1245 lines
91 KiB
Plaintext
1245 lines
91 KiB
Plaintext
{
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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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"# Lecture 3. Projectors. Least squares problem. QR factorization."
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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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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"# Projectors"
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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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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"* A projector is a square matrix $P$ which satisfies\n",
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"$$\n",
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"P^2=P\n",
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"$$"
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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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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"* Lets inspect the geometric meaning of a projection\n",
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""
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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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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"* For any vector $v$ in $\\mathrm{range}(P)$ one has $Pv=v$"
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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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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"* For $S_1 = \\mathrm{range}(P)$ and $S_2=\\mathrm{null}(P)$ it is said that $P$ projects onto $S_1$ along $S_2$"
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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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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Orthogonal projectors\n",
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"* For orthogonal projector: $S_1\\perp S_2$"
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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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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"* One can show (homework) that this is equivalent to $P = P^T$"
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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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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"- Lets inspect the geometric meaning of an orthogonal projection\n",
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""
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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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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Projection with orthonormal basis\n",
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"* Let us consider orthogonal projection onto certain $S_1$"
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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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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"* Let dimension of $S_1$ be $n$; $q_1,q_2,...,q_n$ form an orthonormal basis for $S_1$ and $q_{n+1},...,q_m$ form an orthonormal basis for $S_2$."
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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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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"* Consider \n",
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"$$\n",
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"Q = \\bigg[q_1\\bigg|q_2\\bigg|...\\bigg|q_n\\bigg|q_{n+1}\\bigg|...\\bigg|q_m\\bigg]\n",
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"$$"
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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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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"* Then:\n",
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"$$\n",
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"PQ = \\bigg[q_1\\bigg|q_2\\bigg|...\\bigg|q_n\\bigg|0\\bigg|...\\bigg|0\\bigg]\n",
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"$$"
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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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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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" * We thus have $Q^TPQ =\\mathrm{diag}(1,1,...,1,0,0,0)=\\Sigma$\n",
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"which provides SVD decomposition ot $P$: \n",
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"\n",
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"$$P = Q\\Sigma Q^T$$"
