Good morning, I was reviewing Gaussian mixture models and am struggling to understand how soft clustering works. Based on the image provided in lecture 11a slide 3 it seems to me that it generates a smoother transition at the boundaries of the clusters, but I don't see how this is possible if the \(\pi_{k}\) have a fixed set of probabilities for the k clusters.
Basically how is it possible that the probability of a given assignment is related to the distance from the cluster center if it is now randomly assigned? I assume this relates to the conditioning Martin mentioned but I do not quite understand it. Thanks in advance for your time!
In the soft clustering scheme, instead of assigning each sample to only one cluster (as it is done in hard clustering), we assign each sample to multiple cluster. The sum of assignment for each sample through all clusters is equal to 1 ($\sum{k}{q{kn}}=1$) . For example for samples that are close to one cluster center the assignment is almost 1 to the closes cluster and zero for others. On the other hand if another sample is between two or more cluster centers, it can be assigned 0.5 to one cluster and 0.4 to the other and 0.1 to the rest. In each iteration of updating the clusters' parameters, we use these assignments and weight the contribution of each sample to each cluster's parameters relative to this assignment.
GMM - z_n assignments
Good morning, I was reviewing Gaussian mixture models and am struggling to understand how soft clustering works. Based on the image provided in lecture 11a slide 3 it seems to me that it generates a smoother transition at the boundaries of the clusters, but I don't see how this is possible if the \(\pi_{k}\) have a fixed set of probabilities for the k clusters.
Basically how is it possible that the probability of a given assignment is related to the distance from the cluster center if it is now randomly assigned? I assume this relates to the conditioning Martin mentioned but I do not quite understand it. Thanks in advance for your time!
Hi,
In the soft clustering scheme, instead of assigning each sample to only one cluster (as it is done in hard clustering), we assign each sample to multiple cluster. The sum of assignment for each sample through all clusters is equal to 1 ($\sum{k}{q{kn}}=1$) . For example for samples that are close to one cluster center the assignment is almost 1 to the closes cluster and zero for others. On the other hand if another sample is between two or more cluster centers, it can be assigned 0.5 to one cluster and 0.4 to the other and 0.1 to the rest. In each iteration of updating the clusters' parameters, we use these assignments and weight the contribution of each sample to each cluster's parameters relative to this assignment.
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