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I was at a group study recently, and I think one of the points made was that clustering is like 0-dimensional manifold-learning. Is this right? What is the reason behind it?

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Probably the thinking is something along these lines...

In manifold learning we have data in $R^n$, and we want to learn a lower dimensional manifold that the data is close to lying on.

A set of points with the discrete topology is a zero dimensional manifold.

Some clustering groups points into clusters of points that are close to the centroid of the cluster. The centroids are a discrete set of points. So, the points are close to lying on a zero dimensional manifold.

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