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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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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"* Reduced SVD of a projector $P$ reads:\n",
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"$$\n",
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"P = \\hat Q \\hat Q^T\n",
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"$$\n",
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"where $\\hat Q$ is not orthogonal matrix (as it is not square), but it has orthonormal columns."
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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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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Projection with an arbitrary basis \n",
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"* Suppose that we want to project on a space spanned by columns of arbitrary $A$ (they are not necessarily orthonormal, but assumed to be linearly independent)"
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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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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"* In passing from $v$ to its orthogonal projection $y \\in \\mathrm{range}(A)$, the difference $y-v$ must be orthogonal to $\\mathrm{range}(A)$:\n",
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"$A^T (Ax-v) = 0$, indeed:"
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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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""
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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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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"* If the columns of $A$ are linearly independent:\n",
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"$$\n",
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"x = (A^T A)^{-1}A^T v\n",
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"$$"
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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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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"* The projection is thus given by the operator\n",
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"$$\n",
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"P = A(A^T A)^{-1}A^T\n",
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"$$"
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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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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"# Linear systems\n",
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"\n",
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"- Linear systems appear as:\n",
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"\n",
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" - Linear regression problems\n",
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" - Discretization of partial differential/integral equations\n",
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" - At elementary steps of iterative optimization techniques"
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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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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Linear equations and matrices\n",
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"A linear system of equations can be written in the form\n",
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"\n",
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"\\begin{align*}\n",
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" &2 x + 3 y = 5\\quad &\\longrightarrow \\quad &2x + 3 y + 0 z = 5\\\\\n",
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" &3 x + 2z = 4\\quad &\\longrightarrow\\quad &3 x + 0 y + 2 z = 4\\\\\n",
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" &x + y = 2\\quad &\\longrightarrow\\quad & 1 x + 1 y + 0 z = 2\\\\\n",
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"\\end{align*}"
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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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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Matrix form\n",
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"\n",
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"$$\n",
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"\\begin{pmatrix}\n",
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"2 & 3 & 0 \\\\\n",
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"3 & 0 & 2 \\\\\n",
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"1 & 1 & 0 \\\\\n",
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"\\end{pmatrix}\\begin{pmatrix}\n",
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"x \\\\\n",
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"y \\\\\n",
