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ed4bf0de5077b73819edfb096345d1a8adce1894
FYS-STK4155/doc/Programs/EigenvaluesDeepLearning/eigen_tf1_working.ipynb
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mhjensen ed4bf0de50 added codes
2019-12-11 09:03:43 +01:00

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{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Finding eigenvalues of matrices with neural networks. "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Script for finding the eigenvectors corresponding to the largest eigenvalue of a matrix with a neural network."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"jupyter": {
"outputs_hidden": true
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"A = Tensor(\"Const:0\", shape=(6, 6), dtype=float64)\n",
"x0 = Tensor(\"Const_1:0\", shape=(1, 6), dtype=float64)\n",
"WARNING:tensorflow:From <ipython-input-3-fcdc6fc989db>:39: dense (from tensorflow.python.layers.core) is deprecated and will be removed in a future version.\n",
"Instructions for updating:\n",
"Use keras.layers.Dense instead.\n",
"WARNING:tensorflow:From /usr/local/lib/python3.7/site-packages/tensorflow_core/python/layers/core.py:187: Layer.apply (from tensorflow.python.keras.engine.base_layer) is deprecated and will be removed in a future version.\n",
"Instructions for updating:\n",
"Please use `layer.__call__` method instead.\n",
"dnn_output = Tensor(\"dnn/dense_1/BiasAdd:0\", shape=(1, 6), dtype=float64)\n",
"x_trial = Tensor(\"loss/transpose:0\", shape=(6, 1), dtype=float64)\n",
"Tensor(\"loss/mul:0\", shape=(6, 6), dtype=float64)\n",
"Tensor(\"loss/mul_1:0\", shape=(6, 6), dtype=float64)\n",
"Tensor(\"loss/Tensordot_3:0\", shape=(6, 1), dtype=float64)\n",
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"name": "stdout",
"output_type": "stream",
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"Step: 99100 / 100000 loss: 0.0\n",
"Step: 99200 / 100000 loss: 0.0\n",
"Step: 99300 / 100000 loss: 0.0\n",
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"Step: 99500 / 100000 loss: 0.0\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Step: 99600 / 100000 loss: 0.0\n",
"Step: 99700 / 100000 loss: 0.0\n",
"Step: 99800 / 100000 loss: 0.0\n",
"Step: 99900 / 100000 loss: 0.0\n"
]
}
],
"source": [
"import tensorflow.compat.v1 as tf\n",
"tf.disable_v2_behavior()\n",
"tf.reset_default_graph()\n",
"# tf.set_random_seed(343)\n",
"\n",
"# import tensorflow as tf\n",
"import numpy as np\n",
"from matplotlib import cm\n",
"from matplotlib import pyplot as plt\n",
"from mpl_toolkits.mplot3d import axes3d\n",
"#from lib import compute_dx_dt\n",
"\n",
"matrix_size = 6\n",
"\n",
"A = np.random.random_sample(size=(matrix_size,matrix_size))\n",
"A = (A.T + A)/2.0\n",
"start_matrix = A\n",
"\n",
"eigen_vals, eigen_vecs = np.linalg.eig(A)\n",
"\n",
"A = tf.convert_to_tensor(A)\n",
"print(\"A = \", A)\n",
"\n",
"x_0 = tf.convert_to_tensor(np.random.random_sample(size = (1,matrix_size)))\n",
"print(\"x0 = \", x_0)\n",
"\n",
"## The construction phase\n",
"\n",
"num_iter = 100000\n",
"num_hidden_neurons = [50]\n",
"num_hidden_layers = np.size(num_hidden_neurons)\n",
"\n",
"\n",
"with tf.variable_scope('dnn'):\n",
"\n",
" previous_layer = x_0\n",
"\n",
" for l in range(num_hidden_layers):\n",
