 2*ang(x,y)/pi ang(x,y) is the angle between vector x and y
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Does Kernel Function Only Apply To Support Vector In SVM?

We know that the if α=0 in below equation it is not a support vector, only if α!=0, it is the support vector. L(w, b, α) = Xm i=1 αi − 1 2 Xm i,j=1 y(i)y(j)αiαj(x(i))T x(j). However, for the Kernel ...
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Normal Kernel Estimation in R [closed]

I'm given the following data: ...
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1answer
146 views

Uniqueness of Reproducing Kernel Hilbert Spaces

Digging in the definition of Reproducing Kernel Hilbert Spaces (RKHS) I came across the following example taken from pages 49-51 of [1]: Given the kernel $k(x,y) = \langle x,y\rangle^2$, with $x,y\in ...
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Is a kernel density estimate meaningful if > 25% of my data are duplicates?

The title pretty much says it all. I have data that consists of 80 samples but there are always at least four samples that have exactly the same value. I want to assess, whether the data is unimodal. ...