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part 3: signals
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‎notebooks/03_signals.ipynb‎

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{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# 3 Graph signals\n",
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"\n",
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"A graph signal is a function $\\mathcal{V} \\rightarrow \\mathbb{R}$ that associates a value to each node $v \\in \\mathcal{V}$ of a graph. The signal values can be represented as a vector $f \\in \\mathbb{R}^N$ where $N = |\\mathcal{V}|$ is the number of nodes in the graph."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"import numpy as np\n",
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"from pygsp import graphs"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Let's generate a graph and a random signal."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"graph = graphs.Sensor(N=100)\n",
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"signal = np.random.normal(size=graph.N)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"We can now plot the signal on the graph to visualize it and see that it's indeed random."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"graph.plot_signal(signal)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 3.1 Gradient and divergence\n",
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"\n",
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"The gradient $\\nabla_\\mathcal{G} \\ f$ of the signal $f$ on the graph $\\mathcal{G}$ is a signal on the edges defined as\n",
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"\n",
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"$$(\\nabla_\\mathcal{G})_{(i,j)} \\ f = \\sqrt{W_{ij}} (f_i - f_j)$$"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"graph.compute_differential_operator()\n",
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"gradient = graph.D @ signal\n",
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"assert gradient.size == graph.Ne"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Similarly, we can compute the divergence of an edge signal, which is again a signal on the nodes.\n",
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"\n",
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"$$(\\operatorname{div}_\\mathcal{G} x)_i = \\sum_{j \\sim i} \\sqrt{W_{ij}} x_{(i,j)}$$"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"divergence = graph.D.T @ gradient\n",
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"assert divergence.size == graph.N"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"graph.plot_signal(divergence)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"The Laplacian operator is indeed the divergence of the gradient."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"np.testing.assert_allclose(graph.L @ signal, divergence)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 3.2 Smoothness"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"The smoothness of a signal can be computed by the quadratic form\n",
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"\n",
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"$$ f^\\intercal L f = \\sum_{i \\sim j} W_{ij} (f_i - f_j)^2 $$"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"signal.T @ graph.L @ signal"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 3.3 Exercise\n",
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"\n",
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"What is the smoothest graph signal, i.e. the signal $f$ for which $f^\\intercal L f = 0$? Verify computationally."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"# Your code here."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"What if $L$ is the normalized Laplacian? Verify computationally."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"graph.compute_laplacian('normalized')\n",
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"\n",
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"# Your code here."
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.7.0"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 2
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}

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