How to calculate correlation coefficients using sklearn CCA module?

Question

I need to measure similarity between feature vectors using CCA module. I saw sklearn has a good CCA module available: https://scikit-learn.org/stable/modules/generated/sklearn.cross_decomposition.CCA.html

In different papers I reviewed, I saw that the way to measure similarity using CCA is to calculate the mean of the correlation coefficients, for example as done in this following notebook example: https://github.com/google/svcca/blob/1f3fbf19bd31bd9b76e728ef75842aa1d9a4cd2b/tutorials/001_Introduction.ipynb

How to calculate the correlation coefficients (as shown in the notebook) using sklearn CCA module?

from sklearn.cross_decomposition import CCA
import numpy as np

U = np.random.random_sample(500).reshape(100,5)
V = np.random.random_sample(500).reshape(100,5)

cca = CCA(n_components=1)
cca.fit(U, V)

cca.coef_.shape                   # (5,5)

U_c, V_c = cca.transform(U, V)

U_c.shape                         # (100,1)
V_c.shape                         # (100,1)

This is an example of the sklearn CCA module, however I have no idea how to retrieve correlation coefficients from it.

Answer 1

In reference to the notebook you provided which is a supporting artefact to and implements ideas from the following two papers

1. "SVCCA: Singular Vector Canonical Correlation Analysis for Deep Learning Dynamics and Interpretability". Neural Information Processing Systems (NeurIPS) 2017
2. "Insights on Representational Similarity in Deep Neural Networks with Canonical Correlation". Neural Information Processing Systems (NeurIPS) 2018
   

The authors there calculate 50 = min(A_fake neurons, B_fake neurons) components and plot the correlations between the transformed vectors of each component (i.e. 50).

With the help of the below code, using sklearn CCA, I am trying to reproduce their Toy Example. As we'll see the correlation plots match. The sanity check they used in the notebook came very handy - it passed seamlessly with this code as well.

import numpy as np
from matplotlib import pyplot as plt
from sklearn.cross_decomposition import CCA

# rows contain the number of samples for CCA and the number of rvs goes in columns
X = np.random.randn(2000, 100)
Y = np.random.randn(2000, 50)

# num of components
n_comps = min(X.shape[1], Y.shape[1])
cca = CCA(n_components=n_comps)
cca.fit(X, Y)
X_c, Y_c = cca.transform(X, Y)

# calculate and plot the correlations of all components
corrs = [np.corrcoef(X_c[:, i], Y_c[:, i])[0, 1] for i in range(n_comps)]    
plt.plot(corrs)
plt.xlabel('cca_idx')
plt.ylabel('cca_corr')
plt.show()

Output:

For the sanity check, replace the Y data matrix by a scaled invertible transform of X and rerun the code.

Y = np.dot(X, np.random.randn(100, 100)) 

Output: