Study of connectivity distribution

This notebook is to study the connectivity distrubtion



In [1]:

    
from __future__ import print_function
import sys
sys.path.append('../')

import numpy as np
import matplotlib.pyplot as plt
import matplotlib.gridspec as gridspec
import seaborn as sns

from hopfield import Hopfield

%matplotlib inline
sns.set(font_scale=2.0)



In [2]:

    
prng = np.random.RandomState(seed=100)
normalize = True
T = 1.0
n_store = 7

N_samples = 1000

Dependency in how many states are added

Here we see whether the distribution of synaptic influences depends on the state of the vector.

First we start with only two



In [3]:

    
n_dim = 400

nn = Hopfield(n_dim=n_dim, T=T, prng=prng)
list_of_patterns = nn.generate_random_patterns(n_dim)
nn.train(list_of_patterns, normalize=normalize)

We generate two random patterns and test whether the field (h) is dependent on the initial state. We see very similar normal behavior for both of them



In [4]:

    
x = np.dot(nn.w, np.sign(prng.normal(size=n_dim)))
y = np.dot(nn.w, np.sign(prng.normal(size=n_dim)))



In [5]:

    
fig = plt.figure(figsize=(16, 12))
ax = fig.add_subplot(111)
ax.hist(x)
ax.hist(y)
print(np.std(x))









    



0.9904938033

We now then try this with 10 patterns to see how the distributions behave



In [6]:

    
fig = plt.figure(figsize=(16, 12))
ax = fig.add_subplot(111)

for i in range(10):
    x = np.dot(nn.w, np.sign(prng.normal(size=n_dim)))
    ax.hist(x, alpha=0.5)

We see that the normal distribution is mainted, then we calculate the field h (result of the np.dot(w, s) calculation) for a bunch of different initial random states and concatenate the results to see how the whole distribution looks like



In [7]:

    
n_dim = 400

nn = Hopfield(n_dim=n_dim, T=T, prng=prng)
list_of_patterns = nn.generate_random_patterns(n_dim)
nn.train(list_of_patterns, normalize=normalize)


x = np.empty(n_dim)

for i in range(N_samples):
    h = np.dot(nn.w, np.sign(prng.normal(size=n_dim)))
    x = np.concatenate((x, h))


fig = plt.figure(figsize=(16, 12))
ax = fig.add_subplot(111)
n, bins, patches = ax.hist(x, bins=30)



In [8]:

    
print(np.var(x))
print(nn.sigma)









    



1.00648912771
0.353553390593

Dependence on network size

Now we test test how the histogram looks for different sizes



In [9]:

    
n_dimensions = [200, 800, 2000, 5000]
fig = plt.figure(figsize=(16, 12))
gs = gridspec.GridSpec(2, 2)

for index, n_dim in enumerate(n_dimensions):

    nn = Hopfield(n_dim=n_dim, T=T, prng=prng)
    list_of_patterns = nn.generate_random_patterns(n_store)
    nn.train(list_of_patterns, normalize=normalize)

    x = np.empty(n_dim)
    for i in range(N_samples):
        h = np.dot(nn.w, np.sign(prng.normal(size=n_dim)))
        x = np.concatenate((x, h))
    
    ax = fig.add_subplot(gs[index//2, index%2])
    ax.set_xlim([-1, 1])
    ax.set_title('n_dim = ' + str(n_dim) + ' std = ' + str(np.std(x)))
    
    weights = np.ones_like(x)/float(len(x))
    n, bins, patches = ax.hist(x, bins=30, weights=weights, normed=False)

Now we calculate the variance of the h vector as a function of the dimension



In [10]:

    
n_dimensions = np.logspace(1, 4, num=20)
variances = []
standar_deviations = []

for index, n_dim in enumerate(n_dimensions):
    print('number', index, 'of', n_dimensions.size, '   n_dim =', n_dim)

    n_dim = int(n_dim)
  

    nn = Hopfield(n_dim=n_dim, T=T, prng=prng)
    list_of_patterns = nn.generate_random_patterns(n_store)
    nn.train(list_of_patterns, normalize=normalize)

    x = np.empty(n_dim)
    for i in range(N_samples):
        h = np.dot(nn.w, np.sign(prng.normal(size=n_dim)))
        x = np.concatenate((x, h))
        
    variances.append(np.var(x))
    standar_deviations.append(np.std(x))









    



number 0 of 20    n_dim = 10.0
number 1 of 20    n_dim = 14.3844988829
number 2 of 20    n_dim = 20.6913808111
number 3 of 20    n_dim = 29.7635144163
number 4 of 20    n_dim = 42.8133239872
number 5 of 20    n_dim = 61.5848211066
number 6 of 20    n_dim = 88.586679041
number 7 of 20    n_dim = 127.42749857
number 8 of 20    n_dim = 183.298071083
number 9 of 20    n_dim = 263.665089873
number 10 of 20    n_dim = 379.269019073
number 11 of 20    n_dim = 545.559478117
number 12 of 20    n_dim = 784.759970351
number 13 of 20    n_dim = 1128.83789168
number 14 of 20    n_dim = 1623.77673919
number 15 of 20    n_dim = 2335.72146909
number 16 of 20    n_dim = 3359.81828628
number 17 of 20    n_dim = 4832.93023857
number 18 of 20    n_dim = 6951.92796178
number 19 of 20    n_dim = 10000.0



In [11]:

    
fig = plt.figure(figsize=(16, 12))
ax = fig.add_subplot(111)
ax.semilogx(n_dimensions, variances,'*-', markersize=16, label='var')
ax.semilogx(n_dimensions, standar_deviations, '*-', markersize=16, label='std')
ax.axhline(y=nn.sigma, color='k', label='nn.sigma')
ax.legend()









    Out[11]:





<matplotlib.legend.Legend at 0x7f0d6cf93890>



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