#' Sparse Discriminant Analysis solved via ADMM
#'
#' Applies alternating direction methods of multipliers algorithm to
#' the optimal scoring formulation of sparse discriminant analysis proposed
#' by Clemmensen et al. 2011.
#'
#' @param Xt n by p data matrix, (not a data frame, but a matrix)
#' @param Yt n by K matrix of indicator variables (Yij = 1 if i in class j).
#' This will later be changed to handle factor variables as well.
#' Each observation belongs in a single class, so for a given row/observation,
#' only one element is 1 and the rest is 0.
#' @param Om p by p parameter matrix Omega in generalized elastic net penalty.
#' @param gam Regularization parameter for elastic net penalty.
#' @param lam Regularization parameter for l1 penalty, must be greater than zero.
#' @param mu Penalty parameter for augmented Lagrangian term, must be greater than zero.
#' @param q Desired number of discriminant vectors.
#' @param PGsteps Maximum number if inner proximal gradient algorithm for finding beta.
#' @param PGtol Two stopping tolerances for inner ADMM method, first is absolute tolerance, second is relative.
#' @param maxits Number of iterations to run
#' @param tol Stopping tolerance for proximal gradient algorithm.
#' @param selector Vector to choose which parameters in the discriminant vector will be used to calculate the
#' regularization terms. The size of the vector must be *p* the number of predictors. The
#' default value is a vector of all ones. This is currently only used for ordinal classification.
#' @param initTheta Initial first theta, default value is a vector of ones.
#' @return \code{SDAD} returns an object of \code{\link{class}} "\code{SDAD}" including a list
#' with the following named components: (More will be added later to handle the predict function)
#' \describe{
#' \item{\code{call}}{The matched call.}
#' \item{\code{B}}{p by q matrix of discriminant vectors.}
#' \item{\code{Q}}{K by q matrix of scoring vectors.}
#' \item{\code{subits}}{Total number of iterations in proximal gradient subroutine.}
#' \item{\code{totalits}}{Number coordinate descent iterations for all discriminant vectors}
#' }
#' @seealso \code{SDADcv}, \code{\link{SDAAP}} and \code{\link{SDAP}}
#' @keywords internal
SDAD <- function (x, ...) UseMethod("SDAD")
#' @return \code{NULL}
#'
#' @rdname SDAD
#' @method SDAD default
SDAD.default <- function(Xt, Yt, Om, gam, lam, mu, q, PGsteps, PGtol, maxits, tol, selector = rep(1,dim(Xt)[2]), initTheta, ...){
# TODO: Handle Yt as a factor and generate dummy matrix from it
###
# Initialize training sets etc
###
# Get training data size
nt <- dim(Xt)[1] # num. samples
p <- dim(Xt)[2] # num. features
K <- dim(Yt)[2] # num. classes
# Logging variables
subits <- 0
totalits <- rep(maxits,q)
# Check if Om is diagonal. If so, use matrix inversion lemma in linear
# system solves.
if(norm(diag(diag(Om))-Om,type = "F") < 1e-15 & sum(selector) == length(selector)){
# Flag to use Sherman-Morrison-Woodbury to translate to
# smaller dimensional linear system solves.
SMW <- 1
# Easy to invert diagonal part of Elastic net coefficient matrix.
M <- mu + 2*gam*diag(Om)
Minv = 1/M
# Cholesky factorization for smaller linear system.
RS = chol(diag(nt) + 2*Xt%*%((Minv/nt)*t(Xt)));
} else{ # Use Cholesky for solving linear systems in ADMM step
# Flag to not use SMW
SMW <- 0
A <- mu*diag(p) + 2*(crossprod(Xt)/nt + gam*Om)
R2 <- chol(A)
}
#+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
# Matrices for theta update.
#+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
D <- (1/nt)*(crossprod(Yt))
R <- chol(D) # Cholesky factorization of D.
# Initialize B and Q.
Q <- matrix(1,K,q)
B <- matrix(0,p,q)
#++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
# Call Alternating Direction Method to solve SDA.
#++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
# For j=1,2,..., q compute the SDA pair (theta_j, beta_j).
for(j in 1:q){
###
# Initialization
###
# Compute Qj (K by j, first j-1 scoring vectors, all-ones last col)
Qj <- Q[,1:j]
# Precompute Mj = I - Qj*Qj'*D.
Mj <- function(u){
return(u - Qj%*%(crossprod(Qj,D%*%u)))
}
# Initialize theta
theta <- matrix(stats::runif(K),nrow=K,ncol=1)
theta <- Mj(theta)
if(j == 1 & !missing(initTheta)){
theta=initTheta
}
theta <- theta/as.numeric(sqrt(crossprod(theta,D%*%theta)))
# Initialize coefficient vector for elastic net step
d <- 2*crossprod(Xt,Yt%*%(theta/nt))
# Initialize beta
if(SMW == 1){
btmp <- Xt%*%(Minv*d)/nt
beta <- (Minv*d) - 2*Minv*(crossprod(Xt,backsolve(RS,forwardsolve(t(RS),btmp))))
}else{
beta <- backsolve(R2,forwardsolve(t(R2),d))
}
#+++++++++++++++++++++++++++++++++++++++++++++++++++++
# Alternating direction method to update (theta, beta)
#+++++++++++++++++++++++++++++++++++++++++++++++++++++
for(its in 1:maxits){
# Update beta using alternating direction method of multipliers.
b_old <- beta
if(SMW == 1){
# Use SMW-based ADMM
betaOb <- ADMM_EN_SMW(Minv, Xt, RS, d, beta, lam, mu, PGsteps, PGtol, TRUE, selector)
beta <- betaOb$y
} else{
betaOb <- ADMM_EN2(R2, d, beta, lam, mu, PGsteps, PGtol, TRUE, selector)
beta <- betaOb$y
}
subits <- subits + betaOb$k
# Update theta using the projected solution
if(norm(beta, type="2") > 1e-15){
# Update theta
b <- crossprod(Yt, Xt%*%beta)
y <- forwardsolve(t(R),b)
z <- backsolve(R,y)
tt <- Mj(z)
t_old <- theta
theta <- tt/sqrt(as.numeric(crossprod(tt,D%*%tt)))
# Update changes
db <- norm(beta-b_old, type="2")/norm(beta, type="2")
dt <- norm(theta-t_old, type="2")/norm(theta, type="2")
} else{
# Update b and theta
beta <- beta*0
theta <- theta*0
db <- 0
dt <- 0
}
if(max(db,dt) < tol){
# Converged
totalits[j] <- its
break
}
}
# Make the first argument be positive, this is to make the results
# more reproducible and consistent.
if(theta[1] < 0){
theta <- (-1)*theta
beta <- (-1)*beta
}
# Update Q and B
Q[,j] <- theta
B[,j] <- beta
}
totalits <- sum(totalits)
#Return B and Q in a SDAAP object
retOb <- structure(
list(call = match.call(),
B = B,
Q = Q,
subits = subits,
totalits = totalits),
class = "SDAD")
return(retOb)
}
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