Global Feature Reconstruction Error (GFRE) and Distortion (GFRD)

Example for the usage of the skmatter.metrics.global_reconstruction_error as global feature reconstruction error (GFRE) and skmatter.metrics.global_reconstruction_distortion global feature reconstruction distortion (GFRD). We apply the global reconstruction measures on the degenerate CH4 manifold dataset. This dataset was specifically constructed to be representable by a 4-body features (bispectrum) but not by a 3-body features (power spectrum). In other words the dataset contains environments which are different, but have the same 3-body features. For more details about the dataset please refer to Pozdnyakov 2020 .

The skmatter dataset already contains the 3 and 4-body features computed with librascal so we can load it and compare it with the GFRE/GFRD.

import numpy as np
import matplotlib.pyplot as plt
import matplotlib as mpl

mpl.rc('font', size=20)

from skmatter.datasets import load_degenerate_CH4_manifold
from skmatter.metrics import (
    global_reconstruction_error,
    global_reconstruction_distortion,
)

# load features
degenerate_manifold = load_degenerate_CH4_manifold()
power_spectrum_features = degenerate_manifold.data.SOAP_power_spectrum
bispectrum_features = degenerate_manifold.data.SOAP_bispectrum

gfre_matrix = np.zeros((2, 2))
print("Computing GFRE...")

# reconstruction error of power spectrum features using power spectrum features
gfre_matrix[0, 0] = global_reconstruction_error(power_spectrum_features, power_spectrum_features)

# reconstruction error of bispectrum features using power spectrum features
gfre_matrix[0, 1] = global_reconstruction_error(power_spectrum_features, bispectrum_features)

# reconstruction error of power spectrum features using bispectrum features
gfre_matrix[1, 0] = global_reconstruction_error(bispectrum_features, power_spectrum_features)

# reconstruction error of bispectrum features using bispectrum features
gfre_matrix[1, 1] = global_reconstruction_error(bispectrum_features, bispectrum_features)

print("Computing GFRE finished.")

Computing GFRE...
Computing GFRE finished.

gfrd_matrix = np.zeros((2, 2))
print("Computing GFRD...")

# reconstruction distortion of power spectrum features using power spectrum features
gfrd_matrix[0, 0] = global_reconstruction_distortion(power_spectrum_features, power_spectrum_features)

# reconstruction distortion of power spectrum features using bispectrum features
gfrd_matrix[0, 1] = global_reconstruction_distortion(power_spectrum_features, bispectrum_features)

# reconstruction distortion of bispectrum features using power spectrum features
gfrd_matrix[1, 0] = global_reconstruction_distortion(bispectrum_features, power_spectrum_features)

# reconstruction distortion of bipsectrum features using bispectrum features
gfrd_matrix[1, 1] = global_reconstruction_distortion(bispectrum_features, bispectrum_features)

print("Computing GFRD finished.")

Computing GFRD...
Computing GFRD finished.

fig, (axGFRE, axGFRD, cbar_ax) = plt.subplots(
    1,
    3,
    figsize=(10, 4),
    gridspec_kw=dict(width_ratios=(1, 1, 0.2)),
)


pcm1 = axGFRE.imshow(gfre_matrix, vmin=0, vmax=0.25)
axGFRE.set_ylabel("F")
axGFRE.set_xlabel("F'")
axGFRE.set_title("GFRE(F, F')")

axGFRE.set_xticks([0, 1])
axGFRE.set_xticklabels(["3-body", "4-body"])
axGFRE.set_yticks([0, 1])
axGFRE.set_yticklabels(["3-body", "4-body"])

pcm2 = axGFRD.imshow(gfrd_matrix, vmin=0, vmax=0.25)
axGFRD.set_xlabel("F'")
axGFRD.set_title("GFRD(F, F')")

axGFRD.set_xticks([0, 1])
axGFRD.set_xticklabels(["3-body", "4-body"])
axGFRD.set_yticks([0, 1])
axGFRD.set_yticklabels(["", ""])

cbar = fig.colorbar(pcm2, cax=cbar_ax, label='GFRE or GFRD')
plt.show()

It can be seen that the reconstruction error of 4-body features with 3-body features in the left plot in the upper right corner is large, expressing that the dataset contains 4-body information that cannot be well linearly reconstructed using 3-body information. This is expected, since the dataset was specifically designed for this purpose (for more information please read Pozdnyakov 2020). On the other hand the 3-body features can be perfectly reconstructed from the 4-body features as seen in the left plot in the lower left corner. However, this reconstruction distorts the 4-body features significantly as seen in the right plot in the lower left corner which is a typical behaviour of higher order features and can be also observed for polynomial kernel features.