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For vanilla K-Means clustering algorithm I know that the time complexity is:

Time complexity: O(tknm),

where n is the number of data points, k is the number of clusters, and t is the number of iterations, m is the dimensionality of the vectors.

So, when I studied about Mini-batch K-Means to make the algorithm converge faster, I wanted to find out what is the Space & Time complexity of it?

Essentially so that I understand well, how much we are optimizing over vanilla K-Means.

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Infinite.

Mini-batch k-means never converges, you need to use an iteration limit or similar heuristic, and you can never guarantee to have found a local optimum.

In essence, mini-batch k-means is:

  1. draw a random sample
  2. perform one iteration of k-means using this sample
  3. repeat

Assuming that your sample size is N, 2 takes O(k N m t) time.

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