debiasingMatrix {selectiveInference}

Find an approximate inverse of a non-negative definite matrix.

Description

Find some rows of an approximate inverse of a non-negative definite symmetric matrix by solving optimization problem described in Javanmard and Montanari (2013). Can be used for approximate Newton step from some consistent estimator (such as the LASSO) to find a debiased solution.

Usage

debiasingMatrix(Xinfo, 
                is_wide,			
                nsample, 
                rows, 
		verbose=FALSE, 
		bound=NULL,            
   		linesearch=TRUE,    
   		scaling_factor=1.5, 
		max_active=NULL,    
		max_try=10,         
		warn_kkt=FALSE,     
		max_iter=50,       
		kkt_stop=TRUE,
		parameter_stop=TRUE,
		objective_stop=TRUE,
                kkt_tol=1.e-4,      
		parameter_tol=1.e-4,
		objective_tol=1.e-4)

Arguments

Xinfo

Either a non-negative definite matrix S=t(X) is_wide is TRUE, then Xinfo should be X, otherwise it should be S.

is_wide

Are we solving for rows of the debiasing matrix assuming it is a wide matrix so that Xinfo=X and the non-negative definite matrix of interest is t(X)

nsample

Number of samples used in forming the cross-covariance matrix. Used for default value of the bound parameter.

rows

Which rows of the approximate inverse to compute.

verbose

Print out progress as rows are being computed.

bound

Initial bound parameter for each row. Will be changed if linesearch is TRUE.

linesearch

Run a line search to find as small as possible a bound parameter for each row?

scaling_factor

In the linesearch, the bound parameter is either multiplied or divided by this factor at each step.

max_active

How large an active set to consider in solving the problem with coordinate descent. Defaults to max(50, 0.3*nsample).

max_try

How many tries in the linesearch.

warn_kkt

Warn if the problem does not seem to be feasible after running the coordinate descent algorithm.

max_iter

How many full iterations to run of the coordinate descent for each value of the bound parameter.

kkt_stop

If TRUE, check to stop coordinate descent when KKT conditions are approximately satisfied.

parameter_stop

If TRUE, check to stop coordinate descent based on relative convergence of parameter vector, checked at geometrically spaced iterations 2^k.

objective_stop

If TRUE, check to stop coordinate descent based on relative decrease of objective value, checked at geometrically spaced iterations 2^k.

kkt_tol

Tolerance value for assessing whether KKT conditions for solving the dual problem and feasibility of the original problem.

parameter_tol

Tolerance value for assessing convergence of the problem using relative convergence of the parameter.

objective_tol

Tolerance value for assessing convergence of the problem using relative decrease of the objective.

Details

This function computes an approximate inverse as described in Javanmard and Montanari (2013), specifically display (4). The problem is solved by considering a dual problem which has an objective similar to a LASSO problem and is solvable by coordinate descent. For some values of bound the original problem may not be feasible, in which case the dual problem has no solution. An attempt to detect this is made by stopping when the active set grows quite large, determined by max_active.

Value

M

Rows of approximate inverse of Sigma.

Author(s)

Ryan Tibshirani, Rob Tibshirani, Jonathan Taylor, Joshua Loftus, Stephen Reid

References

Adel Javanmard and Andrea Montanari (2013). Confidence Intervals and Hypothesis Testing for High-Dimensional Regression. Arxiv: 1306.3171

Examples

set.seed(10)
n = 50
p = 100
X = matrix(rnorm(n * p), n, p)
S = t(X) %*% X / n
M = debiasingMatrix(S, FALSE, n, c(1,3,5))
M2 = debiasingMatrix(X, TRUE, n, c(1,3,5))
max(M - M2)