Least Squares using the SVD

In [2]: 
import numpy as np
import numpy.linalg as la
import scipy.linalg as spla
In [19]: 
# tall and skinny w/nullspace
np.random.seed(12)
A = np.random.randn(6, 4)
b = np.random.randn(6)
A[3] = A[4] + A[5]
A[1] = A[5] + A[1]
A[2] = A[3] + A[1]
A[0] = A[3] + A[1]

Part I: Singular least squares using QR

Let's see how successfully we can solve the least squares problem when the matrix has a nullspace using QR:

In [4]: 
Q, R = la.qr(A)
In [5]: 
R.round(3)
Out[5]:
array([[-4.526,  3.492, -0.204, -3.647],
       [ 0.   ,  0.796,  0.034,  0.603],
       [ 0.   ,  0.   , -1.459,  0.674],
       [ 0.   , -0.   , -0.   ,  0.   ]])

We can choose x_qr[3] as we please:

In [6]: 
x_qr = np.zeros(A.shape[1])
In [20]: 
x_qr[3] = 0
In [21]: 
QTbnew = Q.T.dot(b)[:3,] - R[:3, 3] * x_qr[3]
x_qr[:3] = spla.solve_triangular(R[:3,:3], QTbnew, lower=False)

Let's take a look at the residual norm and the norm of x_qr:

In [22]: 
R.dot(x_qr)-Q.T.dot(b)[:4]
Out[22]:
array([ -4.44089210e-16,   0.00000000e+00,  -1.11022302e-16,
        -7.35042876e-01])
In [23]: 
la.norm(A.dot(x_qr)-b, 2)
Out[23]:
2.1267152888030982
In [24]: 
la.norm(x_qr, 2)
Out[24]:
0.82393512974131577

Choose a different x_qr[3] and compare residual and norm of x_qr.

Part II: Solving least squares using the SVD

Now compute the SVD of $A$:

In [25]: 
U, sigma, VT = la.svd(A)

Make a matrix Sigma of the correct size:

In [26]: 
Sigma = np.zeros(A.shape)
Sigma[:4,:4] = np.diag(sigma)

And check that we've actually factorized A:

In [27]: 
(U.dot(Sigma).dot(VT) - A).round(4)
Out[27]:
array([[ 0., -0.,  0.,  0.],
       [ 0., -0.,  0.,  0.],
       [ 0., -0.,  0.,  0.],
       [ 0., -0., -0.,  0.],
       [ 0., -0., -0.,  0.],
       [ 0.,  0.,  0.,  0.]])

Now define Sigma_pinv as the "pseudo-"inverse of Sigma, where "pseudo" means "don't divide by zero":

In [28]: 
Sigma_pinv = np.zeros(A.shape).T
Sigma_pinv[:3,:3] = np.diag(1/sigma[:3])
Sigma_pinv.round(3)
Out[28]:
array([[ 0.147,  0.   ,  0.   ,  0.   ,  0.   ,  0.   ],
       [ 0.   ,  0.624,  0.   ,  0.   ,  0.   ,  0.   ],
       [ 0.   ,  0.   ,  1.055,  0.   ,  0.   ,  0.   ],
       [ 0.   ,  0.   ,  0.   ,  0.   ,  0.   ,  0.   ]])

Now compute the SVD-based solution for the least-squares problem:

In [29]: 
x_svd = VT.T.dot(Sigma_pinv).dot(U.T).dot(b)
In [30]: 
la.norm(A.dot(x_svd)-b, 2)
Out[30]:
2.1267152888030982
In [31]: 
la.norm(x_svd)
Out[31]:
0.77354943014895838
  • What do you observe about $\|\text{x_svd}\|_2$ compared to $\|\text{x_qr}\|_2$?
  • Is $\|\text{x_svd}\|_2$ compared to $\|\text{x_qr}\|_2$?