  Region-specific linear models are widely used in practical applications
because of their non-linear but highly interpretable model representations. One
of the key challenges in their use is non-convexity in simultaneous
optimization of regions and region-specific models. This paper proposes novel
convex region-specific linear models, which we refer to as partition-wise
linear models. Our key ideas are 1) assigning linear models not to regions but
to partitions (region-specifiers) and representing region-specific linear
models by linear combinations of partition-specific models, and 2) optimizing
regions via partition selection from a large number of given partition
candidates by means of convex structured regularizations. In addition to
providing initialization-free globally-optimal solutions, our convex
formulation makes it possible to derive a generalization bound and to use such
advanced optimization techniques as proximal methods and decomposition of the
proximal maps for sparsity-inducing regularizations. Experimental results
demonstrate that our partition-wise linear models perform better than or are at
least competitive with state-of-the-art region-specific or locally linear
models.
