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429 lines
20 KiB
Plaintext
429 lines
20 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 3,
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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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"import networkx as nx\n",
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"import time\n",
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"import math\n",
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"import pandas as pd\n",
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"import scipy as sp\n",
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"import plotly.express as px\n",
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"import plotly.graph_objs as go\n",
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"from scipy.sparse import *\n",
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"from scipy import linalg\n",
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"from scipy.sparse.linalg import norm\n",
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"from scipy.optimize import least_squares"
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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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"## Arnoldi \n",
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"\n",
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"This is a copy of the algorithm defined and tested in the notebook `algo2_testing`. It's an implementation of the Algorithm 2 from the paper. It's needed in this notebook since this function is called by the `algo4` function. It's implemented to return exactly what's needed in the `algo4` function.\n",
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"\n",
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"Everything will be reorganized in the main.py file once everything is working."
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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": 4,
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"metadata": {},
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"outputs": [],
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"source": [
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"def Arnoldi(A,v0,m):\n",
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" v = v0\n",
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" beta = np.linalg.norm(v)\n",
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" v = v/beta\n",
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" H = sp.sparse.lil_matrix((m+1,m)) \n",
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" V = sp.sparse.lil_matrix((A.shape[0],m+1))\n",
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" V[:,0] = v # each column of V is a vector v\n",
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"\n",
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" for j in range(m):\n",
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" w = A @ v \n",
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" for i in range(j):\n",
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" H[i,j] = v.T @ w # tmp is a 1x1 matrix, so it's O(1) in memory\n",
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" w = w - H[i,j]*v \n",
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" \n",
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" H[j+1,j] = np.linalg.norm(w)\n",
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"\n",
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" if H[j+1,j] == 0:\n",
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" # print(\"Arnoldi breakdown\")\n",
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" m = j\n",
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" v = 0\n",
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" break\n",
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" else:\n",
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" if j < m-1:\n",
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" v = w/H[j+1,j]\n",
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" V[:,j+1] = v\n",
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"\n",
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" return V, H, v, beta, j "
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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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"# Algorithm 4 testing\n",
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"\n",
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"This algorithm is based on the \"Algorithm 4\" of the paper, the pseudocode provided by the authors is the following \n",
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"\n",
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"\n",
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"\n",
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"Line 14 is particularly tricky to understand, not working for now. Need to figure out how to solve that linear system. My idea was to do something like that\n",
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"\n",
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"\n",
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"\n",
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"And use the `sp.sparse.linalg.spsolve` function to solve the linear system as $Ax=0$ where $A$ is $[\\bar H_m^i ~ | ~ z]$ but it returns an array of zeros. So the idea it's wrong"
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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": 5,
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"metadata": {},
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"outputs": [],
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"source": [
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"# def Algo4_OLD(Pt, v, m, a: list, tau, maxit: int, x):\n",
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" \n",
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"# # I'm using a non declared variable n here , it's declared in the next cell when I call this function. This will be fixed later in the main.py file\n",
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"\n",
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"# iter , mv = 1, 0\n",
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"# res = np.zeros(len(a)) \n",
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"\n",
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"# # I'm defining 3 canonical vectors of different sizes. It's probably stupid, will be fixed once the algorithm actually works\n",
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"\n",
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"# H_e1 = np.zeros((m+1,1)) # canonical basis vector of size H.shape[0]\n",
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"# H_e1[0] = 1\n",
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"\n",
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"# V_e1 = np.zeros((n,1)) # canonical basis vector of size V.shape[0]\n",
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"# V_e1[0] = 1\n",
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"\n",
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"# s_e1 = np.zeros((len(a),1)) # canonical basis vector of size s.shape[0]\n",
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"# s_e1[0] = 1\n",
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"\n",
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"# def find_k(res): # function to find the index of the largest element in res\n",
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"# k = 0\n",
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"# for i in range(len(a)):\n",
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"# if res[i] == max(res):\n",
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"# k = i\n",
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"# break\n",
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"# return k\n",
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"\n",
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"# def compute_gamma(res, a, k): # function to compute gamma\n",
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"# gamma = np.zeros(len(a))\n",
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"# for i in range(len(a)):\n",
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"# if i != k:\n",
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"# gamma[i] = (res[i]*a[k])/(res[k]*a[i])\n",
