I'm taking a Computational Neuroscience class on Coursera. So far it's been going great! However, I'm getting a little stuck on one of the quiz problems.
I am not taking this class for a certificate or anything. Solely for fun. I already took the quiz and after awhile, I guessed the answer, so this is not even going to be answering the quiz.
The question is framed as the following: Suppose that we had a linear recurrent network of 5 input nodes and 5 output nodes. Let us say that our network's weight matrix W is:
W = [0.6 0.1 0.1 0.1 0.1]
[0.1 0.6 0.1 0.1 0.1]
[0.1 0.1 0.6 0.1 0.1]
[0.1 0.1 0.1 0.6 0.1]
[0.1 0.1 0.1 0.1 0.6]
(Essentially, all 0.1, besides 0.6 on the diagonals.)
Suppose that we have a static input vector u:
u = [0.6]
[0.5]
[0.6]
[0.2]
[0.1]
Finally, suppose that we have a recurrent weight matrix M:
M = [-0.25, 0, 0.25, 0.25, 0]
[0, -0.25, 0, 0.25, 0.25]
[0.25, 0, -0.25, 0, 0.25]
[0.25, 0.25, 0, -0.25, 0]
[0, 0.25, 0.25, 0, -0.25]
Which of the following is the steady state output v_ss of the network? (Hint: See the lecture on recurrent networks, and consider writing some Octave or Matlab code to handle the eigenvectors/values (you may use the "eig" function))'
The notes for the class can be found here. Specifically, the equation for the steady state formula can be found on slides 5 and 6.
I have the following code.
import numpy as np
# Construct W, the network weight matrix
W = np.ones((5,5))
W = W / 10.
np.fill_diagonal(W, 0.6)
# Construct u, the static input vector
u = np.zeros(5)
u[0] = 0.6
u[1] = 0.5
u[2] = 0.6
u[3] = 0.2
u[4] = 0.1
# Connstruct M, the recurrent weight matrix
M = np.zeros((5,5))
np.fill_diagonal(M, -0.25)
for i in range(3):
M[2+i][i] = 0.25
M[i][2+i] = 0.25
for i in range(2):
M[3+i][i] = 0.25
M[i][3+i] = 0.25
# We need to matrix multiply W and u together to get h
# NOTE: cannot use W * u, that's going to do a scalar multiply
# it's element wise otherwise
h = W.dot(u)
print 'This is h'
print h
# Ok then the big deal is:
# h dot e_i
# v_ss = sum_(over all eigens) ------------ e_i
# 1 - lambda_i
eigs = np.linalg.eig(M)
eigenvalues = eigs[0]
eigenvectors = eigs[1]
v_ss = np.zeros(5)
for i in range(5):
v_ss += (np.dot(h,eigenvectors[:, i]))/((1.0-eigenvalues[i])) * eigenvectors[:,i]
print 'This is our steady state v_ss'
print v_ss
The correct answer is:
[0.616, 0.540, 0.609, 0.471, 0.430]
This is what I am getting:
This is our steady state v_ss
[ 0.64362264 0.5606784 0.56007018 0.50057043 0.40172501]
Can anyone spot my bug? Thank you so much! I greatly appreciate it and apologize for the long blog post. Essentially, all you need to look at, is slide 5 and 6 on that top link.
