Question: \(k\) -means with nonnegative, proportions, or Boolean vectors. Suppose that the vectors \(x_{1}, \ldots, x_{N}\) are clustered using \(k\) -means, with group representatives \(z_{1}, \ldots, z_{k}\).

1. Suppose the original vectors \(x_{i}\) are nonnegative, i.e., their entries are nonnegative. Explain why the representatives \(z_{j}\) are also nonnegative.
2. Suppose the original vectors \(x_{i}\) represent proportions, i.e., their entries are nonnegative and sum to one. (This is the case when \(x_{i}\) are word count histograms, for example.) Explain why the representatives \(z_{j}\) also represent proportions, i.e., their entries are nonnegative and sum to one.
3. Suppose the original vectors \(x_{i}\) are Boolean, i.e., their entries are either 0 or 1 . Give an interpretation of \(\left(z_{j}\right)_{i}\), the \(i\) th entry of the \(j\) group representative.

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