[SPARK-19313][ML][MLLIB] GaussianMixture should limit the number of features

## What changes were proposed in this pull request?

The following test will fail on current master

````scala
test("gmm fails on high dimensional data") {
    val ctx = spark.sqlContext
    import ctx.implicits._
    val df = Seq(
      Vectors.sparse(GaussianMixture.MAX_NUM_FEATURES + 1, Array(0, 4), Array(3.0, 8.0)),
      Vectors.sparse(GaussianMixture.MAX_NUM_FEATURES + 1, Array(1, 5), Array(4.0, 9.0)))
      .map(Tuple1.apply).toDF("features")
    val gm = new GaussianMixture()
    intercept[IllegalArgumentException] {
      gm.fit(df)
    }
  }
````

Instead, you'll get an `ArrayIndexOutOfBoundsException` or something similar for MLlib. That's because the covariance matrix allocates an array of `numFeatures * numFeatures`, and in this case we get integer overflow. While there is currently a warning that the algorithm does not perform well for high number of features, we should perform an appropriate check to communicate this limitation to users.

This patch adds a `require(numFeatures < GaussianMixture.MAX_NUM_FEATURES)` check to ML and MLlib algorithms. For the feature limitation, we can limit it such that we do not get numerical overflow to something like `math.sqrt(Integer.MaxValue).toInt` (about 46k) which eliminates the cryptic error. However in, for example WLS, we need to collect an array on the order of `numFeatures * numFeatures` to the driver and we therefore limit to 4096 features. We may want to keep that convention here for consistency.

## How was this patch tested?
Unit tests in ML and MLlib.

Author: sethah <[email protected]>

Closes apache#16661 from sethah/gmm_high_dim.

Author: sethah

mllib/src/main/scala/org/apache/spark/ml/clustering/GaussianMixture.scala

@@ -278,7 +278,9 @@ object GaussianMixtureModel extends MLReadable[GaussianMixtureModel] {
  * While this process is generally guaranteed to converge, it is not guaranteed
  * to find a global optimum.
  *
- * @note For high-dimensional data (with many features), this algorithm may perform poorly.
+ * @note This algorithm is limited in its number of features since it requires storing a covariance
+ * matrix which has size quadratic in the number of features. Even when the number of features does
+ * not exceed this limit, this algorithm may perform poorly on high-dimensional data.
  * This is due to high-dimensional data (a) making it difficult to cluster at all (based
  * on statistical/theoretical arguments) and (b) numerical issues with Gaussian distributions.
  */
@@ -344,6 +346,9 @@ class GaussianMixture @Since("2.0.0") (
 
     // Extract the number of features.
     val numFeatures = instances.first().size
+    require(numFeatures < GaussianMixture.MAX_NUM_FEATURES, s"GaussianMixture cannot handle more " +
+      s"than ${GaussianMixture.MAX_NUM_FEATURES} features because the size of the covariance" +
+      s" matrix is quadratic in the number of features.")
 
     val instr = Instrumentation.create(this, instances)
     instr.logParams(featuresCol, predictionCol, probabilityCol, k, maxIter, seed, tol)
@@ -391,8 +396,8 @@ class GaussianMixture @Since("2.0.0") (
         val (ws, gs) = sc.parallelize(tuples, numPartitions).map { case (mean, cov, weight) =>
           GaussianMixture.updateWeightsAndGaussians(mean, cov, weight, sumWeights)
         }.collect().unzip
-        Array.copy(ws.toArray, 0, weights, 0, ws.length)
-        Array.copy(gs.toArray, 0, gaussians, 0, gs.length)
+        Array.copy(ws, 0, weights, 0, ws.length)
+        Array.copy(gs, 0, gaussians, 0, gs.length)
       } else {
         var i = 0
         while (i < numClusters) {
@@ -486,6 +491,9 @@ class GaussianMixture @Since("2.0.0") (
 @Since("2.0.0")
 object GaussianMixture extends DefaultParamsReadable[GaussianMixture] {
 
