[GitHub] spark pull request #20396: [SPARK-23217][ML] Add cosine distance measure to ...

[srowen <gi...@git.apache.org>]

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

    https://github.com/apache/spark/pull/20396#discussion_r167581903
  
    --- Diff: mllib/src/main/scala/org/apache/spark/ml/evaluation/ClusteringEvaluator.scala ---
    @@ -421,13 +456,220 @@ private[evaluation] object SquaredEuclideanSilhouette {
           computeSilhouetteCoefficient(bClustersStatsMap, _: Vector, _: Double, _: Double)
         }
     
    -    val silhouetteScore = dfWithSquaredNorm
    -      .select(avg(
    -        computeSilhouetteCoefficientUDF(
    -          col(featuresCol), col(predictionCol).cast(DoubleType), col("squaredNorm"))
    -      ))
    -      .collect()(0)
    -      .getDouble(0)
    +    val silhouetteScore = overallScore(dfWithSquaredNorm,
    +      computeSilhouetteCoefficientUDF(col(featuresCol), col(predictionCol).cast(DoubleType),
    +        col("squaredNorm")))
    +
    +    bClustersStatsMap.destroy()
    +
    +    silhouetteScore
    +  }
    +}
    +
    +
    +/**
    + * The algorithm which is implemented in this object, instead, is an efficient and parallel
    + * implementation of the Silhouette using the cosine distance measure. The cosine distance
    + * measure is defined as `1 - s` where `s` is the cosine similarity between two points.
    + *
    + * The total distance of the point `X` to the points `$C_{i}$` belonging to the cluster `$\Gamma$`
    + * is:
    + *
    + * <blockquote>
    + *   $$
    + *   \sum\limits_{i=1}^N d(X, C_{i} ) =
    + *   \sum\limits_{i=1}^N \Big( 1 - \frac{\sum\limits_{j=1}^D x_{j}c_{ij} }{ \|X\|\|C_{i}\|} \Big)
    + *   = \sum\limits_{i=1}^N 1 - \sum\limits_{i=1}^N \sum\limits_{j=1}^D \frac{x_{j}}{\|X\|}
    + *   \frac{c_{ij}}{\|C_{i}\|}
    + *   = N - \sum\limits_{j=1}^D \frac{x_{j}}{\|X\|} \Big( \sum\limits_{i=1}^N
    + *   \frac{c_{ij}}{\|C_{i}\|} \Big)
    + *   $$
    + * </blockquote>
    + *
    + * where `$x_{j}$` is the `j`-th dimension of the point `X` and `$c_{ij}$` is the `j`-th dimension
    + * of the `i`-th point in cluster `$\Gamma$`.
    + *
    + * Then, we can define the vector:
    + *
    + * <blockquote>
    + *   $$
    + *   \xi_{X} : \xi_{X i} = \frac{x_{i}}{\|X\|}, i = 1, ..., D
    + *   $$
    + * </blockquote>
    + *
    + * which can be precomputed for each point and the vector
    + *
    + * <blockquote>
    + *   $$
    + *   \Omega_{\Gamma} : \Omega_{\Gamma i} = \sum\limits_{j=1}^N \xi_{C_{j}i}, i = 1, ..., D
    + *   $$
    + * </blockquote>
    + *
    + * which can be precomputed too for each cluster `$\Gamma$` by its points `$C_{i}$`.
    + *
    + * With these definitions, the numerator becomes:
    + *
    + * <blockquote>
    + *   $$
    + *   N - \sum\limits_{j=1}^D \xi_{X j} \Omega_{\Gamma j}
    + *   $$
    + * </blockquote>
    + *
    + * Thus the average distance of a point `X` to the points of the cluster `$\Gamma$` is:
    + *
    + * <blockquote>
    + *   $$
    + *   1 - \frac{\sum\limits_{j=1}^D \xi_{X j} \Omega_{\Gamma j}}{N}
    + *   $$
    + * </blockquote>
    + *
    + * In the implementation, the precomputed values for the clusters are distributed among the worker
    + * nodes via broadcasted variables, because we can assume that the clusters are limited in number.
    + *
    + * The main strengths of this algorithm are the low computational complexity and the intrinsic
    + * parallelism. The precomputed information for each point and for each cluster can be computed
    + * with a computational complexity which is `O(N/W)`, where `N` is the number of points in the
    + * dataset and `W` is the number of worker nodes. After that, every point can be analyzed
    + * independently from the others.
    + *
    + * For every point we need to compute the average distance to all the clusters. Since the formula
    + * above requires `O(D)` operations, this phase has a computational complexity which is
    + * `O(C*D*N/W)` where `C` is the number of clusters (which we assume quite low), `D` is the number
    + * of dimensions, `N` is the number of points in the dataset and `W` is the number of worker
    + * nodes.
    + */
    +private[evaluation] object CosineSilhouette extends Silhouette {
    +
    +  private[this] var kryoRegistrationPerformed: Boolean = false
    +
    +  private[this] val normalizedFeaturesColName = "normalizedFeatures"
    +
    +  /**
    +   * This method registers the class
    +   * [[org.apache.spark.ml.evaluation.CosineSilhouette.ClusterStats]]
    +   * for kryo serialization.
    +   *
    +   * @param sc `SparkContext` to be used
    +   */
    +  def registerKryoClasses(sc: SparkContext): Unit = {
    +    if (!kryoRegistrationPerformed) {
    +      sc.getConf.registerKryoClasses(
    +        Array(
    +          classOf[CosineSilhouette.ClusterStats]
    +        )
    +      )
    +      kryoRegistrationPerformed = true
    +    }
    +  }
    +
    +  case class ClusterStats(normalizedFeatureSum: Vector, numOfPoints: Long)
    --- End diff --
    
    Back on this -- how about just using a Tuple2 of Vector, Long? no new class to register. 


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