K-Means involves computing distances in input space and assigning data points to the nearest prototype points. What can one say about these prototype points?

I Every cluster has its data assigned to the nearest prototype point.
II They are not restricted to being data points in the training set.
III They are necessarily included in the training set.
IV They are usually referred to as centroids.

Question

K-Means involves computing distances in input space and assigning data points to the nearest prototype points. What can one say about these prototype points?

I	Every cluster has its data assigned to the nearest prototype point.
II	They are not restricted to being data points in the training set.
III	They are necessarily included in the training set.
IV	They are usually referred to as centroids.

BlackTom AI · Accepted Answer

Question restatement: K-Means involves computing distances in input space and assigning data points to the nearest prototype points. The options refer to properties of these prototype points, i.e., the centroids used in K-Means.

Option I: Every cluster has its data assigned to the nearest prototype point. This is essentially describing the assignment step of K-Means, where each data point is placed into the cluster whose centroid is closest in distance. This statement is true because the core mechanism selects the nearest prototype for each point.

Option II: They are not restricted to being data points in the training set. Correct. The prototype (centroid) is typically the mean of the points assigned to that cluster and does not need to coincide with an actual data point from the training set. In many datasets, the centroid lies at a location that is not one of the observed points.

Option III: They are necessarily included in the training set. This is false. As just noted, centroids are computed as means and can lie anywhere in the feature space, not necessarily at an existing training data point. Depending on the data, the centroid may be a point that wasn't observed during training.

Option IV: They are usually referred to as centroids. This is true in standard terminology: the prototype points in K-Means are commonly called centroids, reflecting their role as center points of clusters.

Summary of reasoning for each option:
- I is true because assignment to the nearest prototype is the defining step of K-Means. 
- II is true since centroids are not restricted to data points and are often means of their cluster’s members.
- III is false since centroids need not be training data points.
- IV is true as the conventional name for these prototype points is centroids.

FINTECH 540.01.Fa25 Final Exam

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