  Model selection via penalized likelihood type criteria is a standard task in
many statistical inference and machine learning problems. It has led to
deriving criteria with asymptotic consistency results and an increasing
emphasis on introducing non-asymptotic criteria. We focus on the problem of
modeling non-linear relationships in regression data with potential hidden
graph-structured interactions between the high-dimensional predictors, within
the mixture of experts modeling framework. In order to deal with such a complex
situation, we investigate a block-diagonal localized mixture of polynomial
experts (BLoMPE) regression model, which is constructed upon an inverse
regression and block-diagonal structures of the Gaussian expert covariance
matrices. We introduce a penalized maximum likelihood selection criterion to
estimate the unknown conditional density of the regression model. This model
selection criterion allows us to handle the challenging problem of inferring
the number of mixture components, the degree of polynomial mean functions, and
the hidden block-diagonal structures of the covariance matrices, which reduces
the number of parameters to be estimated and leads to a trade-off between
complexity and sparsity in the model. In particular, we provide a strong
theoretical guarantee: a finite-sample oracle inequality satisfied by the
penalized maximum likelihood estimator with a Jensen-Kullback-Leibler type
loss, to support the introduced non-asymptotic model selection criterion. The
penalty shape of this criterion depends on the complexity of the considered
random subcollection of BLoMPE models, including the relevant graph structures,
the degree of polynomial mean functions, and the number of mixture components.
