Compute the Heat Kernel

This function computes the heat kernel, which is a radial basis function that can be used for smoothing, interpolation, and approximation tasks. The heat kernel is defined as exp(-x^2/(2*sigma^2)), where x is the distance and sigma is the bandwidth. It acts as a similarity measure for points in a space, assigning high values for close points and low values for distant points.

Usage

heat_kernel(x, sigma = 1)

Arguments

x

A numeric vector or matrix representing the distances between data points.

sigma

The bandwidth of the heat kernel, a positive scalar value. Default is 1.

Value

A numeric vector or matrix with the same dimensions as the input `x`, containing the computed heat kernel values.

Details

The heat kernel is widely used in various applications, including machine learning, computer graphics, and image processing. It can be employed in kernel methods, such as kernel PCA, Gaussian process regression, and support vector machines, to capture the local structure of the data. The heat kernel's behavior is controlled by the bandwidth parameter sigma, which determines the smoothness of the resulting function.

Examples

x <- seq(-3, 3, length.out = 100)
y <- heat_kernel(x, sigma = 1)
plot(x, y, type = "l", main = "Heat Kernel")