[GitHub] [spark] zhengruifeng commented on a change in pull request #27758: [SPARK-31007][ML] KMeans optimization based on triangle-inequality

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zhengruifeng commented on a change in pull request #27758: [SPARK-31007][ML] KMeans optimization based on triangle-inequality
URL: https://github.com/apache/spark/pull/27758#discussion_r386809482
 
 

 ##########
 File path: mllib/src/main/scala/org/apache/spark/mllib/clustering/DistanceMeasure.scala
 ##########
 @@ -234,6 +342,39 @@ private[spark] object EuclideanDistanceMeasure {
 }
 
 private[spark] class CosineDistanceMeasure extends DistanceMeasure {
+
+  /**
+   * @return Radii of centers. If distance between point x and center c is less than
+   *         the radius of center c, then center c is the closest center to point x.
+   *         For Cosine distance, it is similar to Euclidean distance. However, here
+   *         radian/angle is used instead of Cosine distance: for center c, finding
+   *         its closest center, computing the radian/angle between them, halving the
+   *         radian/angle, and converting it back to Cosine distance at the end.
+   */
+  override def computeRadii(centers: Array[VectorWithNorm]): Array[Double] = {
+    val k = centers.length
+    if (k == 1) {
+      Array(Double.NaN)
+    } else {
+      val distances = Array.fill(k)(Double.PositiveInfinity)
+      var i = 0
+      while (i < k) {
+        var j = i + 1
+        while (j < k) {
+          val d = distance(centers(i), centers(j))
+          if (d < distances(i)) distances(i) = d
+          if (d < distances(j)) distances(j) = d
+          j += 1
+        }
+        i += 1
+      }
+
+      // d = 1 - cos(x)
+      // r = 1 - cos(x/2) = 1 - sqrt((cos(x) + 1) / 2) = 1 - sqrt(1 - d/2)
+      distances.map(d => 1 - math.sqrt(1 - d / 2))
 
 Review comment:
   Yes, Cosine distance doesn't obey the triangle inequality, but the following lemma should be available to apply:
   
   given a point x, and let b and c be centers. If angle(x, b)<angle(b,c)/2, then angle(x,b)<angle(x,c), cos_distance(x,b)=1-cos(x,b)<cos_distance(x,c)=1-cos(x,c)
   
   That is because: [PRINCIPLES FROM GEOMETRY](http://www.angelfire.com/nt/navtrig/B1.html)
   
   > Each side of a spherical triangle is less than the sum of the other two.
   
   angle(x,b) + angle(x,c) > angle(b,c)
   angle(x,b) < angle(b,c)/2
   
   => angle(x,c) > angle(b,c)/2 > angle(x,b)
   => cos_distance(x,c) > cos_distance(x,b)
   
   
    

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