" current_layer = tf.layers.dense(previous_layer, num_hidden_neurons[l],activation=tf.nn.sigmoid)\n",
" previous_layer = current_layer\n",
"\n",
" dnn_output = tf.layers.dense(previous_layer, matrix_size)\n",
"\n",
"with tf.name_scope('loss'):\n",
" print(\"dnn_output = \", dnn_output)\n",
" \n",
" x_trial = tf.transpose(dnn_output)\n",
" print(\"x_trial = \", x_trial)\n",
" \n",
" temp1 = (tf.tensordot(tf.transpose(x_trial), x_trial, axes=1)*A)\n",
" temp2 = (1- tf.tensordot(tf.transpose(x_trial), tf.tensordot(A, x_trial, axes=1), axes=1))*np.eye(matrix_size)\n",
" func = tf.tensordot((temp1-temp2), x_trial, axes=1)\n",
" \n",
" print(temp1)\n",
" print(temp2)\n",
" print(func)\n",
" \n",
" func = tf.transpose(func)\n",
" x_trial = tf.transpose(x_trial)\n",
" \n",
" loss = tf.losses.mean_squared_error(func, x_trial)\n",
"\n",
"learning_rate = 0.001\n",
"\n",
"with tf.name_scope('train'):\n",
" optimizer = tf.train.GradientDescentOptimizer(learning_rate)\n",
" traning_op = optimizer.minimize(loss)\n",
"\n",
"init = tf.global_variables_initializer()\n",
"\n",
"g_dnn = None\n",
"\n",
"losses = []\n",
"\n",
"with tf.Session() as sess:\n",
" init.run()\n",
" for i in range(num_iter):\n",
" sess.run(traning_op)\n",
"\n",
" if i % 100 == 0:\n",
" l = loss.eval()\n",
" print(\"Step:\", i, \"/\",num_iter, \"loss: \", l)\n",
" losses.append(l)\n",
"\n",
" x_dnn = x_trial.eval()\n",
"x_dnn = x_dnn.T"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Plotting loss over time"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Text(0, 0.5, 'Loss')"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
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\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plt.plot(losses[:5])\n",
"plt.xlabel(\"Iteration\")\n",
"plt.ylabel(\"Loss\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Printing eigenvector and eigenvalues"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Eigenvector NN = \n",
" [[0.39102385]\n",
" [0.41813081]\n",
" [0.4112727 ]\n",
" [0.42373881]\n",
" [0.31962169]\n",
" [0.47054133]] \n",
"\n",
"Eigenvalue NN = \n",
" [[3.32927854]] \n",
" \n",
"\n",
"Eigenvector analytic = \n",
" [[ 0.39102384 0.37327188 0.68249482 0.19314131 0.43158254 -0.13565482]\n",
" [ 0.41813082 -0.21544287 0.0517078 -0.85524846 0.04349033 -0.20672676]\n",
" [ 0.4112727 -0.78668253 0.1648857 0.34671246 -0.02876616 0.25250949]\n",
" [ 0.42373881 0.32112534 0.09676804 0.07916383 -0.83656144 0.04310365]\n",
" [ 0.31962169 -0.03283447 -0.4622512 0.32363834 0.08752416 -0.75543726]\n",
" [ 0.47054132 0.30196602 -0.53037685 0.00531919 0.32174849 0.55005219]]\n",
"\n",
"\n",
"Eigenvalues analytic = \n",
" [ 3.32927854 -0.93859 -0.71386523 0.60372286 0.19411317 -0.14395577]\n"
]
}
],
"source": [
"print(\"Eigenvector NN = \\n\", (x_dnn/(x_dnn**2).sum()**0.5), \"\\n\")\n",
"\n",
"eigen_val_nn = x_dnn.T @ (start_matrix @ x_dnn) / (x_dnn.T @ x_dnn)\n",
"\n",
"print(\"Eigenvalue NN = \\n\", eigen_val_nn, \"\\n \\n\")\n",
"print(\"Eigenvector analytic = \\n\", eigen_vecs)\n",
"print(\"\\n\")\n",
"print(\"Eigenvalues analytic = \\n\",eigen_vals)\n"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.7.5"
}
},
"nbformat": 4,
"nbformat_minor": 4
}
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