Table 2.

Overview of symbols used in the description of the scregclust algorithm

Symbol	Description
Problem setup:
n	Total number of cells in the input matrix
pt	Number of target genes
pr	Number of regulators
Zt, Zr	target gene and regulator expression (n × pt resp. n × pr matrices)
K	Desired number of clusters
Π	K × pt cluster membership matrix for target genes with Π(i, j) ∈ {0, 1} and ${\sum }_{i=1}^K{\mathbf{\Pi }}^{\left(i,j\right)}\le 1$
J	pt × pt indicator matrix describing prior knowledge of biological relationships between target genes (e.g. pathway co-occurence)
For data splits d =  1, 2:
nd	Number of cells in the d-th data split
Zt,d	target gene expression used in the d-th data split (nd × pt matrix)
Zr,d	regulator expression used in the d-th data split (nd × pr matrix)
For each cluster i = 1, …, K:
Ci	Set of target genes in cluster i
Ri	Set of regulators associated with cluster i
Ni	Maximum number of regulators associated with cluster i
si	Sign vectors containing one sign for each regulator in Ri
Bi	non-negative regression coefficients (∣Ri∣ × pt matrix)
${\sigma }_{i,j}^2$	positive variance parameter for each target gene and cluster
Optimization-related (i = 1, …, K, j = 1, …, pt):
λ	Positive penalty parameter used in coop-Lasso
wi	Positive weight vector of length pr for each cluster i in coop-Lasso
Bi,OLS	Ordinary least squares estimates of the regression coefficients in cluster i (pr × ∣Ci∣ matrix)
Bi,CL	Coop-lasso estimates of the regression coefficients in cluster i (pr × ∣Ci∣ matrix)
τ	Non-negative threshold for rag-bag clustering
pi,j	Prior probability of target gene j being in cluster i
Lj	K × n2 likelihood matrix for target gene j
vj	Vector of n2 votes for the cluster assignment of target gene j
μ	Prior strength in [0, 1] for trade-off between likelihood and prior in cluster allocation
Validation measures (i = 1, …, K, j  =  1, …, pt, k = 1, …, pr):
$R_i^2$	Predictive R2 for cluster i
$R_{i,-k}^2$	Predictive R2 for cluster i with regulator k omitted
Ii,k	Importance of regulator k in cluster i
$R_{i,j}^2$	Predictive R2 for target gene j predicted by regulators in Ri
Sj	Silhouette score for target gene j
Output:
T	Regulatory table providing a summary of regulator strength in each cluster (pr × K matrix)