KMeans ignores example weights in centroid updates and default initialization #7876

KMeans accepts an ExampleWeightColumnName, but appears to ignore weight magnitudes during the main training loop and the default K-means|| initialization.

Expected behavior

Centroids should be weighted means of their assigned examples:

centroid = sum(weight * features) / sum(weight)

Initialization should also account for example weights where appropriate.

Actual behavior

Source inspection shows:

• Centroid updates accumulate unweighted feature vectors and divide by the number of assigned rows.
• Default K-means|| initialization samples using distance-based weights without incorporating example weights. Its candidate-reduction step also counts rows rather than summing their weights.
• Explicit K-means++ initialization does use example weights, but the subsequent training loop still computes unweighted centroids.

Consequently, a weight column can affect initialization differently depending on the selected algorithm, without producing weighted centroid updates.

Minimal repro

The following example isolates the centroid calculation by requesting one cluster:

using System;
using Microsoft.ML;
using Microsoft.ML.Data;
using Microsoft.ML.Trainers;

var mlContext = new MLContext(seed: 1);
var data = mlContext.Data.LoadFromEnumerable(new[]
{
    new Input { Features = new[] { 0f }, Weight = 1f },
    new Input { Features = new[] { 10f }, Weight = 9f }
});

var model = mlContext.Clustering.Trainers.KMeans(
    new KMeansTrainer.Options
    {
        NumberOfClusters = 1,
        ExampleWeightColumnName = nameof(Input.Weight),
        InitializationAlgorithm =
            KMeansTrainer.InitializationAlgorithm.KMeansPlusPlus,
        NumberOfThreads = 1
    }).Fit(data);

VBuffer<float>[] centroids = default;
model.Model.GetClusterCentroids(ref centroids, out _);
Console.WriteLine(centroids[0].GetValues()[0]);

public class Input
{
    [VectorType(1)]
    public float[] Features { get; set; } = new float[1];

    public float Weight { get; set; }
}

Expected centroid: 9, from (1 * 0 + 9 * 10) / (1 + 9).

Source inspection predicts 5, the unweighted mean. This repro has not yet been executed.

Relevant source

• Unweighted centroid accumulation
• Division by cluster size
• K-means|| sampling
• Candidate weights calculated as row counts

Found while improving the weight-column documentation for #6746 in #7875.