Table 1 Notations

K

Number of SNPs

N

Number of genes

H

Number of samples

M

Number of group-wise associations

x

Random variables of K SNPs

z

Random variables of N genes

y

Latent variables to model group-wise associaiton

\(\mathbf {X}\in {\mathbb {R}}^{K \times H}\)

SNP matrix data

\(\mathbf {Z} \in {\mathbb {R}}^{N \times H}\)

Gene expression matrix data

\(\mathbf {A} \in \mathbb {R}^{M\times K}\)

Group-wise association coefficient matrix between x and y

\(\mathbf {B} \in \mathbb {R}^{N\times M}\)

Group-wise association coefficient matrix between y and z

\(\mathbf {C} \in \mathbb {R}^{N\times K}\)

Individual association coefficient matrix between x and y

α,β,γ,ρ

Regularization parameters

\(\mathbf {R} \in \mathbb {R}^{N\times K}\)

Indicator matrix showing which elements in C can be nonzero