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"# else:\n",
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"# gamma[i] = 0\n",
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"# return gamma\n",
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"\n",
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"# # compute the residual vector\n",
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"\n",
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"# # create a sp.sparse.eye matrix of size Pt\n",
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"# I = sp.sparse.eye(Pt.shape[0], Pt.shape[1], format='lil')\n",
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"# for i in range(len(a)):\n",
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"# r = ((1-a[i])/a[i])*v - ((1/a[i])*I - Pt) @ x\n",
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"# res[i] = a[i]*norm(r)\n",
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"\n",
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"\n",
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"# while max(res) >= tau and iter <= maxit:\n",
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"# k = find_k(res)\n",
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"# gamma = compute_gamma(res, a, k)\n",
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"# V, H, v, beta, j = Arnoldi((1/a[k])*I - Pt, r, m)\n",
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"\n",
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"# mv = mv + j\n",
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"\n",
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"# # compute y as the minimizer of || beta*e1 - Hy ||_2 using the least squares method\n",
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"# y = sp.sparse.linalg.lsqr(H, beta*H_e1)[0]\n",
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"\n",
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"# # reshape y to be a column vector\n",
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"# y = y.reshape(y.shape[0],1)\n",
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"\n",
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"# # update x \n",
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"# x += V[:,0:y.shape[0]] @ y\n",
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"\n",
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"# # compute the residual vector\n",
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"# res[k] = a[k]*np.linalg.norm(beta*V_e1 - V[:,0:y.shape[0]] @ y)\n",
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" \n",
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"# # for i in range(len(a)) but not k\n",
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"# for i in range(len(a)):\n",
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"# if i != k and res[i] >= tau:\n",
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"# # Compute H as described in the paper\n",
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"# H = H + ((1-a[i])/a[i] - (1-a[k])/a[k])*sp.sparse.eye(H.shape[0], H.shape[1], format='lil') \n",
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"\n",
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"# z = beta*H_e1 - H @ y # define z as in the paper (page 9)\n",
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"# A_tmp = sp.sparse.hstack([H, z]) # stack H and z, as in the paper, to solve the linear system (?)\n",
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"# A_tmp = A_tmp.tocsc() # Convert A to CSC format for sparse solver\n",
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"\n",
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"# # What should I put here? What does it mean in the paper the line 14 of the pseudocode?\n",
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"# result = sp.sparse.linalg.spsolve(A_tmp, np.zeros(A_tmp.shape[0])) # if I solve this, I get a vector of zeros.\n",
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"# print(result)\n",
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" \n",
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"# # I don't know if the code below is correct since I don't get how to solve the linear system above, so I'm unsure about what y and gamma should be. For now it's commented out.\n",
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"\n",
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"# # # update x\n",
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"# # x += V[:,0:y.shape[0]] @ y\n",
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"# # # update the residual vector\n",
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"# # res[i] = (a[i]/a[k])*gamma[k]*res[k] \n",
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"\n",
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"# iter = iter + 1\n",
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"\n",
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"# return x, iter, mv"
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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": 109,
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"metadata": {},
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"outputs": [],
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"source": [
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"def find_k(res): # function to find the index of the largest element in res\n",
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" k = 0\n",
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" for i in range(len(a)):\n",
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" if res[i] == max(res):\n",
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" k = i\n",
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" break\n",
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" return k\n",
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"\n",
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"def compute_gamma(res, a, k): # function to compute gamma\n",
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" gamma = np.zeros(len(a))\n",
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" for i in range(len(a)):\n",
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" if i != k:\n",
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" gamma[i] = (res[i]*a[k])/(res[k]*a[i])\n",
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" else:\n",
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" gamma[i] = 0\n",
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" return gamma"
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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": 123,
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"metadata": {},
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"outputs": [],
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"source": [
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"def Algo4(Pt, v, m, a: list, tau, maxit: int, x):\n",
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"\n",
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" x = sp.sparse.lil_matrix((Pt.shape[0], len(a)))\n",
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" mv = 0\n",
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"\n",
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" # compute the residual vector\n",
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" I = sp.sparse.eye(Pt.shape[0], Pt.shape[1], format='lil')\n",
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" res = np.zeros(len(a))\n",
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" for i in range(len(a)):\n",
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" r = sp.sparse.linalg.spsolve(I - a[i]*Pt, v)\n",
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" res[i] = a[i]*np.linalg.norm(r)\n",
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"\n",
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" iter = 0\n",
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" for _ in range(maxit):\n",
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" print(\"\\niter: \", iter)\n",
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" k = find_k(res)\n",
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" gamma = compute_gamma(res, a, k)\n",
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" \n",
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" # Run Arnoldi\n",
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" V, H, v, beta, j = Arnoldi((1/a[k])*I - Pt, r, m)\n",
