lsh notebook
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{
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"cells": [
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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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"outputs": [],
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"source": [
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"import numpy as np\n",
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"import floky\n",
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"from tqdm import tqdm\n",
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"import matplotlib.pyplot as plt\n",
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"import glob\n",
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"import imageio\n",
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"from skimage.transform import resize\n",
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"from scipy import stats\n",
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"import time"
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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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"# LSH Definition\n",
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"We hash some some vectors $p$ into buckets with $h$ coming from a hash function family $H$ $h: M \\mapsto S$. The hash functions map from vector space to a bucket. We hope to be able to query $q$ approximate nearest neighbors for cosine similarity, maximum inner product, l2 distance etc. We do this by making more similar vectors collide in the same hash buckets. \n",
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"\n",
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"Given that:\n",
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"\n",
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"* The approximation is defined by a factor $c > 1$.\n",
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"* The distance threshold $R > 0$.\n",
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"\n",
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"### It is guaranteed:\n",
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"\n",
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"* If $Distance(p, q) \\le R$ then $h(p) = h(q)$ with a probability $\\ge P_1$. $\\quad$ \n",
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"* If $Distance(p, q) \\ge cR$ then $h(p) = h(q)$ with a probability $\\le P_2$\n",
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"\n",
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"## LSH for Approximate nearest neighbor\n",
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"Only useful if $P_q \\gt P_2$\n",
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"\n",
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"* $K$: Bucket width: (In random projections this is the number of random projections)\n",
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"* $L$: Number of hash tables. When querying the union of the collisions in all the hash tables is returned\n",
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"\n",
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"### Guarantees:\n",
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"\n",
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"* Preprocessing time: $\\mathcal{O}(nLKt)$ ($t$ is the time of evaluating the hash function)\n",
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"* Space: $\\mathcal{O}(nL)$ (and the data points space)\n",
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"* query time: $\\mathcal{O}(LKt + dn P_2^K)$\n",
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"* Succes of finding point $q$ within distance $cR$ (if it exists) w/ probability $\\ge 1 - (1 - P_1^K)^L$\n",
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"\n",
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"So given hyperparameters $P_1, P_2$ we can derive $K = \\frac{\\log n}{\\log 1 / P_2}$\n",
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"\n",
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"and $L = n^{\\rho}$, where $\\rho = \\frac{\\log P_1}{\\log P_2}$\n",
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"\n",
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"## Parameter estimation\n",
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"If $P_1$ is known, see (L2 and SRP below), *L* can be derived for a chosen value of *k* and a success probability $1-\\delta$. \n",
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"\n",
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"$$ L = \\frac{\\log \\delta}{ \\log(1 - P_1^k)} $$\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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"# L2 \n",
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"\n",
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"$$h(x) = \\lfloor \\frac{ w^Tx + b}{r} \\rfloor$$\n",
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"\n",
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"$$ P(h(x) = h(y)) = 1 - 2\\Phi(-r/d) - \\frac{2}{\\sqrt{2 \\pi} r / d } (1 - e^{r^2/ (2d^2)})$$\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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"# Cosine similarity (Signed Random Projections)\n",
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"$$ h(x) = sign(w^Tx) $$\n",
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"$$ P(h(x) = h(y)) = 1 - \\frac{\\theta}{\\pi} $$\n",
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"\n",
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"$\\theta = \\cos^{-1}(\\text{cosine similarity})$\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {},
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"outputs": [],
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"source": [
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"def collision_prob_l2(r, distance):\n",
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" # https://arxiv.org/pdf/1411.3787.pdf eq. 10\n",
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" a = 1 - 2 * stats.norm.cdf(-r / distance)\n",
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" b = 2 / (np.sqrt(2 * np.pi) * r / distance) * (1 - np.exp(-r**2 / (2 * distance**2)))\n",
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" return a - b\n",
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"\n",
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"def collision_prob_cosine(sim):\n",
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" return 1 - np.arccos(sim) / np.pi"
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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": 3,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"0.609548422215397"
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]
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},
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"execution_count": 3,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"collision_prob_l2(2, 1)"
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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 n_hash_tables(p1, p2, n):\n",
