Ridge regression on circulant matrices

ridge_circulant.py

import numpy as np

# Initialize the generating vector
x = np.array([1, 2, 3, 4, 5])
y = np.array([5, 4, 3, 2, 1])
# Initialize the permutation matrix
P = np.array([
    [0, 0, 0, 0, 1],
    [1, 0, 0, 0, 0],
    [0, 1, 0, 0, 0],
    [0, 0, 1, 0, 0],
    [0, 0, 0, 1, 0],
])

# Form the circulant matrix using the generating vector
X = np.vstack((np.dot(np.linalg.matrix_power(P, i), x) for i in range(len(x))))

# Set the regularization coefficient
alpha = 1

# Solve closed form solution
w_spatial = np.linalg.inv(X.T.dot(X) + np.eye(len(x)) * alpha).dot(X.T).dot(y)

# Solve in frequency domain
x_freq = np.fft.fft(x)
y_freq = np.fft.fft(y)
w_freq = (np.conj(x_freq) * y_freq) / (np.conj(x_freq) * x_freq + alpha)
w_fft = np.real(np.fft.ifft(w_freq))