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    "# Theory of LSF"
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    "While this chapter focuses heavily on document similarity--that is, documents expressed as a space of sets of k-shingles therefore making Jaccard distance the appropriate metric, the concepts can be adapted to a different space and/or a different metric for distance (e.g., Euclidian space and distance). The choice of space and, hence, distance measure are dependent on the similarity taks being solved. The core requirements for a family of functions to be considered locality-sensitive are:\n",
    "\n",
    "1. The functions must be statistically independent--that is, the probability for some response to be given by two or more functions $P(A \\cap B)$ can be estimated to be the product of the probabilities of each of the functions yielding that response, $P(A)P(B)$.\n",
    "2. The functions must--with high probability--make similar pairs to be candidate pairs as opposed to dissimilar pairs.\n",
    "3. The functions must be efficient in two aspects:\n",
    "    * candidate pairs are identified faster than an *exhaustive* comparison of all pairs.\n",
    "    * functions can be combined (i.e., according to som OR- and/or AND-construction) to reduce false positives and false negatives.\n",
    "    \n",
    "Furthermore, quoting directly from {cite:ps}`rajaraman2011mining`, if we let \"$d_1 < d_2$ be two distances according to some distance measure d. A family $F$ of functions is said to be $(d_1, d_2, p_1, p_2)$-*sensitive* if for every $f$ in $F$:\n",
    "\n",
    "1. If $d(x, y) ≤ d_1$, then the probability that $f(x) = f(y)$ is at least $p_1$.\n",
    "2. If $d(x, y) ≥ d_2$, then the probability that $f(x) = f(y)$ is at most $p_2$.\n",
    "\n",
    "\n",
    "Now, we recall that the minhash family of functions estimate the Jaccard similarity--which is the probability that two sets are similar, $h(S_1) = h(S_2)$, given their items. Relating it to the above definition, we can view such similarity as to be indicative of two sets being close in distance on a space of sets. In such a space, the probability that two sets are similar--that is, their Jaccard similarity, $\\text{SIM}(S_1, S_2)$ can be transformed into the Jaccard distance measure by $1 - \\text{SIM}(S_1, S_2)$. Hence, the minhash family of functions is said to be $(d_1, d_2, 1 - d_1, 1 - d_2)$-*sensitive* where $d$ is the Jaccard distance."
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