{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Alternating least squares for Tucker model\n",
    "\n",
    "```\n",
    "Copyright 2022 National Technology & Engineering Solutions of Sandia,\n",
    "LLC (NTESS). Under the terms of Contract DE-NA0003525 with NTESS, the\n",
    "U.S. Government retains certain rights in this software.\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function `pyttb.tucker_als()` computes the best rank-$(R_1,R_2,\\ldots,R_n)$ approximation of tensor $\\mathcal{X}$, according to the specified dimensions in the vector $(R_1,R_2,\\ldots,R_n)$. The form of $\\mathcal{X}$ can be a `tensor`, `sptensor`, `ktensor`, or `ttensor`. The result is a `ttensor`.\n",
    "\n",
    "The method is originally from Tucker (1966) and later revisited in De Lathauwer et al. (2000).\n",
    "\n",
    "* L. R. Tucker, Some mathematical notes on three-mode factor analysis, Psychometrika, 31:279-311, 1966, http://dx.doi.org/10.1007/BF02289464\n",
    "* L. De Lathauwer, B. De Moor, J. Vandewalle, On the best rank-1 and rank-(R_1, R_2, R_N) approximation of higher-order tensors, SIAM J. Matrix Analysis and Applications, 21:1324-1342, 2000, http://doi.org/10.1137/S0895479898346995\n",
    "\n",
    "Note: Oftentimes it's better to use `pyttb.hosvd()` instead."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import os\n",
    "import sys\n",
    "import pyttb as ttb\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Create a data tensor of shape (5, 4, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.random.seed(0)  # Set seed for reproducibility\n",
    "X = ttb.sptenrand(\n",
    "    [5, 4, 3], nonzeros=10\n",
    ")  # Create a tensor with 10 nonzeros using the 'nonzeros' param.\n",
    "X"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Create an approximation with all ranks equal to 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "T = ttb.tucker_als(X, 2)  # best rank(2,2,2) approximation\n",
    "T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Create an approximation with specific ranks of [2, 2, 1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "T = ttb.tucker_als(X, [2, 2, 1])  # best rank(2,2,1) approximation\n",
    "T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Use a different ordering of the dimensions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "T = ttb.tucker_als(X, 2, dimorder=[2, 1, 0])\n",
    "T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Use the `\"nvecs\"` initialization method\n",
    "This initialization is more expensive but generally works very well."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "T = ttb.tucker_als(X, 2, dimorder=[0, 1, 2], init=\"nvecs\")\n",
    "T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Specify the initial guess manually"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "U0 = [np.random.rand(5, 2), np.random.rand(4, 2), np.random.rand(3, 2)]\n",
    "T = ttb.tucker_als(X, 2, dimorder=[0, 1, 2], init=U0)\n",
    "T"
   ]
  }
 ],
 "metadata": {},
 "nbformat": 4,
 "nbformat_minor": 2
}