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{
"cells": [
{
"cell_type": "markdown",
"id": "ordered-morocco",
"metadata": {},
"source": [
"Given the matrix $\\mathbf\\Phi$ defined by"
]
},
{
"cell_type": "markdown",
"id": "funny-genome",
"metadata": {},
"source": [
"$\\mathbf\\Phi(\\mathbf x, M) =\n",
"\\begin{bmatrix}\n",
" (x_0)^0 & (x_0)^1 & \\cdots & (x_0)^{M - 1} \\\\\n",
" (x_1)^0 & (x_1)^1 & \\cdots & (x_1)^{M - 1} \\\\\n",
" \\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
" (x_{N - 1})^0 & (x_{N - 1})^1 & \\cdots & (x_{N - 1})^{M - 1}\n",
"\\end{bmatrix}\n",
"\\qquad \\mathbf x \\in \\mathbb{R}^N \\qquad M \\in \\mathbb{N} \\qquad \\mathbf\\Phi(\\mathbf x, M) \\in \\mathbb{R}^{N x M}$"
]
},
{
"cell_type": "markdown",
"id": "urban-texture",
"metadata": {},
"source": [
"implement a function in python using numpy which computes $\\mathbf\\Phi(\\mathbf x, M)$ without the use of `for` loops."
]
},
{
"cell_type": "markdown",
"id": "collectible-literacy",
"metadata": {},
"source": [
"To do so, note that\n",
"\n",
"$\\mathbf\\Phi(\\mathbf x, M) = \\mathbf A \\hspace{0.2em} \\hat\\diamond \\hspace{0.2em} \\mathbf B$\n",
"$\\qquad \\mathbf A =\\begin{bmatrix}\n",
" x_0 & x_0 & \\cdots & x_0 \\\\\n",
" x_1 & x_1 & \\cdots & x_1 \\\\\n",
" \\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
" x_{N - 1} & x_{N - 1} & \\cdots & x_{N - 1}\n",
"\\end{bmatrix}\n",
"\\qquad \\mathbf B = \\begin{bmatrix}\n",
" 0 & 1 & \\cdots & M - 1 \\\\\n",
" 0 & 1 & \\cdots & M - 1 \\\\\n",
" \\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
" 0 & 1 & \\cdots & M - 1\n",
"\\end{bmatrix}$"
]
},
{
"cell_type": "markdown",
"id": "figured-scoop",
"metadata": {},
"source": [
"where $\\hat\\diamond$ denotes element wise exponentiation."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "cloudy-athletics",
"metadata": {},
"outputs": [],
"source": [
"import numpy as np"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "recorded-female",
"metadata": {},
"outputs": [],
"source": [
"# Reference implementation with for loops\n",
"def phi(x, M):\n",
" phi = np.zeros((x.size, M))\n",
" for n in range(x.size):\n",
" for m in range(M):\n",
" phi[n, m] = x[n]**m\n",
" return phi"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "macro-jamaica",
"metadata": {},
"outputs": [],
"source": [
"phi(10 * np.arange(4), 3)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "furnished-index",
"metadata": {},
"outputs": [],
"source": [
"# Using elementwise exponentiation of arrays that have the same shape, wastes time and memory on copies\n",
"def phi(x, M):\n",
" A = np.repeat(x.reshape(x.size, 1), M, axis = 1)\n",
" B = np.repeat(np.arange(M).reshape(1, M), x.size, axis = 0)\n",
" return np.power(A, B)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "japanese-setting",
"metadata": {},
"outputs": [],
"source": [
"phi(10 * np.arange(4), 3)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "finite-jesus",
"metadata": {},
"outputs": [],
"source": [
"# Using elementwise exponentiation with broadcasting\n",
"def phi(x, M):\n",
" return np.power(x.reshape(x.size, 1), np.arange(M).reshape(1, M))"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "liberal-community",
"metadata": {},
"outputs": [],
"source": [
"phi(10 * np.arange(4), 3)"
]
}
],
"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": "3.7.9"
}
},
"nbformat": 4,
"nbformat_minor": 5
}