Add: SVD

15.SVD/SVD.py

@@ -0,0 +1,66 @@
+import numpy as np
+import sys
+import os
+from pathlib import Path
+sys.path.append(str(Path(os.path.abspath(__file__)).parent.parent))
+from utils import *
+
+def svd(A):
+    """
+    given an m x n matrix,
+    return the result of SVD,
+    as a tuple of (U, Sigma, V)
+    """
+    m , n = A.shape
+
+    symmetry = A.T @ A
+    rank = np.linalg.matrix_rank(symmetry)
+    eigen_values, eigen_vectors = np.linalg.eig(symmetry)
+    eigen_order = eigen_values.argsort()[::-1]
+    eigen_values = eigen_values[eigen_order]
+
+    eigen_values = eigen_values[: rank]
+    eigen_vectors = eigen_vectors[:, eigen_order]
+    # V is of shape [n, n]
+    V = eigen_vectors
+    eigen_vectors = eigen_vectors[:, : rank]
+
+    singular_values = np.sqrt(eigen_values)
+    singular_matrix = np.zeros_like(A)
+    for i, v in enumerate(singular_values):
+        singular_matrix[i][i] = v
+
+    U1 = A @ eigen_vectors / singular_values
+    U2 = get_solution_domain(row_echelon(A.T))
+    U = np.concatenate([U1, U2], axis=-1)
+    return U, singular_matrix, V
+
+
+if __name__ == '__main__':
+    def demonstrate(A, desc):
+        print(desc)
+        U, singular_matrix, V = svd(A)
+        print("U is:")
+        print(U)
+        print("Singular matrix is:")
+        print(singular_matrix)
+        print("V is:")
+        print(V)
+        print("The reconstructed matrix is:")
+        print(U @ singular_matrix @ V.T)
+
+    A = np.array([[1, 1],
+                  [2, 2],
+                  [0, 0]]).astype(float)
+    demonstrate(A, 'Example 1')
+
+    A = np.array([[1, 0, 0, 0],
+                  [0, 0, 0, 4],
+                  [0, 3, 0, 0],
+                  [0, 0, 0, 0],
+                  [2, 0, 0, 0]]).astype(float)
+    demonstrate(A, 'Example 2')
+
+    A = np.array([[3, 1],
+                  [2, 1]]).astype(float)
+    demonstrate(A, 'Example 3')