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" return n**(np.log(p1) / np.log(p2))\n",
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" \n",
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"def n_projections(p2, n):\n",
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" return np.log(n) / (np.log(1 / p2))\n",
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" \n",
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"def p_query(p1, k, L):\n",
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" return (1 - (1 - p1**k)**L)\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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"# Simhash for Cosine similarity and hamming distance\n",
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"\n",
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"https://www.cs.princeton.edu/courses/archive/spr04/cos598B/bib/CharikarEstim.pdf\n",
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"\n",
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"https://arxiv.org/pdf/1411.3787.pdf\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"metadata": {},
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"outputs": [],
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"source": [
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"def cosine_sim(a, b):\n",
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" return np.dot(a, b) / (np.linalg.norm(a) * np.linalg.norm(b))"
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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": 6,
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"metadata": {},
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"outputs": [],
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"source": [
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"def hash_l2(v, r, seed=1):\n",
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" np.random.seed(seed)\n",
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" w = np.random.randn(*v.shape)\n",
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" b = np.random.uniform(0., r)\n",
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" h = np.floor((w.dot(v) + b) / r) \n",
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" return h\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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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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"0.7468166888933649\n"
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]
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},
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{
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"data": {
|
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"text/plain": [
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"0.8466788083683645"
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]
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},
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"execution_count": 7,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"p1 = collision_prob_cosine(0.7)\n",
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"print(p1)\n",
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"p_query(p1, 9, 25)"
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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": 8,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"image/png": 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truncated
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"text/plain": [
|
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"<Figure size 432x288 with 2 Axes>"
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]
|
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},
|
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"metadata": {
|
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"needs_background": "light"
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},
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"output_type": "display_data"
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},
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{
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"data": {
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"image/png": 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|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 2 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
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||||
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|
||||
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||||
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"data": {
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|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 2 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
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|
||||
},
|
||||
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|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 2 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light"
|
||||
},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"for sim in [0.5, 0.7, 0.8, 0.9]:\n",
|
||||
" p1 = collision_prob_cosine(sim)\n",
|
||||
" k = np.linspace(1, 20)\n",
|
||||
" L = np.linspace(1, 25)\n",
|
||||
"\n",
|
||||
" kk, ll = np.meshgrid(k, L)\n",
|
||||
"\n",
|
||||
" plt.title(f\"sim: {sim}\")\n",
|
||||
" plt.contourf(kk, ll, p_query(p1, kk, ll), levels=10)\n",
|
||||
" plt.xlabel(\"hash projections: k\")\n",
|
||||
" plt.ylabel(\"hash tables: L\")\n",
|
||||
" plt.colorbar()\n",
|
||||
" plt.show()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Gap amplification\n",
|
||||
"\n",
|
||||
"$$P(h(x) = h(y)) = 1 - (1 - P_h^k)^L$$\n",
|
||||
"\n",
|
||||
"where $P_h$ is similarity function."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 11,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"def collision_prob(k, L, ph):\n",
|
||||
" return 1 - (1 - ph**k)**L"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 31,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"<matplotlib.legend.Legend at 0x7ffb155ee780>"
|
||||
]
|
||||
},
|
||||
"execution_count": 31,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"<Figure size 1008x576 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light"
|
||||
},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"k = 1\n",
|
||||
"L = 1\n",
|
||||
"ph = np.linspace(0, 1)\n",
|
||||
"plt.figure(figsize=(14, 8))\n",
|
||||
"plt.title(\"Gap amplification\")\n",
|
||||
"plt.xlabel(\"similarity\")\n",
|
||||
"plt.ylabel(\"collision probability\");\n",
|
||||
"plt.plot(ph, collision_prob(k, L, ph), label=f\"k={k} L={L}\")\n",
|
||||
"k = 10\n",
|
||||
"L = 1\n",
|
||||
"plt.plot(ph, collision_prob(k, L, ph), label=f\"k={k} L={L}\")\n",
|
||||
"k = 10\n",
|
||||
"L = 13\n",
|
||||
"plt.plot(ph, collision_prob(k, L, ph), label=f\"k={k} L={L}\")\n",
|
||||
"plt.legend()"
|
||||
]
|
||||
}
|
||||
],
|
||||
"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": "3.7.3"
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 4
|
||||
}
|
||||
Reference in new issue
Block a user