sortedL1Prox {SLOPE}

Sorted L1 Proximal Operator

Description

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.

Usage

sortedL1Prox(x, lambda, method = c("stack", "pava"))

Arguments

x

A vector. In SLOPE, this is the vector of coefficients.

lambda

A non-negative and decreasing sequence of weights for the Sorted L1 Norm. Needs to be the same length as x.

method

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.).

Value

An evaluation of the proximal operator at x and lambda.

Source

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.