kernels
A kernel formed by addition of two kernels k(.,.) = k1(.,.) + k2(.,.)
Cauchy Kernel given by the expression K(x,y) = 1/(1 + ||x-y||2/σ2)
A (symmeteric positive definite) covariance function.
A (symmeteric positive definite) covariance function. Covariance functions are central to Gaussian/Student T Process Models as well as SVMs.
The index set over which K(.,.) is defined K: T × T -> V
The value outputted by the kernel
The type of the kernel matrix object.
Automatic relevance determination (ARD) version of the cubic spline kernel.
Automatic relevance determination (ARD) version of the cubic spline kernel.
Input domain
Implementation of the cubic spline kernel/covariance function, for arbitrary domains I over which a field and norm are defined as implicits.
Implementation of the cubic spline kernel/covariance function, for arbitrary domains I over which a field and norm are defined as implicits.
The index set/domain over which the kernel is defined
A kernel/covariance function which can be seen as a combination of base kernels over a subset of the input space.
A kernel/covariance function which can be seen as a combination of base kernels over a subset of the input space.
for example K((x1, y1), (x1, y2)) = k1(x1,x2) + k2(y1,y2)
Abstract class which can be extended to implement various Multivariate Density Kernels.
Dirac kernel is equivalent to the classical Dirac delta function scaled by a hyper-parameter called the noise level.
Dirac kernel is equivalent to the classical Dirac delta function scaled by a hyper-parameter called the noise level.
K(x,y) = noise*DiracDelta(x,y)
Created by mandar on 28/09/2016.
Implementation of the Normalized Exponential Kernel
Implementation of the Normalized Exponential Kernel
K(x,y) = exp(β*(x.y))
Created by mandar on 20/11/15.
Fractional Brownian Kernel:
Fractional Brownian Kernel:
Fractional Brownian Motion is a stochastic process first studied by Mandelbrot and Von Ness its covariance function generalized to multivariate index sets is.
K(x,y) = 1/2 × (||x||2H + ||y||2H - ||x-y||2H)
Implements the gaussian spectral mixture kernel as outlined in Wilson et.
Implements the gaussian spectral mixture kernel as outlined in Wilson et. al.
The kernel is defined as the inverse fourier transform of a gaussian spectral density as is shown by Bochner's theorem.
K(d) = exp(-2π2 dTΣ-1d) × cos(2πμTd)
The domain over which the kernel is defined
An exponential family covariance function between space time coordinates
Covariance expressed as exponential of some scaled distance metric between arguments.
Covariance expressed as exponential of some scaled distance metric between arguments.
K(x, y) = σ2 exp(-1/2 × ||x-y||/θ)
Implementation of the half integer Matern-ARD covariance function
Implementation of the half integer Matern covariance function, for arbitrary domains.
Implementation of the half integer Matern covariance function, for arbitrary domains.
RBF Kernels of the form K(x,y) = exp(-||x - y||2/2 × l2)
RBF Kernels of the form K(x,y) = exp(-||x - y||2/2 × l2)
Created by mandar on 23/11/15.
Defines a base class for kernels defined on arbitrary objects.
Defines a base class for kernels defined on arbitrary objects.
The domain over which kernel function k(x, y) is defined. i.e. x,y belong to T
The type of value returned by the kernel function k(x,y)
Defines a trait which outlines the basic functionality of Kernel Matrices.
Implementation of the Normalized Exponential Kernel
Implementation of the Normalized Exponential Kernel
K(x,y) = exp(-||x-y||/b)
18/8/15.
Kernels for linear PDE operator equations.
Kernels with a locally stored matrix in the form of a breeze DenseMatrix instance.
Kernels with a locally stored matrix in the form of a breeze DenseMatrix instance. Optionally a kernel matrix stored as a PartitionedPSDMatrix can also be generated.
Scalar Kernel defines algebraic behavior for kernels of the form K: Index x Index -> Double, i.e.
Scalar Kernel defines algebraic behavior for kernels of the form K: Index x Index -> Double, i.e. kernel functions whose output is a scalar/double value. Generic behavior for these kernels is given by the ability to add and multiply valid kernels to create new valid scalar kernel functions.
Implementation of locally stationary kernels as defined in http://jmlr.csail.mit.edu/papers/volume2/genton01a/genton01a.pdf
Implementation of locally stationary kernels as defined in http://jmlr.csail.mit.edu/papers/volume2/genton01a/genton01a.pdf
K(x,y) = K1(x+y/2)×K2(x-y)
The index set or input domain over which the kernel function is evaluated.
Mahalanobis kernel is an anisotropic generalization of the RBF Kernel, its definition is based on the so called Mahalanobis distance between two vectors x and y.
Mahalanobis kernel is an anisotropic generalization of the RBF Kernel, its definition is based on the so called Mahalanobis distance between two vectors x and y.
K(x,y) = h*exp(-(x - y)T . M . (x - y))
In this implementation the symmetric positive semi-definite matrix M is assumed to be diagonal.
Created by mandar on 22/08/16.
A kernel formed by multiplication of two kernels k(.,.) = k1(.,.) * k2(.,.)
Standard Polynomial SVM Kernel of the form K(x,y) = (xT .
Standard Polynomial SVM Kernel of the form K(x,y) = (xT . y + 1.0)r
Rational Quadratic Kernel given by the expression K(x,y) = (1 + ||x-y||2/2μl2)-0.5 × μ
Rational Quadratic Kernel given by the expression K(x,y) = (1 + ||x-y||2/2μl2)-0.5 × μ
Squared Exponential Kernel is a generalized RBF Kernel K(x,y) = h2*exp(-||x - y||2/2 × l2)
Squared Exponential Kernel is a generalized RBF Kernel K(x,y) = h2*exp(-||x - y||2/2 × l2)
Defines an abstract class outlines the basic functionality requirements of an SVM Kernel
Represents covariance function of a GP which is scaled version of a base GP.
Represents covariance function of a GP which is scaled version of a base GP.
z ~ GP(m(.), K(.,.))
y = g(x)*z
y ~ GP(g(x)*m(x), g(x)K(x,x')g(x'))
Implementation of separable stationary kernels as defined in http://jmlr.csail.mit.edu/papers/volume2/genton01a/genton01a.pdf
Implementation of separable stationary kernels as defined in http://jmlr.csail.mit.edu/papers/volume2/genton01a/genton01a.pdf
K(x,y) = K1(x)×K2(y)
The index set or input domain over which the kernel function is evaluated.
A kernel type which extends StationaryKernel as well as LocalScalarKernel
C(x,y) = Cs(||x-y||)
C(x,y) = Cs(||x-y||)
An abstract representation of stationary kernel functions, this requires an implicit variable which represents how addition, subtraction etc are carried out for the input domain T
T-Student Kernel K(x,y) = 1/(1 + ||x - y||d)
T-Student Kernel K(x,y) = 1/(1 + ||x - y||d)
Represents a kernel on a product space R × S
Defines a global singleton object SVMKernel having functions which can construct kernel matrices.
Cauchy Kernel given by the expression K(x,y) = 1/(1 + ||x-y||2/σ2)