机器学习基础：奇异值分解（SVD）

from numpy import array
from numpy import diag
from numpy import zeros
from scipy.linalg import svd
# define a matrix
A = array([
 [1,2,3,4,5,6,7,8,9,10],
 [11,12,13,14,15,16,17,18,19,20],
 [21,22,23,24,25,26,27,28,29,30]])
print(A)

>>> A
array([[ 1,  2,  3,  4,  5,  6,  7,  8,  9, 10],
       [11, 12, 13, 14, 15, 16, 17, 18, 19, 20],
       [21, 22, 23, 24, 25, 26, 27, 28, 29, 30]])

# Singular-value decomposition
U, s, VT = svd(A)
# create m x n Sigma matrix
Sigma = zeros((A.shape[0], A.shape[1]))
# populate Sigma with n x n diagonal matrix
Sigma[:A.shape[0], :A.shape[0]] = diag(s)
# select
n_elements = 2
Sigma = Sigma[:, :n_elements]
VT = VT[:n_elements, :]
# reconstruct
B = U.dot(Sigma.dot(VT))
print(B)

>>> B
array([[ 1.,  2.,  3.,  4.,  5.,  6.,  7.,  8.,  9., 10.],
       [11., 12., 13., 14., 15., 16., 17., 18., 19., 20.],
       [21., 22., 23., 24., 25., 26., 27., 28., 29., 30.]])

# transform
T = U.dot(Sigma)
print(T)

>>> T
array([[-18.52157747,   6.47697214],
       [-49.81310011,   1.91182038],
       [-81.10462276,  -2.65333138]])

T = A.dot(VT.T)
print(T)

[[-18.52157747   6.47697214]
 [-49.81310011   1.91182038]
 [-81.10462276  -2.65333138]]