K-Means Update Step
Implement the update step of the K-Means algorithm.
Given data points, current assignments, and the number of clusters k, compute new centroids as the mean of all points assigned to each cluster.
Return the new centroids as a 2D list. If a cluster has no points, keep its centroid at the origin (all zeros).
Round each centroid coordinate to 4 decimal places.
Example:
X = [[1, 0], [3, 0], [8, 0], [10, 0]] assignments = [0, 0, 1, 1] k = 2
[[2.0, 0.0], [9.0, 0.0]]
- The data points are grouped based on their current assignments: cluster 0 has points [[1, 0], [3, 0]] and cluster 1 has points [[8, 0], [10, 0]].
- For each cluster, the new centroid is calculated as the mean of all points assigned to it: for cluster 0, the centroid is [(1+3)/2,(0+0)/2]=[2,0] and for cluster 1, the centroid is [(8+10)/2,(0+0)/2]=[9,0].
- The calculated centroids are then rounded to 4 decimal places, resulting in [[2.0, 0.0], [9.0, 0.0]].
- Since both clusters have points assigned to them, there's no need to handle the case where a cluster has no points.
Constraints:
- X: 2D list of data points (n x d)
- assignments: list of n cluster indices (0 to k-1)
- k: number of clusters
- Return 2D list of k centroids, each rounded to 4 decimal places
- Empty clusters get centroid at origin
Background Knowledge
The K-Means algorithm is a widely used unsupervised learning technique for clustering data points into k distinct groups based on their similarities. The algorithm iteratively updates the centroids of the clusters and reassigns the data points to the closest cluster. The update step is a crucial part of the K-Means algorithm, where the centroids are updated based on the current assignments of the data points.
In the context of the K-Means algorithm, a centroid is the mean position of all the points in a cluster. The update step involves calculating the new centroids by taking the average of all the points assigned to each cluster. This process is repeated until the centroids converge or a stopping criterion is met. The K-Means algorithm is sensitive to the initial placement of the centroids and may get stuck in local optima.
The K-Means algorithm is based on the concept of Euclidean distance, which is used to measure the similarity between data points. The algorithm aims to minimize the sum of squared errors (SSE) between the data points and their assigned centroids. The update step is essential in minimizing the SSE and improving the overall clustering quality.
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