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"\n",
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" H = H[:-1,:]\n",
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" V = V[:,:-1]\n",
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" mv = mv + j\n",
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"\n",
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" # compute y as the minimizer of || beta*e1 - Hy ||_2 using the least squares method\n",
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" H_e1 = np.zeros(H.shape[0])\n",
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" H_e1[0] = 1\n",
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" y = sp.sparse.linalg.lsqr(H, beta*H_e1)[0]\n",
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" \n",
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" # Update res[k]\n",
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" y = y.reshape(y.shape[0],1)\n",
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" V_e1 = np.zeros(V.shape[0])\n",
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" V_e1[0] = 1\n",
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"\n",
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" res[k] = a[k]*np.linalg.norm(beta*V_e1 - V @ y) \n",
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" print(\"res[\", k, \"] = \", res[k]) \n",
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" \n",
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" for i in range(len(a)):\n",
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" print(\"res = \", res)\n",
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" if i != k and res[i] >= tau:\n",
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" print(\"res[\", i, \"] is greater than tau = \", tau, \" at iteration \", iter, \" entering the for loop\")\n",
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"\n",
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" # Compute H as described in the paper\n",
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" H_new = H + ((1-a[i])/a[i] - (1-a[k])/a[k])*sp.sparse.eye(H.shape[0], H.shape[1], format='lil')\n",
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"\n",
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" # Compute z as described in the paper\n",
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" tmp = H_e1*beta \n",
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" z1, z2 = tmp.reshape(tmp.shape[0],1), H @ y\n",
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" z = z1 - z2\n",
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"\n",
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" # Solve the linear system \n",
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" A = sp.sparse.hstack([H_new, z])\n",
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" b = gamma[i]*(beta*H_e1)\n",
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" b = b.reshape(b.shape[0],1)\n",
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" \n",
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" # use the least squares method to solve the linear system\n",
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" y_to_split = sp.sparse.linalg.lsqr(A, b)[0]\n",
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" \n",
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" # the last element of y_to_split is the last element of gamma[i], the other elements are the elements of y\n",
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" y = y_to_split[:-1].reshape(y_to_split[:-1].shape[0],1)\n",
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" gamma[i] = y_to_split[-1]\n",
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"\n",
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" x_new = x\n",
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" x_new[:,[k]] = x[:,[k]] + V @ y\n",
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"\n",
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" # update x\n",
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" x_new[:,i] = x[:,i] + V @ y\n",
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"\n",
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" # update the residual vector\n",
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" res[i] = (a[i]/a[k])*gamma[k]*res[k]\n",
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" print(\"res[\", i, \"] = \", res[i])\n",
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" \n",
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" else:\n",
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" if res[i] < tau:\n",
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" print(\"res[\", i, \"] = \", res[i], \" is smaller than tau = \", tau, \" at iteration \", iter)\n",
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"\n",
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" err = np.absolute(res).max()\n",
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" print(\"err = \", err)\n",
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" if err < tau:\n",
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" return x_new, iter, mv\n",
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" \n",
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" iter = iter + 1\n",
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" x = x_new\n",
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"\n",
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" raise Exception('Maximum number of iterations reached')\n",
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" \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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"Basic test case with random numbers to test the algorithm."
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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": 124,
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"metadata": {},
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"outputs": [],
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"source": [
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"n = 1000\n",
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"m = 1100\n",
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"tau = 1e-5\n",
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"a = [0.85, 0.9, 0.95, 0.99]\n",
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"\n",
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"x = sp.sparse.lil_matrix((n,1))\n",
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"x[0,0] = 1\n",
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"\n",
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"# generate a random graph\n",
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"G = nx.gnp_random_graph(n, 0.01)\n",
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"v = np.repeat(1.0 / 1000, 1000) # p is the personalization vector\n",
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"v = v.reshape(v.shape[0],1)\n",
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"\n",
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"A = nx.to_scipy_sparse_array(G, dtype=float)\n",
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"S = A.sum(axis=1) # S[i] is the sum of the weights of edges going out of node i\n",
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"S[S != 0] = 1.0 / S[S != 0] # S[i] is now the sum of the weights of edges going into node i\n",
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"Q = sp.sparse.csr_array(sp.sparse.spdiags(S.T, 0, *A.shape)) # Q is the matrix of edge weights"
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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": 125,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"\n",
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"iter: 0\n",
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"res[ 3 ] = 1.9500347533542841\n",
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"res = [0.02974037 0.03169717 0.03368163 1.95003475]\n",
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"res[ 0 ] is greater than tau = 1e-05 at iteration 0 entering the for loop\n",
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"res[ 0 ] = 0.0\n",
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"res = [0. 0.03169717 0.03368163 1.95003475]\n",
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"res[ 1 ] is greater than tau = 1e-05 at iteration 0 entering the for loop\n",
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"res[ 1 ] = 0.0\n",
