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, point 3](http://www.angelfire.com/nt/navtrig/B1.html) > Each side of a spherical triangle is less than the sum of the other two. [Triangle_inequality:](https://en.wikipedia.org/wiki/Triangle_inequality) > In spherical geometry, the shortest distance between two points is an arc of a great circle, but the triangle inequality holds provided the restriction is made that the distance between two points on a sphere is the length of a minor spherical line segment (that is, one with central angle in [0, π]) with those endpoints.[4][5] 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) angle(x,b) < angle(b,c)/2 <=> cos(x,b) > sqrt{ (cos(b,c) + 1)/2 } <=> cos_distance(x,b) < 1 - sqrt{ (cos(b,c) + 1)/2 } = 1 - sqrt{ 1 - cos_distance(b,c) / 2 } => Give two centers b and c, if point x has cos_distance(x,b) < 1 - sqrt{ 1 - cos_distance(b,c) / 2 }, then point x belongs to center b. ---------------------------------------------------------------- This is an automated message from the Apache Git Service. To respond to the message, please log on to GitHub and use the URL above to go to the specific comment. For queries about this service, please contact Infrastructure at: us...@infra.apache.org With regards, Apache Git Services --------------------------------------------------------------------- To unsubscribe, e-mail: reviews-unsubscr...@spark.apache.org For additional commands, e-mail: reviews-h...@spark.apache.org