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From: Yunni <git@git.apache.org>
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Subject: [GitHub] spark pull request #15795: [SPARK-18081][ML][DOCS] Add user guide for Locali...
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Message-Id: <20161202231529.5E97FE04BB@git1-us-west.apache.org>
Date: Fri,  2 Dec 2016 23:15:29 +0000 (UTC)
archived-at: Fri, 02 Dec 2016 23:15:31 -0000

Github user Yunni commented on a diff in the pull request:

    https://github.com/apache/spark/pull/15795#discussion_r90736831
  
    --- Diff: docs/ml-features.md ---
    @@ -1478,3 +1478,139 @@ for more details on the API.
     {% include_example python/ml/chisq_selector_example.py %}
     </div>
     </div>
    +
    +# Locality Sensitive Hashing
    +[Locality Sensitive Hashing(LSH)](https://en.wikipedia.org/wiki/Locality-sensitive_hashing) is an important class of hashing techniques, which is commonly used in clustering, approximate nearest neighbor search and outlier detection with large datasets.
    +
    +The general idea of LSH is to use a family of functions (we call them LSH families) to hash data points into buckets, so that the data points which are close to each other are in the same buckets with high probability, while data points that are far away from each other are very likely in different buckets. A formal definition of LSH family is as follows:
    +
    +In a metric space `(M, d)`, where `M` is a set and `d` is a distance function on `M`, an LSH family is a family of functions `h` that satisfy the following properties:
    +`\[
    +\forall p, q \in M,\\
    +d(p,q) < r1 \Rightarrow Pr(h(p)=h(q)) \geq p1\\
    +d(p,q) > r2 \Rightarrow Pr(h(p)=h(q)) \leq p2
    +\]`
    +This LSH family is called `(r1, r2, p1, p2)`-sensitive.
    +
    +In this section, we call a pair of input features a false positive if the two features are hashed into the same hash bucket but they are far away in distance, and we define false negative as the pair of features when their distance are close but they are not in the same hash bucket.
    +
    +## Bucketed Random Projection for Euclidean Distance
    +
    +[Bucketed Random Projection](https://en.wikipedia.org/wiki/Locality-sensitive_hashing#Stable_distributions) is the LSH family in `spark.ml` for Euclidean distance. The Euclidean distance is defined as follows:
    +`\[
    +d(\mathbf{x}, \mathbf{y}) = \sqrt{\sum_i (x_i - y_i)^2}
    +\]`
    +Its LSH family projects features onto a random unit vector and divide the projected results to hash buckets:
    +`\[
    +h(\mathbf{x}) = \lfloor \frac{\mathbf{x} \cdot \mathbf{v}}{r} \rfloor
    +\]`
    +where `v` is a normalized random unit vector and `r` is user-defined bucket length. The bucket length can be used to control the average size of hash buckets. A larger bucket length means higher probability for features to be in the same bucket.
    +
    +Bucketed Random Projection accepts arbitrary vectors as input features, and supports both sparse and dense vectors.
    +
    +<div class="codetabs">
    +<div data-lang="scala" markdown="1">
    +
    +Refer to the [RandomProjection Scala docs](api/scala/index.html#org.apache.spark.ml.feature.RandomProjection)
    +for more details on the API.
    +
    +{% include_example scala/org/apache/spark/examples/ml/BucketedRandomProjectionLSHExample.scala %}
    +</div>
    +
    +<div data-lang="java" markdown="1">
    +
    +Refer to the [RandomProjection Java docs](api/java/org/apache/spark/ml/feature/RandomProjection.html)
    +for more details on the API.
    +
    +{% include_example java/org/apache/spark/examples/ml/JavaBucketedRandomProjectionLSHExample.java %}
    +</div>
    +</div>
    +
    +## MinHash for Jaccard Distance
    +[MinHash](https://en.wikipedia.org/wiki/MinHash) is the LSH family in `spark.ml` for Jaccard distance where input features are sets of natural numbers. Jaccard distance of two sets is defined by the cardinality of their intersection and union:
    +`\[
    +d(\mathbf{A}, \mathbf{B}) = 1 - \frac{|\mathbf{A} \cap \mathbf{B}|}{|\mathbf{A} \cup \mathbf{B}|}
    +\]`
    +As its LSH family, MinHash applies a random hash function `g` to each elements in the set and take the minimum of all hashed values:
    +`\[
    +h(\mathbf{A}) = \min_{a \in \mathbf{A}}(g(a))
    +\]`
    +
    +The input sets for MinHash are represented as binary vectors, where the vector indices represent the elements themselves and the non-zero values in the vector represent the presence of that element in the set. While both dense and sparse vectors are supported, typically sparse vectors are recommended for efficiency. For example, `Vectors.sparse(10, Array[(2, 1.0), (3, 1.0), (5, 1.0)])` means there are 10 elements in the space. This set contains elem 2, elem 3 and elem 5. All non-zero values are treated as binary "1" values.
    +
    +**Note:** Empty sets cannot be transformed by MinHash, which means any input vector must have at least 1 non-zero entry.
    +
    +<div class="codetabs">
    +<div data-lang="scala" markdown="1">
    +
    +Refer to the [MinHash Scala docs](api/scala/index.html#org.apache.spark.ml.feature.MinHash)
    +for more details on the API.
    +
    +{% include_example scala/org/apache/spark/examples/ml/MinHashLSHExample.scala %}
    +</div>
    +
    +<div data-lang="java" markdown="1">
    +
    +Refer to the [MinHash Java docs](api/java/org/apache/spark/ml/feature/MinHash.html)
    +for more details on the API.
    +
    +{% include_example java/org/apache/spark/examples/ml/JavaMinHashLSHExample.java %}
    +</div>
    +</div>
    +
    +## Feature Transformation
    --- End diff --
    
    This doc is in L1509 and L1516


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