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"res = [0. 0. 0.03368163 1.95003475]\n",
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"res[ 2 ] is greater than tau = 1e-05 at iteration 0 entering the for loop\n",
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"res[ 2 ] = 0.0\n",
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"res = [0. 0. 0. 1.95003475]\n",
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"err = 1.9500347533542841\n",
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"\n",
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"iter: 1\n",
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"res[ 3 ] = 1.9500347533542841\n",
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"res = [0. 0. 0. 1.95003475]\n",
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"res[ 0 ] = 0.0 is smaller than tau = 1e-05 at iteration 1\n",
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"res = [0. 0. 0. 1.95003475]\n",
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"res[ 1 ] = 0.0 is smaller than tau = 1e-05 at iteration 1\n",
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"res = [0. 0. 0. 1.95003475]\n",
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"res[ 2 ] = 0.0 is smaller than tau = 1e-05 at iteration 1\n",
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"res = [0. 0. 0. 1.95003475]\n",
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"err = 1.9500347533542841\n",
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"\n",
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"iter: 2\n",
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"res[ 3 ] = 1.9500347533542841\n",
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"res = [0. 0. 0. 1.95003475]\n",
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"res[ 0 ] = 0.0 is smaller than tau = 1e-05 at iteration 2\n",
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"res = [0. 0. 0. 1.95003475]\n",
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"res[ 1 ] = 0.0 is smaller than tau = 1e-05 at iteration 2\n",
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"res = [0. 0. 0. 1.95003475]\n",
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"res[ 2 ] = 0.0 is smaller than tau = 1e-05 at iteration 2\n",
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"res = [0. 0. 0. 1.95003475]\n",
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"err = 1.9500347533542841\n",
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"\n",
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"iter: 3\n"
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]
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},
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{
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"ename": "KeyboardInterrupt",
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"evalue": "",
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"output_type": "error",
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"traceback": [
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"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
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"\u001b[0;31mKeyboardInterrupt\u001b[0m Traceback (most recent call last)",
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"\u001b[0;32m/tmp/ipykernel_85082/3677688099.py\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0miter\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mmv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mAlgo4\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mQ\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mv\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mm\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mtau\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m100\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
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"\u001b[0;32m/tmp/ipykernel_85082/4220477307.py\u001b[0m in \u001b[0;36mAlgo4\u001b[0;34m(Pt, v, m, a, tau, maxit, x)\u001b[0m\n\u001b[1;32m 18\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 19\u001b[0m \u001b[0;31m# Run Arnoldi\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 20\u001b[0;31m \u001b[0mV\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mH\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mv\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mbeta\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mj\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mArnoldi\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m/\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mI\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0mPt\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mm\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 21\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 22\u001b[0m \u001b[0mH\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mH\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
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"\u001b[0;32m/tmp/ipykernel_85082/1990871025.py\u001b[0m in \u001b[0;36mArnoldi\u001b[0;34m(A, v0, m)\u001b[0m\n\u001b[1;32m 10\u001b[0m \u001b[0mw\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mA\u001b[0m \u001b[0;34m@\u001b[0m \u001b[0mv\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 11\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mi\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mj\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 12\u001b[0;31m \u001b[0mH\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mj\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mv\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mT\u001b[0m \u001b[0;34m@\u001b[0m \u001b[0mw\u001b[0m \u001b[0;31m# tmp is a 1x1 matrix, so it's O(1) in memory\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 13\u001b[0m \u001b[0mw\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mw\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0mH\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mj\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mv\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 14\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
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"\u001b[0;32m~/.local/lib/python3.10/site-packages/scipy/sparse/_lil.py\u001b[0m in \u001b[0;36m__setitem__\u001b[0;34m(self, key, x)\u001b[0m\n\u001b[1;32m 320\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_set_arrayXarray\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mrow\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcol\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 321\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 322\u001b[0;31m \u001b[0;32mdef\u001b[0m \u001b[0m__setitem__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mkey\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 323\u001b[0m \u001b[0;31m# Fast path for simple (int, int) indexing.\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 324\u001b[0m if (isinstance(key, tuple) and len(key) == 2 and\n",
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"\u001b[0;31mKeyboardInterrupt\u001b[0m: "
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]
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}
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],
|
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"source": [
|
|
"x, iter, mv = Algo4(Q, v, m, a, tau, 100, x)"
|
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]
|
|
}
|
|
],
|
|
"metadata": {
|
|
"kernelspec": {
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"display_name": "Python 3.10.6 64-bit",
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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": {
|
|
"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",
|
|
"name": "python",
|
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"nbconvert_exporter": "python",
|
|
"pygments_lexer": "ipython3",
|
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"version": "3.10.6"
|
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},
|
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"orig_nbformat": 4,
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"vscode": {
|
|
"interpreter": {
|
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"hash": "916dbcbb3f70747c44a77c7bcd40155683ae19c65e1c03b4aa3499c5328201f1"
|
|
}
|
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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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}
|