  It is well-established that many iterative sparse reconstruction algorithms
can be unrolled to yield a learnable neural network for improved empirical
performance. A prime example is learned ISTA (LISTA) where weights, step sizes
and thresholds are learned from training data. Recently, Analytic LISTA
(ALISTA) has been introduced, combining the strong empirical performance of a
fully learned approach like LISTA, while retaining theoretical guarantees of
classical compressed sensing algorithms and significantly reducing the number
of parameters to learn. However, these parameters are trained to work in
expectation, often leading to suboptimal reconstruction of individual targets.
In this work we therefore introduce Neurally Augmented ALISTA, in which an LSTM
network is used to compute step sizes and thresholds individually for each
target vector during reconstruction. This adaptive approach is theoretically
motivated by revisiting the recovery guarantees of ALISTA. We show that our
approach further improves empirical performance in sparse reconstruction, in
particular outperforming existing algorithms by an increasing margin as the
compression ratio becomes more challenging.