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"z \n",
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"\\end{pmatrix} = \n",
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"\\begin{pmatrix}\n",
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"5 \\\\\n",
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"4 \\\\\n",
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"2\n",
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"\\end{pmatrix}\n",
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"$$\n",
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"\n",
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"or simply\n",
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"\n",
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"$$ A u = b $$"
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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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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Over/under determined linear systems\n",
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"\n",
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"If the system $Ax = b$ has\n",
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"\n",
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"- more equations than unknowns, it is called overdetermined system (typically no solution)\n",
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"\n",
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"- less equations than unknowns, it is called underdetermined system (typically many solutions)"
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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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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Overdetermined linear systems\n",
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"\n",
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"- The number of equations is > than the number of unknowns. \n",
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"\n",
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"- The simplest example is linear fitting, fitting a set of 2D points by a line.\n",
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"\n",
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"- Then, a typical way is to minimize the residual (least squares)\n",
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"\n",
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"$$\\Vert A x - b \\Vert_2 \\rightarrow \\min$$"
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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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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"## Underdetermined linear systems\n",
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"\n",
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"- The number of equations is > than the number of unknowns. \n",
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"\n",
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"- In this case, there could be infinitely many solutions $x$\n",
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"\n",
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"- Typical problem in Machine Learning: employ some kind of regularization. For example, one may find the solution to $Ax=b$ with a minimal norm \n",
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"$$\\Vert x \\Vert_2\\to\\min$$"
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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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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Example of least squares\n",
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"Consider a two-dimensional example. Suppose we have a linear model \n",
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"\n",
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"$$y = ax + b$$\n",
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"\n",
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"and noisy data $(x_1, y_1), \\dots (x_n, y_n)$. Then the linear system on coefficients will look as follows\n",
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"\n",
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"$$\n",
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"\\begin{split}\n",
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"a x_1 &+ b &= y_1 \\\\\n",
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"&\\vdots \\\\\n",
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"a x_n &+ b &= y_n \\\\\n",
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"\\end{split}\n",
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"$$\n",
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"\n",
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"or in a matrix form\n",
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"\n",
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"$$\n",
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"\\begin{pmatrix}\n",
