The sparse module has been used in:
The sparse module is less mature than the tensor module. This documentation is also not mature.
The sparse module provides two kinds of sparse tensors are supported: CSC matrices and CSR matrices. Operations that are implemented:
Subtensor selection (aka. square-bracket notation, aka indexing) is not implemented, but the CSR and CSC datastructures support effecient implementations.
There are no GPU implementations for sparse matrices implemented in Theano.
Some documentation for sparse has been written here.