I have read that the k-means algorithm tries to minimize the within cluster sum of squares (or variance). With some brainstorming, a question popped up. Why is it that k-means or any other clustering algorithm that has within cluster variance as its objective to minimize, chose this as the objective function to minimize? What is it about within cluster variance that helps you decide that this is what you want to focus while clustering? - And especially clustering?
Let me put the question in another way (This question can be a sub-question or another way of putting the same question). Why would you say that minimizing within cluster variance is the right way of clustering (referring to the algorithms that minimize it)? Can there be other objective functions that can be minimized (or maximized or anything) for clustering?