  It is promising to solve linear inverse problems by unfolding iterative
algorithms (e.g., iterative shrinkage thresholding algorithm (ISTA)) as deep
neural networks (DNNs) with learnable parameters. However, existing ISTA-based
unfolded algorithms restrict the network architectures for iterative updates
with the partial weight coupling structure to guarantee convergence. In this
paper, we propose hybrid ISTA to unfold ISTA with both pre-computed and learned
parameters by incorporating free-form DNNs (i.e., DNNs with arbitrary feasible
and reasonable network architectures), while ensuring theoretical convergence.
We first develop HCISTA to improve the efficiency and flexibility of classical
ISTA (with pre-computed parameters) without compromising the convergence rate
in theory. Furthermore, the DNN-based hybrid algorithm is generalized to
popular variants of learned ISTA, dubbed HLISTA, to enable a free architecture
of learned parameters with a guarantee of linear convergence. To our best
knowledge, this paper is the first to provide a convergence-provable framework
that enables free-form DNNs in ISTA-based unfolded algorithms. This framework
is general to endow arbitrary DNNs for solving linear inverse problems with
convergence guarantees. Extensive experiments demonstrate that hybrid ISTA can
reduce the reconstruction error with an improved convergence rate in the tasks
of sparse recovery and compressive sensing.
