networks into $16$ and $12$ clusters. With both data sets, we set the number of iterations of the NNMF algorithm at $1000$. We applied the NNMF algorithm for $10000$ times, and used the best result in terms of the approximation error. Initial values of the two low-dimensional matrices were randomly chosen according to the normal distribution $\mathcal{N}(0.5,0.5)$.

Figures \[fig:mnist\_nnmf\_lnn\], \[fig:mnist\_nnmf\_roles\], \[fig:food\_nnmf\_lnn\], and \[fig:food\_nnmf\_roles\] show the resulting cluster structures and the representative roles of the clusters. Comparing these figures with the results in Figures \[fig:mnist\_align\_lnn\], \[fig:mnist\_roles\], \[fig:food\_align\_lnn\], and \[fig:food\_roles\], we can observe that the previous NNMF-based method could not capture the structures of the input and output dimension values in as much detail as our proposed method, since it does not take the sign information into account. Furthermore, with the NNMF-based method, we should define the number of clusters in advance, and we cannot observe the hierarchical structure of clusters to find the optimal resolution for interpreting the roles of partial structures of an LNN.

Appendix 2: Effect of Sign Alignment of Feature Vectors {#appendix-2-effect-of-sign-alignment-of-feature-vectors .unnumbered}
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Here, we discuss the effect of the sign alignment of the feature vectors based on cosine similarity (Algorithm \[alg:alignment\]).

Figures \[fig:mnist\_v\] shows the effect 