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2.15 Matrix Approximation

SVD is a way to factorize $A=U\Sigma V^T \in \mathbb{R}^{m \times n}$ into the product of three matrices, where $U \in \mathbb{R}^{m \times m}$ and $V \in \mathbb{R}^{n \times n}$ are orthogonal and $\Sigma$ contains the singular values on its main diagonal.Instead of doing the full SVD factorization, we will now investigate how the SVD allows us to represent matrix A as a sum of simpler(low-rank) matrices $A_i$, which lends itself to a matrix approximation scheme that is cheaper to compute than the full SVD.

We consider a rank-1 matrix $A_i \in \mathbb{R}^{m \times n}$ as
                    $A_i=u_iv_i^T$
which is formed by the outer product of the $i^{th}$ orthogonal column vector of $U$ and $V$.

A matrix $A \in \mathbb{R}^{m \times n}$ of rank $r$ can be written as a sum of rank-1 matrices A, so that
$A=\sum_{i=1}^{r}\sigma_iu_iv_i^{T}=\sum_{i=1}^{r}\sigma_iA_i$

where the outer-product matrices $A_i$ are weighted by the $i^{th}$ singular value $\sigma_i$.We summed up the $r$ individual rank-1 matrices to obtain the rank-r matrix $A$.If the sum does not run over all matrices $A_i, i=1,\ldots,r,$but only up to an intermediate value $k<r$, we obtain a rank-k approximation.

$\hat{A}(k)=\sum_{i=1}^{k} \sigma_i u_i v_i^T= \sum_{i=1}^{k} \sigma_iA_i$ 

of $A$ with $rk(\hat{A}(k))=k$.

We can use SVD to reduce a rank-r matrix $A$ to a rank-k matrix $\hat{A}$ in a principled optimal manner.We can interpret the approximation of $A$ by a rank-k matrix as a form of lossy compression. Therefore, the low rank approximation of a matrix appears in many machine learning applications, e.g., image processing, noise filtering and regularization of ill-posed problems.Furthermore, it plays a key role in dimensionality reduction and principal component analysis(PCA).

Example:
import numpy as np
from numpy.linalg import eig
from scipy.linalg import svd
# define a matrix
A = np.array([[2,1],[1,2]])
print("Original matrix A 2X2")
print(A)
# factorize
U, S, V = svd(A)
print("Left Singular Vectors 2x2")
print(U)
print("Singular value matrix Sigma 2x3")
print(S)
print("Right singular vectors 3x3")
print(V)
# create m x n Sigma matrix with zeros
Sigma =np.zeros((A.shape[0], A.shape[1]))
# populate Sigma with n x n diagonal matrix
Sigma[:A.shape[0], :A.shape[0]] = np.diag(S)
B = U.dot(Sigma.dot(V))
print("Reconstructed Matrix")
print(round(B))

u0=U[0]
v0=V[0]
u1=U[1]
v1=V[1]
print("Rank-1 Approximation using one rank-1 matrix")
print(np.outer(S[0]*u0,v0))
print("Rank-2 Approximation using two rank-1 matrix")
print(np.outer(S[1]*u1,v1)+np.outer(S[0]*u0,v0))

O/P:
Original matrix A 2X2
[[2 1]
 [1 2]]
Left Singular Vectors 2x2
[[-0.70710678 -0.70710678]
 [-0.70710678  0.70710678]]
Singular value matrix Sigma 2x3
[3. 1.]
Right singular vectors 3x3
[[-0.70710678 -0.70710678]
 [-0.70710678  0.70710678]]
Reconstructed Matrix
[[2. 1.]
 [1. 2.]]
Rank-1 Approximation using one rank-1 matrix
[[1.5 1.5]
 [1.5 1.5]]
Rank-2 Approximation using two rank-1 matrix
[[2. 1.]
 [1. 2.]]

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