The proximal operator for the Sorted L1 Norm, which is the penalty function in SLOPE. It solves the problem $$ \arg\,\min_x \Big(J(x, \lambda) + \frac{1}{2} ||x - v||_2^2\Big) $$ where \(J(x, \lambda)\) is the Sorted L1 Norm.
sortedL1Prox(x, lambda, method = c("stack", "pava"))M. Bogdan, E. van den Berg, Chiara Sabatti, Weijie Su, and Emmanuel J. Candès, “SLOPE – adaptive variable selection via convex optimization,” Ann Appl Stat, vol. 9, no. 3, pp. 1103–1140, 2015, doi: 10.1214/15-AOAS842.
A vector. In SLOPE, this is the vector of coefficients.
A non-negative and decreasing sequence
of weights for the Sorted L1 Norm. Needs to be the same length as
x.
Method used in the prox. "stack" is a stack-based algorithm
(Algorithm 4 in Bogdan et al.). "pava" is the PAVA algorithm used
in isotonic regression (also Algorithm 3 in Bogdan et al.).
An evaluation of the proximal operator at x and lambda.