MGSDA (version 1.6.1)

dLDA: Estimate the matrix of discriminant vectors using L_1 penalty on the rows

Description

Solve Multi-Group Sparse Discriminant Anlalysis problem for the supplied value of the tuning parameter lambda.

Usage

dLDA(xtrain, ytrain, lambda, Vinit = NULL,eps=1e-6,maxiter=1000,rho=1)

Value

Returns a px(G-1) matrix of canonical vectors V.

Arguments

xtrain

A Nxp data matrix; N observations on the rows and p features on the columns.

ytrain

A N-vector containing the group labels. Should be coded as 1,2,...,G, where G is the number of groups.

lambda

Tuning parameter.

Vinit

A px(G-1) optional initial value for the optimization algorithm; the default value is NULL.

eps

Tolerance level for the convergence of the optimization algorithm; the default value is 1e-6.

maxiter

Maximal number of iterations for the optimization algorithm; the default value is 1000.

rho

A scalar that ensures the objective function is bounded from below; the default value is 1.

Author

Irina Gaynanova

Details

Solves the following optimization problem: $$\min_V \frac12 Tr(V^tWV+\rho V^tDD^tV)-Tr(D^tV)+\lambda\sum_{i=1}^p\|v_i\|_2$$ Here W is the within-group sample covariance matrix and D is the matrix of orthogonal contrasts between the group means, both are constructed based on the supplied values of xtrain and ytrain.

When \(G=2\), the row penalty reduces to vector L_1 penalty.

References

I.Gaynanova, J.Booth and M.Wells (2016) "Simultaneous Sparse Estimation of Canonical Vectors in the p>>N setting", JASA, 111(514), 696-706.

Examples

# Example 1
n=10
p=100
G=3
ytrain=rep(1:G,each=n)
set.seed(1)
xtrain=matrix(rnorm(p*n*G),n*G,p)
V=dLDA(xtrain,ytrain,lambda=0.1)
sum(rowSums(V)!=0) # number of non-zero rows