Kernel Functions
Written on July 20th, 2021 by control lab
Kernel Functions
Firstly, we realized the KernelFunctions to compute TK matrices and a monomial basis, given \(x\) and degree \(q\) one can call
import numpy as np
from PMKLpy import KernelFunctions
x = np.random.rand(100, 1)
q = 1
Lower, Upper = x.min(axis = 0) - 0.1 , x.max(axis = 0) + 0.1
Kernel = KernelFunctions.Kernel(x, Lower, Upper, q)
Kernel is a special class for fast computing of TK. The first argument of is data points, the second and third are low and upper bounds for samples and the last one is degree of monomial basis. This object has 2 attributes. The first one is Kernel.Z, that is a vector of monomial basis. It stores numpy.array object with the shape (n_samples, q) where q The second attribute is Kernel.K is a interior structure for fast computing of TK.
If you need only monomial basis, then you can use the monomials(x,d) function takes a matrix of inputs x and computes the monomial basis Z of degree d
from PMKLpy import Transformation
Z = Transformation.monomials(x, 1)
To compute TK, the user need to call makeK function
import numpy as np
from PMKLpy import KernelFunctions
x = np.random.rand(100, 1)
q = 1
Lower, Upper = x.min(axis = 0) - 0.1 , x.max(axis = 0) + 0.1
Kernel = KernelFunctions.Kernel(x, Lower, Upper, q)
TK = KernelFunctions.makeK(Kernel, P)
makeK is a function that returns a Kernel matrix (n_samples, n_samples). It requires a P matrix of the form (2q, 2q) and an object Kernel.
For general Kernel matrices K(X1, X2) we realized a function TKtest. This function takes two matrices of inputs, as well as monomial basis of the inputs Z1, Z2 and a lower a and upper b bound over which we integrate and a matrix P that parameterizes the TK kernel function. Computes the test kernel matrix for a TK kernel.
from PMKLpy import Transformation
from PMKLpy import KernelFunctions
x1 = x[:100]
x2 = x[100:]
Lower, Upper = x.min(axis = 0) - 0.1 , x.max(axis = 0) + 0.1
P = np.eye(6)
Z1 = Transformation.monomials(x1, 1)
Z2 = Transformation.monomials(x2, 1)
TK = KernelFunctions.TKtest(x1,x2,Z1,Z2,Lower, Upper,P)
