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Taken from the appendix to the paper (Yongning Wang & Ruey S. Tsay) of this (2019) paper Clustering Multiple Time Series with Structural Breaks. Appendix to be downloaded her Appendix

To fix label switching, the following loss function is defined. The most probable thing is that I misunderstand the definition of $G_{sk}$. s is indicating a given regime, $k\epsilon\{1,\cdots,K\}$ is giving the group and $n\epsilon\{1,\cdots,N\}$ is the individual. If I want to select n s.t. $n\epsilon G_{sk}$, I imagine $G_{sk}$ looks somewhat like $\left(0,0,0,0,1,1,1,0,...\right)^{T}$ So in case of K=3 and N=50, you get $dim(G_s)=50\times3$, but then I would need discrete optimization with $K^N$ possibilities of a $KN$, matrix, which will never get stored.

This does not seem reasonable. For context, it pertains to a loss function for label-switching. The main question remains, there is no definition of $G_{sk}$ given anywhere, and as the "action" of the loss function, the use of $n\epsilon G_{sk}$ in the summation of the loss function confuses me. I guess my question is, what do the authors mean by it.

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