  Recent breakthrough results in compressed sensing (CS) have established that
many high dimensional objects can be accurately recovered from a relatively
small number of non- adaptive linear projection observations, provided that the
objects possess a sparse representation in some basis. Subsequent efforts have
shown that the performance of CS can be improved by exploiting the structure in
the location of the non-zero signal coefficients (structured sparsity) or using
some form of online measurement focusing (adaptivity) in the sensing process.
In this paper we examine a powerful hybrid of these two techniques. First, we
describe a simple adaptive sensing procedure and show that it is a provably
effective method for acquiring sparse signals that exhibit structured sparsity
characterized by tree-based coefficient dependencies. Next, employing
techniques from sparse hierarchical dictionary learning, we show that
representations exhibiting the appropriate form of structured sparsity can be
learned from collections of training data. The combination of these techniques
results in an effective and efficient adaptive compressive acquisition
procedure.
