What hypothesis should I have for k-means clustering? I am wondering if we can just run k-means clustering without any hypothesis to find the optimal clusters or finding the optimal clusters are the hypothesis for K-means clustering? What hypothesis should I have for clustring using K-means?
 A: The hypothesis of k-means is that the data is generated from k exact centers, and some Gaussian noise.
Consider the initial use case: transmitting signals over phone or radio. Due to physical limitations, if you send a 200 Hz signal, you may measure a 190 Hz signal on the receiving side. And if you send a 190 Hz, maybe you get 180 Hz. On another wire, it may be different. But the error will be somewhat similar for the signald that you use. So if you take all of them, assume all k different signals were used, and the distortion for all of them is similar, then k-means will work quite well to identify the centers of the recieved signals.
Gaussian error is not as explicit as in Gaussian mixture modeling. But the way we estimate the mean and perform a least-squares estimation in k-means does assume the errors are symmetric (so only symmetric disteibutions work), small errors are common and large error are infrequent (so not uniform), translation invariant (i.e. all Gaussians have the same variance).
