  Recent technological advancements have led to the rapid generation of
high-throughput biological data, which can be used to address novel scientific
questions in broad areas of research. These data can be thought of as a large
matrix with covariates annotating both rows and columns of this matrix. Matrix
linear models provide a convenient way for modeling such data. In many
situations, sparse estimation of these models is desired. We present fast,
general methods for fitting sparse matrix linear models to structured
high-throughput data. We induce model sparsity using an L$_1$ penalty and
consider the case when the response matrix and the covariate matrices are
large. Due to data size, standard methods for estimation of these penalized
regression models fail if the problem is converted to the corresponding
univariate regression scenario. By leveraging matrix properties in the
structure of our model, we develop several fast estimation algorithms
(coordinate descent, FISTA, and ADMM) and discuss their trade-offs. We evaluate
our method's performance on simulated data, E. coli chemical genetic screening
data, and two Arabidopsis genetic datasets with multivariate responses. Our
algorithms have been implemented in the Julia programming language and are
available at https://github.com/senresearch/MatrixLMnet.jl.