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"x_1 & 1 \\\\\n",
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"\\vdots & \\vdots \\\\\n",
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"x_n & 1 \\\\\n",
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"\\end{pmatrix}\n",
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"\\begin{pmatrix}\n",
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"a \\\\\n",
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"b\n",
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"\\end{pmatrix} =\n",
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"\\begin{pmatrix}\n",
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"y_1 \\\\\n",
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"\\vdots \\\\\n",
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"y_n \\\\\n",
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"\\end{pmatrix},\n",
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"$$\n",
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"\n",
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"which represents overdetermined system."
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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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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Solving the least squares\n",
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"- Suppose we need to solve\n",
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"\n",
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"$$ \\Vert A x - b \\Vert_2 \\rightarrow \\min_x, $$\n",
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"\n",
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"where $A$ is $m \\times n$, $m \\geq n$."
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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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""
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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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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"- The residual $r = A x - b$ should be orthogonal to $\\mathrm{range} A$, and\n",
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"or equivalently:\n",
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"$$\n",
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"P b = A x,\n",
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"$$\n",
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"where $P$ is a projector onto the range of $A$, that is $P = A(A^T A)^{-1}A^T$"
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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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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"- If $A$ is of a full rank, then \n",
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"$$\n",
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"x = (A^T A)^{-1} A^T b\n",
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"$$\n",
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"- The matrix $(A^T A)^{-1} A^T$ is known as a pseudoinverse, denoted as $A^+$. Can be easily computed in numpy."
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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": 1,
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"metadata": {
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"data": {
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"image/png": "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",
|
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"text/plain": [
|
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"<Figure size 432x288 with 1 Axes>"
|
|
]
|
|
},
|
|
"metadata": {
|
|
"needs_background": "light"
|
|
},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"%matplotlib inline\n",
|
|
"import numpy as np\n",
|
|
"import matplotlib.pyplot as plt\n",
|
|
"\n",
|
|
"a0 = 1; b0 = 2; n = 10\n",
|
|
"\n",
|
|
"xi = np.arange(n)\n",
|
|
"yi = a0 * xi + b0 + np.random.random(n)\n",
|
|
"A = np.hstack([xi[:,None], np.ones(n)[:,None]])\n",
|
|
"coef = np.linalg.pinv(A) @ yi\n",
|
|
"\n",
|
|
"plt.plot(xi, yi, 'o', label='$(x_i, y_i)$')\n",
|
|
"plt.plot(xi, coef[0]*xi + coef[1], label='Least Squares')\n",
|
|
"plt.legend(loc='best')"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## What happens inside pinv?\n",
|
|
"We dont know yet -- computing pseudoinverse might be tough. Let us inspect a computationally cheaper way."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## Solving the linear least squares via $QR$ decomposition.\n",
|
|
"\n",
|
|
"- Any matrix can be factored into a product \n",
|
|
"$$ A = Q R$$\n",
|
|
"where $Q$ is unitary, and $R$ is upper triangular.\n",
|
|
"- Finding optimal $x$ is equivalent to solving\n",
|
|
"$$\n",
|
|
"A^T A x = A^T b\n",
|
|
"$$\n",
|
|
"If $A$ is of full rank, it is the same as\n",
|
|
"$$ Rx = Q^T b. $$\n",
|
|
"- Since $R$ is upper triangular, the solving of this linear system costs $\\mathcal{O}(n^2)$. \n",
|
|
