@@ -17,7 +17,7 @@
},
{
"cell_type" : " code"
"execution_count" : 1 ,
"execution_count" : 7 ,
"metadata" : {
"collapsed" : true
},
@@ -48,9 +48,15 @@
"metadata" : {},
"source" : [
" ### Solution 1\n "
" 1. $E(x) = -\\ frac{1}{2}\\ sum_{1,2}{-x_1x_2} - (0.5 + 0.5)$ ?\n "
" 1. Write answer here.\n "
" 1. Find local minimum"
" 1. $E(x) = \\ frac{1}{2}(x_1x_2 - x_1 - x_2)$\n "
" 1. \\ begin{array}{ l c r }\n "
" x_1 & x_2 & E(x) \\\\\n "
" 0 & 0 & 0 \\\\\n "
" 0 & 1 & -0.5 \\\\\n "
" 1 & 0 & -0.5 \\\\\n "
" 1 & 1 & -0.5 \\\\\n "
" \\ end{array}\n "
" 1. The stable states are the local minima. If we look at the previous question, these states are: ($x_1=0$ and $x_2=1, x_1=1 $and $x_2=0, x_1=1$ and $x_2=1$)"
]
},
{
@@ -108,27 +114,52 @@
},
{
"cell_type" : " code"
"execution_count" : 3 ,
"execution_count" : 8 ,
"metadata" : {
"collapsed" : true
},
"outputs" : [],
"source" : [
" # The optimization function\n "
" def optimize(n):\n "
" node_states = np.zeroes(n)\n "
" return node_states"
" node_states = np.random.randint(2, size=n)\n "
" print node_states\n "
" weights = np.full((n,n),-2)\n "
" for i in range(n):\n "
" weights[i,i] = 0\n "
" biases = np.repeat(1,n)\n "
" E = 1\n "
" a = np.zeros(n)\n "
" while E > 0:\n "
" for i in range(n):\n "
" #print weights[i]\n "
" a[i] = np.sum(np.dot(weights[:,i],node_states) + biases[i])\n "
" if a[i] >= 0:\n "
" node_states[i] = 1\n "
" else:\n "
" node_states[i] = 0\n "
" E = -0.5*np.dot(np.dot(node_states.T,weights),node_states) - np.dot(node_states.T, biases)\n "
" #print E, node_states\n "
" return node_states\n "
]
},
{
"cell_type" : " code"
"execution_count" : 4 ,
"metadata" : {
"collapsed" : true
},
"outputs" : [],
"execution_count" : 25 ,
"metadata" : {},
"outputs" : [
{
"name" : " stdout"
"output_type" : " stream"
"text" : [
" [0 0 0 1 1 0 1 0 0 1 1 1 1 1 1 0 1]\n "
" [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1]\n "
]
}
],
"source" : [
" # Print solutions\n "
" # Print solutions\n "
" print optimize(17)"
]
},
{
@@ -144,20 +175,11 @@
},
{
"cell_type" : " code"
"execution_count" : 7 ,
"metadata" : {},
"outputs" : [
{
"name" : " stderr"
"output_type" : " stream"
"text" : [
" /Users/jordythielen/anaconda2/lib/python2.7/site-packages/ipykernel_launcher.py:8: DeprecationWarning: `imresize` is deprecated!\n "
" `imresize` is deprecated in SciPy 1.0.0, and will be removed in 1.2.0.\n "
" Use ``skimage.transform.resize`` instead.\n "
" \n "
]
}
],
"execution_count" : 12 ,
"metadata" : {
"collapsed" : true
},
"outputs" : [],
"source" : [
" # The source image\n "
" f = urllib2.urlopen(\" https://homepages.cae.wisc.edu/~ece533/images/watch.png\" )\n "
@@ -181,24 +203,24 @@
},
{
"cell_type" : " code"
"execution_count" : 8 ,
"execution_count" : 13 ,
"metadata" : {},
"outputs" : [
{
"data" : {
"text/plain" : [
" Text(0.5,1,u'x2') "
" <matplotlib.text.Text at 0xb85b860> "
]
},
"execution_count" : 8 ,
"execution_count" : 13 ,
"metadata" : {},
"output_type" : " execute_result"
},
{
"data" : {
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"image/png": 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xvbTzQ44e/02JjbvLd7DMEACQWveZobE7sJ6j0zX2ZIuWfh5xtLpzX/o5vWcK3vXy8vLB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\nSE0wBACk9v+bQ8IHz+YMKgAAAABJRU5ErkJggg==\n",
"text/plain" : [
" <matplotlib.figure.Figure at 0x10db66710 >"
" <matplotlib.figure.Figure at 0xb58fbe0 >"
]
},
"metadata" : {},
@@ -238,7 +260,7 @@
},
{
"cell_type" : " code"
"execution_count" : 9 ,
"execution_count" : 53 ,
"metadata" : {
"collapsed" : true
},
@@ -247,13 +269,17 @@
