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Questions tagged [rbf]

Radial Basis Functions (RBFs) are commonly used for interpolating scattered data, in numerical meshfree simulation methods, and in artificial neural networks

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Solution of ill conditioned RBF kernel to be used in Quadratic programming

This problem is similar to the one described here: Large-and-ill-conditioned-quadratic-convex-problem RBF kernel: Equation Reason for RBF being ill-conditioned: The kernel matrix K is singular when ...
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How interpolate a huge cloud of scattered 3D points?

I have a cloud consist of a million scattered 3d points. I want to get a uniform cloud of 3d points. I think to interpolate in blocks. However, as shown in the figure grid, there is a problem of ...
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1answer
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RBF kernel mapping

I was reading that the Gaussian/RBF kernel maps its input onto the surface of normalized hypersphere. Our RBF kernel given by: $k(x,z) = exp(\frac{- ||x-z||^2}{2\sigma^2})$ Can anyone explain why ...
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General Dimension of The RBF Interpolation

I am reading about the RBF interpolation in order to apply it to interpolate a bunch of data over a three dimensional space. This may sound as a simple question but it is annoying me not knowing it. ...
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1answer
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How to find the particular solution for Augmented Thin Plate Splines in the context of the Dual Reciprocity Boundary Element Method

In the dual reciprocity boundary element method (DRBEM) the non-homogeneous terms are expanded in terms of radial basis functions. This expansion involves approximating the solution to the linear ...
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1answer
96 views

Matrix operations for RBF solver

GOAL: I got from someone the python code for an RBF solver. The solver stores 9 transformation matrix (each of which, once decomposed, have tx, ty and tz set at 0 and sx, sy and sz set to 1, so only ...
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2answers
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How does one approximate a second derivative with ATPS interpolation

When using the Dual Reciprocity Boundary Element Method ( or any radial basis function method ) to solve a nonlinear differential equation it is necessary to approximate some derivatives of a ...
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28 views

Radial Basis Function

I have this question and I need help; What is the effect of using a more hidden layer on the performance of approximation of the RBF network?
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How can I use RBF interpolation on a highly stretched rectangular domain?

I performed a 2D parametric analysis where one variable is much larger than the other. Basically I sampled a function in many points: let's say 5 points for $x_1$ and 5 points for $x_2$, where the ...
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Confusion between Wendland RBF functions - missing Wendland functions

I need to compute Wendland functions for a project, and got confused between the formula to construct Wendland functions $\phi(d,k)$ where d is the dimension So in the original paper introducing ...
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Radial Basis Fn vs. Wavelets (not Neural Networks)

I am interested in parameterizing a surface without a mesh. One technique used in the field of optics is to use Radial Basis Functions (e.g. Gaussians). From a naive point of view, the decomposition ...
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1answer
229 views

How to understand min max constraints?

I am trying to understand how to minimize the given function. In the paper [1] t says, ${{f}_{i}}\left( x \right)$ should be minimized in subject to: $\left\| x-{{x}_{j}} \right\|\ge \beta \text{ }{{...
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How are basis functions discovered using e.g. prototype-based clustering (e.g. K-means)?

How are basis functions (for e.g. RBF) discovered using e.g. prototype-based clustering (e.g. K-means)? Because my notes say that the basis function are discovered using prototype-based methods. I ...
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Understanding the use of Radial Basis Function in Linear Regression

I am attempting to understand the use of Radial Basis Functions (RBFs) as used in linear regression. Building the problem: RBFs can be used as a means of separating data which is not linearly ...
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63 views

Integral calculation of Fourier transform of a function

In this paper, an oscillatory radial basis function is introduced and its properties are considered. The RBF is as follows: $$\phi_d=\frac{J_{d/2-1}(\epsilon r)}{(\epsilon r)^{d/2-1}},\quad d=2,3,4,......
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Absolute value of an RBF distance is less than the absolute value of an actual distance

I have a radial basis function with a linear kernel $f(r)=r$ in $3D.$ I constructed the surface based on this RBF and noticed that the absolute value of actual distance from any point to the ...
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218 views

Gradient descent in $n$-dimensional space in the context of an RBF network

I am trying to implement an algorithm to perform gradient descent in a $n$-dimensional space in the context of an RBF network. My network has 5 inputs and 1 output. It has the following Gaussian ...
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1answer
50 views

Positive weights in Radial Basis Functions

Let $\phi$ be a positive definite radial kernel in $\mathbb{R}^d$. I have a point cloud with positions $(\mathbf{x}_i)_{i\in I}$. RBF interpolation of real valued data $f = (f_i)_{i\in I}$ on this ...
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1answer
295 views

Radial Basis Function RBF Gaussian based Interpolation

Based on short description below (an image), how do I find the highlighted f function value? I understand that it is a value associated with the vertex, sorry I am not a good math student to ...
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159 views

Is normalized RBF always better than RBF

The question is as the title. Mathematically, I want to know does the following inequation always hold for any vector $\mathbf b$? $\mathbf b^T \mathbf B \mathbf B^+ \mathbf b \, \ge \, \mathbf b^T \...
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1answer
132 views

Interpolation with RBF

I have a function that is continuous and differentiable over $\mathbb{R}$ and its support is the whole real line. I want to approximate it through a linear combination of Gaussian functions. I know ...
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304 views

Radial Basis Functions Interpolation

$ \let\oldcdot\cdot \renewcommand{\cdot}{\!\oldcdot\!} \newcommand{\e}{\varepsilon} \renewcommand{\p}{\varphi} \renewcommand{\p}{\varphi} \renewcommand{\vp}{\vec{\boldsymbol\p}(x)} \newcommand{\P}{\...
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3answers
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Symmetry Of Differentiation Matrix

I have a problem computing numerically the eigenvalues of Laplace-Beltrami operator. I use meshfree Radial Basis Functions (RBF) approach to construct differentiation matrix $D$. Testing my code on ...
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1answer
271 views

Showing that a thin-plate spline RBF approximation is real analytic

I am finishing my Ph.D. dissertation in engineering and I would like to show a simple proof. I am having troubles formalizing my ideas into a proof though. I think in a mathematics paper this concept ...
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The positive-definite-ness of RBF kernel

In Micchelli's paper Interpolation of Scattered Data: Distance Matrices and Conditionally Positive Definite Functions it mentioned that the RBF kernel $e^{-\alpha^2\|x^i-x^j\|^2/2}$ is positive ...
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1answer
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Radial Basis Function and Neural Networks

I need a simple explanation about what is the radial basis function? And what is the relationship between the radial basis function and neural networks? And are there any simple examples to explain ...