admm_bp: Fitting A Basis Pursuit Model Using ADMM Algorithm
In yixuan/ADMM: Solving Statistical Optimization Problems Using the ADMM Algorithm

Description Usage Arguments Additional Options Model Fitting Author(s) Examples

View source: R/10_admm_bp.R

Basis Pursuit is an optimization problem that minimizes ||β||_1 subject to y = X * β. Here X is an n by p matrix with p > n. Basis Pursuit is broadly applied in Compressed Sensing to recover a sparse vector β from the transformed lower dimensional vector y.

This function will not directly conduct the computation, but rather returns an object of class "ADMM_BP" that contains several memeber functions to actually constructs and fits the model.

Member functions that are callable from this object are listed below:

`$opts()`	Setting additional options. See section Additional Options for details.
`$fit()`	Fit the model and do the actual computation. See section Model Fitting for details.

1	admm_bp(x, y, ...)

`x`	The transformation matrix
`y`	The transformed vector to recover from

Additional options related to ADMM algorithm can be set through the $opts() member function of an "ADMM_BP" object. The usage of this method is

1 2	model$opts(maxit = 10000, eps_abs = 1e-4, eps_rel = 1e-4, rho = NULL)

Here model is the object returned by admm_bp(). Explanation of the arguments is given below:

maxit: Maximum number of iterations.
eps_abs: Absolute tolerance parameter.
eps_rel: Relative tolerance parameter.
rho: ADMM step size parameter. If set to NULL, the program will compute a default one.

This member function will implicitly return the "ADMM_BP" object itself.

Model will be fit after calling the $fit() member function. This is no argument that needs to be set. The function will return an object of class "ADMM_BP_fit", which contains the following fields:

beta: The recovered β vector in sparse form.
niter: Number of ADMM iterations.

Class "ADMM_BP_fit" also contains a $plot() member function, which plots the coefficients against their indices. See the examples below.

Yixuan Qiu <http://statr.me>

## An Compressed Sensing example ##

## Create a sparse signal vector
set.seed(123)
n = 50
p = 100
nsig = 15
beta_true = c(runif(nsig), rep(0, p - nsig))
beta_true = sample(beta_true)

## Generate the transformation matrix and the compressed vector
x = matrix(rnorm(n * p), n, p)
y = drop(x %*% beta_true)

## Build the model
model = admm_bp(x, y)

## Request a higher precision
model$opts(eps_rel = 1e-5)

## Fit the model
res = model$fit()
res

## Plot for the recovered vector
res$plot()

## The steps above can be accomplished using a chainable call
admm_bp(x, y)$opts(eps_rel = 1e-5)$fit()$plot()

## Compare the true beta and the recovered one
library(ggplot2)
g = res$plot()
d = data.frame(ind = seq_along(beta_true),
               coef = beta_true)
g + geom_segment(aes(x = ind + 0.5, xend = ind + 0.5,
                     y = coef, yend = 0), data = d, color = "red")