" # Hopfield training\n "
" def hopfield_train(X):\n "
" m, n = X.shape\n "
" \n "
" #print m \n "
" # Initialize weights\n "
" \n "
" w = np.zeros((m,m))\n "
" #print w \n "
" # Hebbian learning\n "
" \n "
" # Avoid self-connections (diagonal)\n "
" \n "
" for i in range(m):\n "
" for j in range(m):\n "
" # Avoid self-connections (diagonal)\n "
" if i!=j:\n "
" w[i,j] = w[i,j] + np.dot(X[i,:], X[j,:]) \n "
" #per input pattern??\n "
" return w"
]
},
@@ -274,23 +300,23 @@
},
{
"cell_type" : " code"
"execution_count" : 11 ,
"metadata" : {
"collapsed" : true
},
"execution_count" : 43 ,
"metadata" : {},
"outputs" : [],
"source" : [
" # Hopfield testing\n "
" def hopfield_test(X, w, n_epochs=10):\n "
" \n "
" #???????????????????????????????????????????????????????????????????????????? \n "
" # Loop over epochs\n "
" \n "
" for e in range(n_epochs): \n "
" # Loop over examples\n "
" \n "
" for example in range(X.shape[1]):\n "
" count = 0\n "
" # Loop over nodes\n "
" \n "
" for node in w[:,example]: \n "
" # Update node\n "
" \n "
" Y = node + X[count, example]\n "
" count +=1\n "
" return Y"
]
},
@@ -311,13 +337,28 @@
},
{
"cell_type" : " code"
"execution_count" : 13 ,
"metadata" : {
"collapsed" : true
},
"outputs" : [],
"execution_count" : 54 ,
"metadata" : {},
"outputs" : [
{
"name" : " stdout"
"output_type" : " stream"
"text" : [
" [[ 0. 2. 2. ..., 0. 0. 0.]\n "
" [ 2. 0. 2. ..., 0. 0. 0.]\n "
" [ 2. 2. 0. ..., 0. 0. 0.]\n "
" ..., \n "
" [ 0. 0. 0. ..., 0. 2. 2.]\n "
" [ 0. 0. 0. ..., 2. 0. 2.]\n "
" [ 0. 0. 0. ..., 2. 2. 0.]]\n "
]
}
],
"source" : [
" # Run hopfield training\n "
" # Run hopfield training\n "
" weights = hopfield_train(X)\n "
" print weights\n "
" #so it prints out 2's which is wrong"
]
},
{
@@ -336,7 +377,7 @@
},
{
"cell_type" : " code"
"execution_count" : 15 ,
"execution_count" : 36 ,
"metadata" : {
"collapsed" : true
},
@@ -360,17 +401,31 @@
},
{
"cell_type" : " code"
"execution_count" : 16 ,
"metadata" : {
"collapsed" : true
},
"outputs" : [],
"execution_count" : 41 ,
"metadata" : {},
"outputs" : [
{
"data" : {
"image/png": 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NtoX1ZnZbsI3kPjZ3j82LxMMhnweOA4YCTwIz8lDve4AGYH2PYd8GrgreXwVc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\nIiLA/wd82WU3AAAABElEQVSHirvLfYibmAAAAABJRU5ErkJggg==\n",
"text/plain" : [
" <matplotlib.figure.Figure at 0xb5b6ef0>"
]
},
"metadata" : {},
"output_type" : " display_data"
}
],
"source" : [
" # Corrupt images\n "
" \n "
" X10 = corrupt_images(X, 10) \n "
" # Test associative memory properties\n "
" \n "
" # Plot results"
" pred = hopfield_test(X10, weights)\n "
" # Plot results\n "
" plt.figure()\n "
" plt.plot(pred, label = \" predicted\" )\n "
" plt.plot(X, label = \" real\" )\n "
" plt.legend()\n "
" plt.show() "
]
},
{
@@ -382,7 +437,7 @@
},
{
"cell_type" : " code"
"execution_count" : 17 ,
"execution_count" : null ,
"metadata" : {
"collapsed" : true
},
@@ -406,7 +461,7 @@
},
{
"cell_type" : " code"
"execution_count" : 18 ,
"execution_count" : null ,
"metadata" : {
"collapsed" : true
},
@@ -430,7 +485,7 @@
},
{
"cell_type" : " code"
"execution_count" : 19 ,
"execution_count" : null ,
"metadata" : {
"collapsed" : true
},
@@ -459,18 +514,6 @@
"display_name" : " Python 2"
"language" : " python"
"name" : " python2"
},
"language_info" : {
"codemirror_mode" : {
"name" : " ipython"
"version" : 2
},
"file_extension" : " .py"
"mimetype" : " text/x-python"
"name" : " python"
"nbconvert_exporter" : " python"
"pygments_lexer" : " ipython2"
"version" : " 2.7.13"
}
},
"nbformat" : 4 ,