[WIP] add example that shows how eigenvectors capture structure

examples/eigenvector_localization.py

@@ -2,10 +2,11 @@
 Localization of Fourier modes
 =============================
 
-The Fourier modes (the eigenvectors of the graph Laplacian) can be localized in
-the spacial domain. As a consequence, graph signals can be localized in both
-space and frequency (which is impossible for Euclidean domains or manifolds, by
-the Heisenberg's uncertainty principle).
+The Fourier modes :attr:`pygsp.graphs.Graph.U` (the eigenvectors of the graph
+Laplacian :attr:`pygsp.graphs.Graph.L`) can be localized in the spacial domain.
+As a consequence, graph signals can be localized in both space and frequency
+(which is impossible for Euclidean domains or manifolds, by the Heisenberg's
+uncertainty principle).
 
 This example demonstrates that the more isolated a node is, the more a Fourier
 mode will be localized on it.

examples/fourier_capture_structure.py

@@ -0,0 +1,51 @@
+r"""
+Graph structure captured by the Fourier modes
+=============================================
+
+As an eigendecomposition of the graph Laplacian :attr:`pygsp.graphs.Graph.L`,
+the Fourier modes :attr:`pygsp.graphs.Graph.U` capture the structure of the
+graph. Similarly to
+`PCA <https://en.wikipedia.org/wiki/Principal_component_analysis>`_
+(which is an eigendecomposition of a covariance matrix), the larger-scale
+structure is captured by the first modes (ordered by eigenvalue). The later
+ones capture more and more details. That is why a signal that is well explained
+by a graph has a low-frequency content.
+
+`Laplacian eigenmaps <https://en.wikipedia.org/wiki/Nonlinear_dimensionality_reduction#Laplacian_eigenmaps>`_
+use this for dimensionality reduction.
+"""
+
+import numpy as np
+from scipy import sparse
+import pygsp as pg
+from matplotlib import pyplot as plt
+
+N_VERTICES = 100
+
+def error(L, U, e):
+    normalization = sparse.linalg.norm(L, ord='fro')
+    return np.linalg.norm(L - U @ np.diag(e) @ U.T, ord='fro') / normalization
+
+def errors(graph):
+    graph.compute_fourier_basis()
+    return [error(graph.L, graph.U[:, :k], graph.e[:k]) for k in range(N_VERTICES+1)]
+
+graphs = [
+    pg.graphs.FullConnected(N_VERTICES),
+    pg.graphs.StochasticBlockModel(N_VERTICES),
+    pg.graphs.Sensor(N_VERTICES),
+    pg.graphs.ErdosRenyi(N_VERTICES),
+    pg.graphs.Grid2d(int(N_VERTICES**0.5)),
+    pg.graphs.Community(N_VERTICES),
+    pg.graphs.Comet(N_VERTICES),
+    pg.graphs.SwissRoll(N_VERTICES),
+    pg.graphs.BarabasiAlbert(N_VERTICES),
+]
+
+fig, ax = plt.subplots(1, 1, figsize=(5, 5))
+for graph in graphs:
+    ax.plot(errors(graph), '-', label=graph.__class__.__name__)
+ax.set_xlabel('number of Fourier modes $k$')
+ax.set_ylabel('reconstruction error');
+ax.set_title(r'Laplacian reconstruction error $\frac{\| L - U_k \Lambda_k U_k^\top \| }{\|L\|}$', fontsize=16)
+ax.legend(loc='lower left')