"- It is more stable than solving the normal equations"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"$$R^T Q^T Q R x = R^T Q^T b$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"$$R x = Q^T b$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"# QR decomposition\n",
|
|
"\n",
|
|
"- The name implies that a matrix is represented as a product \n",
|
|
"$$\n",
|
|
" A = Q R, \n",
|
|
"$$\n",
|
|
"where $Q$ is a column orthonormal matrix and $R$ is upper triangular. "
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "fragment"
|
|
}
|
|
},
|
|
"source": [
|
|
"- The matrix sizes: $Q$ is $m \\times n$, $R$ is $n \\times n$ if $m\\geq n$."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "fragment"
|
|
}
|
|
},
|
|
"source": [
|
|
"- QR decomposition is defined for any rectangular matrix."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "fragment"
|
|
}
|
|
},
|
|
"source": [
|
|
"- QR decomposition can be computed in a $O(mn^2)$ (for $m>n$) (via so called direct methods)."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## Full QR vs reduced QR\n",
|
|
""
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## QR decomposition: applications\n",
|
|
"\n",
|
|
"This decomposition plays a crucial role in many problems:\n",
|
|
"- Computing orthonormal bases in a linear space"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "fragment"
|
|
}
|
|
},
|
|
"source": [
|
|
"- Solving overdetermined systems of linear equations"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "fragment"
|
|
}
|
|
},
|
|
"source": [
|
|
"- QR-algorithm for the computation of eigenvectors and eigenvalues ([one of the 10 most important algorithms of the 20th century](https://archive.siam.org/pdf/news/637.pdf)) is based on the QR decomposition"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## Existence of QR decomposition\n",
|
|
"Every rectangular $m \\times n$ matrix has a QR decomposition. \n",
|
|
"\n",
|
|
"\n",
|
|
"There are several ways to prove it and compute it:\n",
|
|
"\n",
|
|
"- Geometrical: using the Gram-Schmidt orthogonalization\n",
|
|
"- Practical: using Householder/Givens transformations "
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## Gram-Schmidt orthogonalization\n",
|
|
"\n",
|
|
"- QR decomposition is a mathematical way of writing down the Gram-Schmidt orthogonalization process. \n",
|
|
"- Given a sequence of vectors $a_1, \\ldots, a_m$ we want to find orthogonal basis $q_1, \\ldots, q_m$ such that every $a_i$ is a linear combination of such vectors. "
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "fragment"
|
|
}
|
|
},
|
|
"source": [
|
|
"Gram-Schmidt:\n",
|
|
"- $q_1 := a_1/\\Vert a_1\\Vert$\n",
|
|
"- $q_2 := a_2 - (a_2, q_1) q_1, \\quad q_2 := q_2/\\Vert q_2 \\Vert$\n",
|
|
"- $q_3 := a_3 - (a_3, q_1) q_1 - (a_3, q_2) q_2, \\quad q_3 := q_3/\\Vert q_3 \\Vert$\n",
|
|
"\n",
|
|
"And so on.\n",
|
|
"\n",
|
|
"Note that the transformation from $Q$ to $A$ has triangular structure, since from the $k$-th vector we subtract only the previous ones. It follows from the fact that the product of triangular matrices is a triangular matrix."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## QR decomposition: the practical way\n",
|
|
"\n",
|
|
"* Gram-Schmidt orthogonalization is not very practical as just described for numerical issues (will see later). One typically does it differently: Householder rotations (implemented in numpy)."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## More on Pseudoinverse\n",
|
|
"Recall that we introduced pseudoinverse as\n",
|
|
"$$\n",
|
|
"A^+ = (A^T A)^{-1} A^T\n",
|
|
"$$\n",
|
|
"- Matrix $A^T A$ can be singular in general case.\n",
|
|
"- Therefore, we need to introduce the concept of pseudoinverse matrix $A^{\\dagger}$.\n",
|
|
"- The matrix $$A^{\\dagger} = \\lim_{\\alpha \\rightarrow 0}(\\alpha I + A^T A)^{-1} A^T$$ is called Moore-Penrose pseudoinverse of the matrix $A$."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## Pseudoinverse and linear systems\n",
|
|
"Consider a system of equations $Ax\\approx b$\n",
|
|
"- $\\forall x$, we have $\\left\\|Ax-b\\right\\|_{2}\\geq \\left\\|Az-b\\right\\|_{2}$, where \n",
|
|
"$z=A^{+}b$\n",
|
|
"- If $A x = b$ is satisfiable, the vector $z=A^{+}b$ is a solution, and satisfies $\\|z\\|_{2}\\leq \\|x\\|_{2}$ for all other solutions\n",
|
|
"\n",
|
|
"As a result, the solution to the linear least squares problem (including least norm solutions) can formally be written as\n",
|
|
"$$x = A^{\\dagger} b.$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## Simple cases\n",
|
|
"\n",
|
|
"- If matrix $A$ is squared and non-singular we get standard inverse of $A$:\n",
|
|
"$$A^{\\dagger} = \\lim_{\\alpha \\rightarrow 0}(\\alpha I + A^T A)^{-1} A^T = (A^T A)^{-1} A^T = A^{-1} (A^{T})^{-1} A^T = A^{-1}$$\n",
|
|
"- If matrix $A$ has full column rank, then $A^T A$ is non-singular and we get \n",
|
|
"$$A^{\\dagger} = \\lim_{\\alpha \\rightarrow 0}(\\alpha I + A^T A)^{-1} A^T = (A^T A)^{-1} A^T. $$\n",
|
|
"- Pseudoinverse is uniquely defined in all other cases, too! "
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## Least squares via SVD \n",
