Graph multiresolution: Kron pyramid

In this demonstration file, we show how to reduce a graph using the PyGSP. Then we apply the pyramid to simple signal. To start open a python shell (IPython is recommended here) and import the required packages. You would probably also import numpy as you will need it to create matrices and arrays.

>>> import numpy as np
>>> from pygsp import graphs, reduction

For this demo we will be using a sensor graph with 512 nodes.

>>> G = graphs.Sensor(512, distribute=True)
>>> G.compute_fourier_basis()

The function graph_multiresolution computes the graph pyramid for you:

>>> levels = 5
>>> Gs = reduction.graph_multiresolution(G, levels, sparsify=False)

Next, we will compute the fourier basis of our different graph layers:

>>> for gr in Gs:
...     gr.compute_fourier_basis()

Those that were already computed are returning with an error, meaning that nothing happened. Let’s now create two signals and a filter, resp f, f2 and g:

>>> f = np.ones((G.N))
>>> f[np.arange(G.N//2)] = -1
>>> f = f + 10*Gs[0].U[:, 7]
>>> f2 = np.ones((G.N, 2))
>>> f2[np.arange(G.N//2)] = -1
>>> g = [lambda x: 5./(5 + x)]

We will run the analysis of the two signals on the pyramid and obtain a coarse approximation for each layer, with decreasing number of nodes. Additionally, we will also get prediction errors at each node at every layer.

>>> ca, pe = reduction.pyramid_analysis(Gs, f, h_filters=g, method='exact')
>>> ca2, pe2 = reduction.pyramid_analysis(Gs, f2, h_filters=g, method='exact')

Given the pyramid, the coarsest approximation and the prediction errors, we will now reconstruct the original signal on the full graph.

>>> f_pred, _ = reduction.pyramid_synthesis(Gs, ca[levels], pe, method='exact')
>>> f_pred2, _ = reduction.pyramid_synthesis(Gs, ca2[levels], pe2, method='exact')

Here are the final errors for each signal after reconstruction.

>>> err = np.linalg.norm(f_pred-f)/np.linalg.norm(f)
>>> err2 = np.linalg.norm(f_pred2-f2)/np.linalg.norm(f2)
>>> assert (err < 1e-10) & (err2 < 1e-10)