ring example: check that DFT is an eigenbasis (by Nathanaël)

examples/eigenvalue_concentration.py

@@ -6,13 +6,15 @@
 graph becomes full.
 """
 
+import numpy as np
 from matplotlib import pyplot as plt
 import pygsp as pg
 
 n_neighbors = [1, 2, 5, 8]
 fig, axes = plt.subplots(4, len(n_neighbors), figsize=(15, 10))
 
 for k, ax in zip(n_neighbors, axes.T):
+
     graph = pg.graphs.Ring(17, k=k)
     graph.compute_fourier_basis()
     graph.plot(graph.U[:, 1], ax=ax[0])
@@ -23,4 +25,14 @@
     graph.set_coordinates('line1D')
     graph.plot(graph.U[:, :4], ax=ax[3], title='')
 
+    # Check that the DFT matrix is an eigenbasis of the Laplacian.
+    U = np.fft.fft(np.identity(graph.n_vertices))
+    LambdaM = (graph.L.todense().dot(U)) / (U + 1e-15)
+    # Eigenvalues should be real.
+    assert np.all(np.abs(np.imag(LambdaM)) < 1e-10)
+    LambdaM = np.real(LambdaM)
+    # Check that the eigenvectors are really eigenvectors of the laplacian.
+    Lambda = np.mean(LambdaM, axis=0)
+    assert np.all(np.abs(LambdaM - Lambda) < 1e-10)
+
 fig.tight_layout()