  In machine learning and data science, we often consider efficiency for
solving problems. In sparse estimation, such as fused lasso and convex
clustering, we apply either the proximal gradient method or the alternating
direction method of multipliers (ADMM) to solve the problem. It takes time to
include matrix division in the former case, while an efficient method such as
FISTA (fast iterative shrinkage-thresholding algorithm) has been developed in
the latter case. This paper proposes a general method for converting the ADMM
solution to the proximal gradient method, assuming that the constraints and
objectives are strongly convex. Then, we apply it to sparse estimation
problems, such as sparse convex clustering and trend filtering, and we show by
numerical experiments that we can obtain a significant improvement in terms of
efficiency.
