  Consider the problem of estimating the entries of an unknown mean matrix or
tensor given a single noisy realization. In the matrix case, this problem can
be addressed by decomposing the mean matrix into a component that is additive
in the rows and columns, i.e.\ the additive ANOVA decomposition of the mean
matrix, plus a matrix of elementwise effects, and assuming that the elementwise
effects may be sparse. Accordingly, the mean matrix can be estimated by solving
a penalized regression problem, applying a lasso penalty to the elementwise
effects. Although solving this penalized regression problem is straightforward,
specifying appropriate values of the penalty parameters is not. Leveraging the
posterior mode interpretation of the penalized regression problem, moment-based
empirical Bayes estimators of the penalty parameters can be defined. Estimation
of the mean matrix using these these moment-based empirical Bayes estimators
can be called LANOVA penalization, and the corresponding estimate of the mean
matrix can be called the LANOVA estimate. The empirical Bayes estimators are
shown to be consistent. Additionally, LANOVA penalization is extended to
accommodate sparsity of row and column effects and to estimate an unknown mean
tensor. The behavior of the LANOVA estimate is examined under misspecification
of the distribution of the elementwise effects, and LANOVA penalization is
applied to several datasets, including a matrix of microarray data, a three-way
tensor of fMRI data and a three-way tensor of wheat infection data.