|
|
"Lets inspect SVD approach to over- and under- determined problems."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"Consider solving $Ax \\approx b$ for $A$ of arbitrary shape and possibly not of a full rank (the system cen be under- or overdetermined).\n",
|
|
"\n",
|
|
"Let $A = U \\Sigma V^T$ be SVD decomposition of $A$, then we need to solve"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"$$\n",
|
|
"U\\Sigma V^T x \\approx b\n",
|
|
"$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"Since $U$ and $V$ are orthogonal, we can just switch to new variables, $\\bar x = V^T x$ and $\\bar b = U^T b$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"$$\n",
|
|
"\\Sigma \\bar x \\approx \\bar b\n",
|
|
"$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"Up to now, its not important if the system is under- or overdetermined. Here it becomes important. Lets consider underdetermined case."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"$$\n",
|
|
"\\begin{pmatrix}\n",
|
|
"\\sigma_1 & 0 & 0\\\\\n",
|
|
"0 & \\sigma_2 & 0 \\\\\n",
|
|
"0 & 0 & 0 \\\\\n",
|
|
"\\end{pmatrix}\n",
|
|
"\\begin{pmatrix}\n",
|
|
"\\bar x_1 \\\\\n",
|
|
"\\bar x_2 \\\\\n",
|
|
"\\bar x_3\n",
|
|
"\\end{pmatrix} \\approx\n",
|
|
"\\begin{pmatrix}\n",
|
|
"\\bar b_1 \\\\\n",
|
|
"\\bar b_2 \\\\\n",
|
|
"\\bar b_3\n",
|
|
"\\end{pmatrix},\n",
|
|
"$$\n"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"$$\n",
|
|
"\\bar x_1 = \\bar b_1 / \\sigma_1, \\bar x_2 = \\bar b_2 / \\sigma_2, \\bar x_3 = 0\n",
|
|
"$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"# Example: Signal recovery "
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 1,
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "fragment"
|
|
}
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"import numpy as np\n",
|
|
"import matplotlib.pyplot as plt\n",
|
|
"import pandas as pd"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 2,
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "fragment"
|
|
}
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"with np.load('data.npz') as data:\n",
|
|
" A, C = data['A'], data[\"C\"]"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 3,
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "fragment"
|
|
}
|
|
},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"(25, 60)"
|
|
]
|
|
},
|
|
"execution_count": 3,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
},
|
|
{
|
|
"data": {
|
|
"image/png": "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",
|
|
"text/plain": [
|
|
"<Figure size 432x288 with 1 Axes>"
|
|
]
|
|
},
|
|
"metadata": {
|
|
"needs_background": "light"
|
|
},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"plt.imshow(A)\n",
|
|
"A.shape"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"The image, stored in a matrix $A$ is obtained from a certain original $A_0$ via convolution with a filter $C$ addition of a noise. The filter $C$ 'blurs' the image, simultaneously changing the image size from $16\\times 51$ to $25\\times 60$."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 38,
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"def mat2vec(A):\n",
|
|
" return np.reshape(np.flipud(A), np.prod(A.shape))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 39,
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "fragment"
|
|
}
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"def vec2mat(a, shape):\n",
|
|
" return np.flipud(np.reshape(a, shape))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "fragment"
|
|
}
|
|
},
|
|
"source": [
|
|
"Representing the images as vectors, we can write down the filtering as follows:\n",
|
|
"$$\n",
|
|
"a_0\\to a = C a_0 + \\epsilon,\n",
|
|
"$$\n",
|
|
"where $\\epsilon$ is a noise vector (consisting from iid normal variables)."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"Lets inspect how the filter $C$ acts on the images"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 40,
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "fragment"
|
|
}
|
|
},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"image/png": "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",
|
|
"text/plain": [
|
|
"<Figure size 432x288 with 1 Axes>"
|
|
]
|
|
},
|
|
"metadata": {
|
|
"needs_background": "light"
|
|
},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"X = np.zeros((16,51))\n",
|
|
"X[5:15,15] = 1; X[5:15,30] = 1; X[5:15,40] = 1\n",
|
|
"X[5,10:20] = 1; X[5,30:41] = 1; X[15,30:41] = 1\n",
|
|
"plt.imshow(X);"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 41,
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "fragment"
|
|
}
|
|
},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"image/png": "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",
|
|
"text/plain": [
|
|
"<Figure size 432x288 with 1 Axes>"
|
|
]
|
|
},
|
|
"metadata": {
|
|
"needs_background": "light"
|
|
},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"x = mat2vec(X)\n",
|
|
"plt.imshow(vec2mat(C @ x, (25, 60)));"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 42,