+  /** Limit number of features such that numFeatures^2^ < Int.MaxValue */
+  private[clustering] val MAX_NUM_FEATURES = math.sqrt(Int.MaxValue).toInt
+
   @Since("2.0.0")
   override def load(path: String): GaussianMixture = super.load(path)
 

mllib/src/main/scala/org/apache/spark/mllib/clustering/GaussianMixture.scala

@@ -46,7 +46,9 @@ import org.apache.spark.util.Utils
  *                       is considered to have occurred.
  * @param maxIterations Maximum number of iterations allowed.
  *
- * @note For high-dimensional data (with many features), this algorithm may perform poorly.
+ * @note This algorithm is limited in its number of features since it requires storing a covariance
+ * matrix which has size quadratic in the number of features. Even when the number of features does
+ * not exceed this limit, this algorithm may perform poorly on high-dimensional data.
  * This is due to high-dimensional data (a) making it difficult to cluster at all (based
  * on statistical/theoretical arguments) and (b) numerical issues with Gaussian distributions.
  */
@@ -170,6 +172,9 @@ class GaussianMixture private (
 
     // Get length of the input vectors
     val d = breezeData.first().length
+    require(d < GaussianMixture.MAX_NUM_FEATURES, s"GaussianMixture cannot handle more " +
+      s"than ${GaussianMixture.MAX_NUM_FEATURES} features because the size of the covariance" +
+      s" matrix is quadratic in the number of features.")
 
     val shouldDistributeGaussians = GaussianMixture.shouldDistributeGaussians(k, d)
 
@@ -211,8 +216,8 @@ class GaussianMixture private (
         val (ws, gs) = sc.parallelize(tuples, numPartitions).map { case (mean, sigma, weight) =>
           updateWeightsAndGaussians(mean, sigma, weight, sumWeights)
         }.collect().unzip
-        Array.copy(ws.toArray, 0, weights, 0, ws.length)
-        Array.copy(gs.toArray, 0, gaussians, 0, gs.length)
+        Array.copy(ws, 0, weights, 0, ws.length)
+        Array.copy(gs, 0, gaussians, 0, gs.length)
       } else {
         var i = 0
         while (i < k) {
@@ -272,6 +277,10 @@ class GaussianMixture private (
 }
 
 private[clustering] object GaussianMixture {
+
+  /** Limit number of features such that numFeatures^2^ < Int.MaxValue */
+  private[clustering] val MAX_NUM_FEATURES = math.sqrt(Int.MaxValue).toInt
+
   /**
    * Heuristic to distribute the computation of the `MultivariateGaussian`s, approximately when
    * d is greater than 25 except for when k is very small.

mllib/src/test/scala/org/apache/spark/ml/clustering/GaussianMixtureSuite.scala

@@ -53,6 +53,20 @@ class GaussianMixtureSuite extends SparkFunSuite with MLlibTestSparkContext
     rDataset = rData.map(FeatureData).toDF()
   }
 
+  test("gmm fails on high dimensional data") {
+    val df = Seq(
+      Vectors.sparse(GaussianMixture.MAX_NUM_FEATURES + 1, Array(0, 4), Array(3.0, 8.0)),
+      Vectors.sparse(GaussianMixture.MAX_NUM_FEATURES + 1, Array(1, 5), Array(4.0, 9.0)))
+      .map(Tuple1.apply).toDF("features")
+    val gm = new GaussianMixture()
+    withClue(s"GMM should restrict the maximum number of features to be < " +
+      s"${GaussianMixture.MAX_NUM_FEATURES}") {
+      intercept[IllegalArgumentException] {
+        gm.fit(df)
+      }
+    }
+  }
+
   test("default parameters") {
     val gm = new GaussianMixture()
 

mllib/src/test/scala/org/apache/spark/mllib/clustering/GaussianMixtureSuite.scala

@@ -25,6 +25,20 @@ import org.apache.spark.mllib.util.TestingUtils._
 import org.apache.spark.util.Utils
 
 class GaussianMixtureSuite extends SparkFunSuite with MLlibTestSparkContext {
+
+  test("gmm fails on high dimensional data") {
+    val rdd = sc.parallelize(Seq(
+      Vectors.sparse(GaussianMixture.MAX_NUM_FEATURES + 1, Array(0, 4), Array(3.0, 8.0)),
+      Vectors.sparse(GaussianMixture.MAX_NUM_FEATURES + 1, Array(1, 5), Array(4.0, 9.0))))
+    val gm = new GaussianMixture()
+    withClue(s"GMM should restrict the maximum number of features to be < " +
+      s"${GaussianMixture.MAX_NUM_FEATURES}") {
+      intercept[IllegalArgumentException] {
+        gm.run(rdd)
+      }
+    }
+  }
+
   test("single cluster") {
     val data = sc.parallelize(Array(
       Vectors.dense(6.0, 9.0),