  In this paper, we revisit the class of iterative shrinkage-thresholding
algorithms (ISTA) for solving the linear inverse problem with sparse
representation, which arises in signal and image processing. It is shown in the
numerical experiment to deblur an image that the convergence behavior in the
logarithmic-scale ordinate tends to be linear instead of logarithmic,
approximating to be flat. Making meticulous observations, we find that the
previous assumption for the smooth part to be convex weakens the least-square
model. Specifically, assuming the smooth part to be strongly convex is more
reasonable for the least-square model, even though the image matrix is probably
ill-conditioned. Furthermore, we improve the pivotal inequality tighter for
composite optimization with the smooth part to be strongly convex instead of
general convex, which is first found in [Li et al., 2022]. Based on this
pivotal inequality, we generalize the linear convergence to composite
optimization in both the objective value and the squared proximal subgradient
norm. Meanwhile, we set a simple ill-conditioned matrix which is easy to
compute the singular values instead of the original blur matrix. The new
numerical experiment shows the proximal generalization of Nesterov's
accelerated gradient descent (NAG) for the strongly convex function has a
faster linear convergence rate than ISTA. Based on the tighter pivotal
inequality, we also generalize the faster linear convergence rate to composite
optimization, in both the objective value and the squared proximal subgradient
norm, by taking advantage of the well-constructed Lyapunov function with a
slight modification and the phase-space representation based on the
high-resolution differential equation framework from the implicit-velocity
scheme.