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"image/png": 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",
|
|
"text/plain": [
|
|
"<Figure size 432x288 with 1 Axes>"
|
|
]
|
|
},
|
|
"metadata": {
|
|
"needs_background": "light"
|
|
},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"a = mat2vec(A)\n",
|
|
"a0 = np.linalg.pinv(C) @ a\n",
|
|
"A0 = vec2mat(a0, (16, 51))\n",
|
|
"plt.imshow(A0);"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 43,
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"u, s, vh = np.linalg.svd(C, full_matrices=False)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 44,
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "fragment"
|
|
}
|
|
},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"((1500, 816), (816,))"
|
|
]
|
|
},
|
|
"execution_count": 44,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"C.shape, s.shape"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 45,
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "fragment"
|
|
}
|
|
},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"image/png": "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",
|
|
"text/plain": [
|
|
"<Figure size 432x288 with 1 Axes>"
|
|
]
|
|
},
|
|
"metadata": {
|
|
"needs_background": "light"
|
|
},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"# equivalent to least squares\n",
|
|
"n = 816\n",
|
|
"s0 = np.zeros(816)\n",
|
|
"s0[:n] = s[:n]\n",
|
|
"C0 = np.dot(u, np.dot(np.diag(s0), vh))\n",
|
|
"A0_guess = vec2mat(np.linalg.pinv(C0) @ a, (16, 51))\n",
|
|
"plt.imshow(A0_guess)\n",
|
|
"plt.show()"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 46,
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
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"image/png": 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",
|
|
"text/plain": [
|
|
"<Figure size 432x288 with 1 Axes>"
|
|
]
|
|
},
|
|
"metadata": {
|
|
"needs_background": "light"
|
|
},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"plt.plot(s0)\n",
|
|
"plt.yscale('log');"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 47,
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "fragment"
|
|
}
|
|
},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAXQAAACJCAYAAAA8GMMgAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjUuMSwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy/YYfK9AAAACXBIWXMAAAsTAAALEwEAmpwYAAAUyElEQVR4nO2dXYwk11XH/6eqq7vna2e/1/baiUNkiUQWMdLKCjIPToDIgIXDQ1AsIfkBsTwQKUggZPISQIrECx8vCMkQy5YgCZbAxEIWiWWCzBN4DYE42AHLcuzVzu7sx8zOV39Udx8epgcm9v2fmemenXZq/j9pNd339q26fU7Vqdq6/z7H3B1CCCF++MkmPQEhhBD7gwK6EEJUBAV0IYSoCAroQghRERTQhRCiIiigCyFERRgroJvZQ2b2PTN7w8we369JCSGE2Ds2qg7dzHIA/w3gZwBcBPAygEfd/b/YmHo+7VPFkWSfF3myfVDj1xxPD6HtAOBkc258zChYZNbB3sfYCGPYfjbHHczvD9yIYQN7U1+MMmaHcfsKMWm4+8gNbHsjjdlff8f23rvB993nbHP7bG/rpztZOwBYj5yYvR4ds9K/ds3dT/GZbFLb6QMB9wN4w93fBAAz+xqARwDQgD5VHMFPfPCxZF9523yyvX2qTifQnk9H53KOe7rfJO0FHRKfkcRvGfcN8na6vdbmBwEd0wnGdHlfVpIDcbD3E98zbqBBLd03KKIx6fb+CGMAwEnfSBeBwDxZn2yKtANAVgZ9vfTOcuK7aF9sWwD4d4qCaXDTNMiJz8mxAAADcv6FY4I5MNiNEcDtGp2XxVp6g8XNLh1Tu76W7li8Rsd8Y+nL36ed2xjnkctZAO9se39x2CaEEGICjBPQU5fO91zKzOy8mV0wswvdfmuM3QkhhIgYJ6BfBHDXtvd3Arj07g+5+xPufs7dz9XzqTF2J4QQImKcgP4ygHvM7ENmVgfwWQDP7c+0hBBC7JWRF0XdvWdmnwPwDQA5gCfd/bvhmCJHeUd68XPtjkayvXWKX3M6x9Lt5Rxf+ejPkL4iWC2JLntsWMkHZa10X97mC0C1VrovGpPxdRm6GDfSoihZBAP4YiVbBIv6Bnx9HIOCz5uqoUZYFI0VD6S9N6KPyLg8GGNkMT5amB1FAXNgPg/GeJ6ed+TXrB/5gp1jfHv11fS53Jjj5/9UM31A1vMg0Czxru2Mo3KBuz8P4PlxtiGEEGJ/0C9FhRCiIiigCyFERVBAF0KIiqCALoQQFWGsRdG9MigytMhP+TdOp68trTN8Bb48kV7SL+Y7dMz8THrJeqbOf4OdZ4FqZkDmXfLl+bVWWtHT2eBjylbaVdkGvybnZNUeAIyqXAKJAJF4MLUBECge6sGYBlMiBT95r3MZR1ZLj8sCvzIGgYRi0Ev7wgPFE7qBGqqT7mNqDICrXKJUFFQWEkh6Qp8TVVHkc2+QY6se+Ki2d/8hULkwZVq+zn1UEpVLb4rnJWDpDGwwy+dGE6r8ILpDF0KIiqCALoQQFUEBXQghKoICuhBCVAQFdCGEqAgK6EIIUREOVraYA+1j6WtI53h6THmS663mTqcrf9w5f5OOOTud7jteX6djmkFZmTbJHrRc8lTBV1rpMnxXWzN0zM2N9PbaLZ6xqmwHJV2YlC7K08QUX4GEjckJa4HMcLqZtvdMg2elmiq4j6Zq6b7aCLLFHpGpAkCrlz4WIgnreof7r9NOj+t1uV+d9UVyPWaGKHlZ4HOW6C5vcJ83G2kfNQM5cbNIx4Zo2pH/Nrppe2+skzJnANrT6TGDGveRkTnUgmNht+gOXQghKoICuhBCVAQFdCGEqAgK6EIIUREU0IUQoiIcqMrFc6A7l16DLufTK+PNY7z+093H0nWZPnLkMh3z4eZisv222jIdUw/qd3VJfbOrvbSSBQAuNtKSnrdJOwAsNNLbW5riappIQVGWadcPguRcWZZWNtRq3D5TRKUw1+AJ1I41NtLt9RYdc7RIjwGA2Ty9r0i9xGCqJgBY6aXVEMvlNB1zvcOVTcudtG9XA7+2iVKj1wuUMcznQXKuWpAYq0HUJ5FK6UgjfZ4fJ8cCAMzU0n4tjM+tw7LFAVjqpu19ucnP5au1dEKt0vl52emkfVGsjX9/rTt0IYSoCAroQghRERTQhRCiIiigCyFERVBAF0KIiqCALoQQFWEs2aKZvQVgFUAfQM/dz0Wf9wzoETXPYCYtfTs6y2VLH5gmssWpS3TMj9YXku2nci6JawTZfjpE2XW9xhOEzWR7l9E18rQUbJrIpgDgZp1Lp1jCqCg3V05kbCz5FQDMN9J2PdngydBO1VeT7aeLFT6mxvuOZGlJXJMVVgXQJyme1gfperAAsNxPyxOvBRLWhfo87bvWSPs2kjqudNLSyXaPn+qsLq4FssV6zqWqs/X08R1JEE820on2bqtzv87X0tsrAplxJ5CdXuul7T1d43JLxkLJZaLlWtpH5WyUVmx37IcO/RPufm0ftiOEEGIM9MhFCCEqwrgB3QF808xeMbPz+zEhIYQQozHuI5cH3P2SmZ0G8IKZve7uL23/wDDQnweA2pFjY+5OCCEEY6w7dHe/NPy7COBZAPcnPvOEu59z93P5DF/MEUIIMR4jB3QzmzGzua3XAD4F4NX9mpgQQoi9Mc4jlzMAnjWzre18xd3/IRxhwIAkijNSb/AIkUABXN52WyAZPEPkiWdynsGuYdxMHU/LCQvj824Xabll27mkaoMYru9BdsRAdjZVI7UQg+3Vs/R3nS34dz1eT0vLbq9zH91O7HOWtAPAqZzLIOcsPe9moBJjwrfVoCblzTwtaZzJuOwtktgx/2WBuLRGtrdecrllz9PfKTp+mIwWAI6SrJhMmggAZxtp395RLNMxJ/L09gribyA+x+ZIbOgT+wDAGrHr8jSXDG9Mp8/l3lRQA3iXjBzQ3f1NAB8bewZCCCH2BckWhRCiIiigCyFERVBAF0KIiqCALoQQFeFAa4oCoHUKWSKgPOP1AXNSOzADH1MQZUNhfIU5N37dK0DqA9IRQE7mFykeGmTlPqqfWM/49iI1C6NJlA2zQfKi+VpaOTCf80RNJ2pp9cKJjI85ThQ4ADCXMR9xnw+Ij6L6sgOk7TCX8cRv0xmvNzpN1DGRwoQlzeoO+LyZpCc695jiCQAapI99n6gvsh3ra4YqF56QrczT4ZAlAQO4wqte43NYL9KxLih3umt0hy6EEBVBAV0IISqCAroQQlQEBXQhhKgICuhCCFERFNCFEKIiHKxs0QHrpeVyg1762rJR8qRZq/10bb7lAZeC3RikE3rVLV13EgAagWyx42lp140Bl8Sx+bGalABws59O9tPqc4Fku8/d2x1BI8WSNZXBd2WJjfrBvQQfw6WWXGAHDJzIxALJZ0n82iXb2hyTnncZyCOjZFEd4qPQ5710Xyc4Fsp+en45scFOtPppyR5LMBf1RfaJ+hilB+eEp+0wCJJzsT6PZMFR4d4x0R26EEJUBAV0IYSoCAroQghRERTQhRCiIiigCyFERVBAF0KIinCgskVzIGflJ1tpydDSBq/N987GsWT7PKkNCAA50Qwt11bomKbxDG1MOnWjP0vHvNM9kWx/u3Ocjllozyfbl7vcPuuB5JNJ1eI6pOnDJcrcmBFpYJRZktmbZdcEgD7SclQAWKfb43NgtVpXndfmvNqfS7YzfwPAQvco7bvSOZLeT5sfW8vt9PHQ6fFTvddP39exDKgA0Ch4NsF+UHeVwTKQsvMVANqkLm4z4+drJEG82kvbe7Gb9ivAz792l0sqrUtquPJp7xrdoQshREVQQBdCiIqggC6EEBVBAV0IISqCAroQQlSEHVUuZvYkgIcBLLr7vcO24wD+GsDdAN4C8EvuvrTjtvpAQcQIxc206mJ1hqs43qyn1QODIInTUi+dAOtkka5jCcSKjJIk9LnRm6FjFjvpVfMrLb6afrOTTkS21uaqi7IX1MwcpG2UZVxVUNTSdlhp8Dmslem+9V4wpp/uu9HnNj2er9O+OaJ6ihQUzK8rA348XivT/rvcTasnAOBym/ddbaXVLMuB8qvVSasreiU/1WkOriC/VE6OBQBYI8fDWqC6WiPHw0qPf9ejRVr5NZ3z2qUs8RsALJfp2HCpxX10ZS3t8846/661jbRhg9Klu2Y3d+hPAXjoXW2PA3jR3e8B8OLwvRBCiAmyY0B395cA3HhX8yMAnh6+fhrAp/d3WkIIIfbKqM/Qz7j7AgAM/57evykJIYQYhVu+KGpm583sgpld6LX4c04hhBDjMWpAv2JmtwPA8O8i+6C7P+Hu59z9XG2KL2oJIYQYj1ED+nMAHhu+fgzA1/dnOkIIIUZlN7LFrwJ4EMBJM7sI4IsA/gDAM2b2KwDeBvCZ3ezM+kBzOa2R6s2mry2DOpf/LCItJ9oIEuNcnk6Pma2zrGFALUgK1SMyqCgx1gqRGm4EEsRumyTG6nJpIkj9VgAAS6gVJGRqFem+9Saf92qT1H1tBvK/Zvp/cscbR+mYuYLXhJ3KSXKuoBIpky1GcsslkqhpqcNrxUYSxPWN9L5KciwAADrpeVuXHwtGJKxRWcxeLZB8NtKSxtYUt93NqbQdrk9z280WaXlio8YTh0WJ5Ng5G/loZSXdZ8s8BhUr6TnU18YvNrpjQHf3R0nXT429dyGEEPuGfikqhBAVQQFdCCEqggK6EEJUBAV0IYSoCAdagi7rOZo30ivgvSZZac/4NadDEj+tbvCvtTaVXjXP6jzZUFSWzcmqeb/k8/Y2USJ0+JiM9BVB2aogpxgnuMQPiFkHDW7v9WZaObBB1C8AcKNJkiQ1eZKkZlASrZ7v3RCsBF0nSHLFyo512lzxMAiOVSNlGWstrtRgeamyMpCsjJCcy4PI0a+nD6J+MzhOptI22pjix0lOztkocRg7XwGgT5LZ9QMfZavpMY0lfiI1ltLxpLk0ygn7rvmMvQUhhBDvCxTQhRCiIiigCyFERVBAF0KIiqCALoQQFUEBXQghKsKByhatN0DjSrpwnudpqZr1+RRzIt8qV/mY/lRaZsQkeQAQlCGkkq8iUCBlnfS8g1KIyKgcjY8JcopRou86IOq7QT2QgjWYhI0nFes30jtaa3D56GoRfNl8hKRHJGEV+kGSKyItzdt8TBH01dKlUJHzPGT0OMnLwAYjyBaj84UdD/1GcJw007brkfMV4MdjGSQOC8rIIiPJ7IK8fSjW0mMay3xHU9fTBm/cCALALtEduhBCVAQFdCGEqAgK6EIIUREU0IUQoiIooAshREVQQBdCiIpwsLLFsof88vVk35SnZT55m9fzq6+lp1/O8OtUn5Q17BeBRivoYjKorMdlS3k33ZcF0jJSFjPcTyTRokNC2SKRowW2o/aOJGykb8DLtGJQ4/I2Uh40hEk+jSd1pBLSPJC91dqBz4k8sdbmEk16bEXHySjZFvPAfyPIFlm2VXb8RPuJ5haREd/mHW672ka6r7HCdcvNa2l5Yu3KTT65XaI7dCGEqAgK6EIIUREU0IUQoiIooAshREVQQBdCiIqwo8rFzJ4E8DCARXe/d9j2uwB+FcDV4ce+4O7P77QtL3voLVxO9uVlWiLQWJ+n2yuW0wqY/jSXQ7BkUYMiuLaNonLp85XxrJuWFWQlVy9YmV41t36QlCpKzsW+blDDldloUA8SKLH6ksQPQKCSCJKARUoWz4gaIhI2EduF6iWiUopUEnknUKy00j7POlxqk3WJuqIXHFsDJukJ7F0LauYWJAFeUHuWJWuLjhOWBGwwqsqFnLNZ4L9iI+2LfIUn2sqXVpPtg8Vrwex2x27u0J8C8FCi/Y/d/b7hvx2DuRBCiFvLjgHd3V8CcOMA5iKEEGIMxnmG/jkz+08ze9LMju3bjIQQQozEqAH9zwB8GMB9ABYA/CH7oJmdN7MLZnahRPCTOSGEEGMxUkB39yvu3nf3AYA/B3B/8Nkn3P2cu58rEPyOVwghxFiMFNDN7PZtb38RwKv7Mx0hhBCjYk6SYv3fB8y+CuBBACcBXAHwxeH7+7Ap2nsLwK+5+8KOOzO7CuD7w7cnAYyv0/nhR3bYRHaQDbaQHd5rgw+6+6mdBu0Y0G8VZnbB3c9NZOfvI2SHTWQH2WAL2WF0G+iXokIIUREU0IUQoiJMMqA/McF9v5+QHTaRHWSDLWSHEW0wsWfoQggh9hc9chFCiIowkYBuZg+Z2ffM7A0ze3wSc5gEwzQJi2b26ra242b2gpn9z/BvpdMomNldZvYtM3vNzL5rZp8fth82OzTN7F/N7D+Gdvi9YfuhsgMAmFluZv9uZn8/fH8YbfCWmX3HzL5tZheGbXu2w4EHdDPLAfwpgJ8F8FEAj5rZRw96HhPiKbw3c+XjAF5093sAvDh8X2V6AH7T3T8C4OMAfn3o/8Nmhw6AT7r7x7D5m46HzOzjOHx2AIDPA3ht2/vDaAMA+MQwe+2WXHHPdpjEHfr9AN5w9zfdvQvgawAemcA8DhySufIRAE8PXz8N4NMHOaeDxt0X3P3fhq9XsXkin8Xhs4O7+9rwbTH85zhkdjCzOwH8PIC/2NZ8qGwQsGc7TCKgnwXwzrb3F4dth5UzW7+yHf49PeH5HBhmdjeAHwfwLziEdhg+avg2gEUAL7j7YbTDnwD4bfxgOZbDZgNg82L+TTN7xczOD9v2bIcdKxbdAlLlRCS1OWSY2SyAvwHwG+6+YkF1nKri7n0A95nZUQDPmtm9E57SgWJmW5XQXjGzByc8nUnzgLtfMrPTAF4ws9dH2cgk7tAvArhr2/s7AVyawDzeL1zZSnY2/Ls44fnccsyswGYw/yt3/9th86Gzwxbuvgzgn7C5vnKY7PAAgF8ws7ew+ej1k2b2lzhcNgAAuPul4d9FAM9i89H0nu0wiYD+MoB7zOxDZlYH8FkAz01gHu8XngPw2PD1YwC+PsG53HJs81b8ywBec/c/2tZ12OxwanhnDjObAvDTAF7HIbKDu/+Ou9/p7ndjMw78o7v/Mg6RDQDAzGbMbG7rNYBPYTOD7Z7tMJEfFpnZz2Hz2VkO4El3/9KBT2ICkMyVfwfgGQAfAPA2gM+4e2VL/pnZTwL4ZwDfwf8/N/0CNp+jHyY7/Bg2F7pybN5YPePuv29mJ3CI7LDF8JHLb7n7w4fNBmb2I9i8Kwc2H4N/xd2/NIod9EtRIYSoCPqlqBBCVAQFdCGEqAgK6EIIUREU0IUQoiIooAshREVQQBdCiIqggC6EEBVBAV0IISrC/wL089BcaKPlCQAAAABJRU5ErkJggg==",
|
|
"text/plain": [
|
|
"<Figure size 432x288 with 1 Axes>"
|
|
]
|
|
},
|
|
"metadata": {
|
|
"needs_background": "light"
|
|
},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"n = 10\n",
|
|
"s0 = np.zeros(816)\n",
|
|
"s0[:n] = s[:n]\n",
|
|
"C0 = np.dot(u, np.dot(np.diag(s0), vh))\n",
|
|
"A0_guess = vec2mat(np.linalg.pinv(C0) @ a, (16, 51))\n",
|
|
"plt.imshow(A0_guess)\n",
|
|
"plt.show()"
|
|
]
|
|
}
|
|
],
|
|
"metadata": {
|
|
"celltoolbar": "Slideshow",
|
|
"kernelspec": {
|
|
"display_name": "Python 3.10.8 ('nummethods')",
|
|
"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": "3.10.8"
|
|
},
|
|
"vscode": {
|
|
"interpreter": {
|
|
"hash": "8d489487a004dbbc79692a52da1077223bea5d5bb7772308e4a4df2310821984"
|
|
}
|
|
}
|
|
},
|
|
"nbformat": 4,
|
|
"nbformat_minor": 4
|
|
}
|