# Example 1 from Candes: Highly Robust Error Correction by Convex Programming # Revised Version: June 23, 2007 RWK. DSelector <- function(X,y,K, sparse = FALSE){ # This is a simple implementation of the Dantzig selector procedure of Candes-Tao. # The basic idea of the Dantzig selector is to solve problems that look like this: # # min || b ||_1 such that || A'r ||_oo < K # # where r = y - Ab. But in this version we've implement a slightly more general #version for: # # min || b ||_1 such that -K <= A'r <= K # # for vector valued K. We can formulate this in the usual QR setup as # # rq.fit.fnc(X,y,R=R, r=r) # # which solves problems of the form: # # min || y - Xb ||_1 s.t. Rb >= r # # If X is sparse it would be good to use the sparse version, but this needs work. # require(quantreg) n <- nrow(X) p <- ncol(X) A <- crossprod(X) R <- rbind(A, -A) a <- crossprod(X,y) r <- c(a-K, -a-K) zp <- rep(0,p) if(sparse){ Ip <- as(p,"matrix.diag.csr") f <- rq.fit.sfnc(Ip, zp, R=R, r=r) } else{ Ip <- diag(p) f <- rq.fit.fnc(Ip, zp, R=R, r=r) } return(f) } LPDecoderRing <- function(A,y,sigma,Kval = 3){ # R Implementation of the Candes-Gauss-Dantzig Decoder Ring NullSpace <- function(A){ S <- svd(A,nrow(A)) r <- length(S\$d) S\$u[,(r+1):nrow(A),drop=FALSE] } Q <- t(NullSpace(A)) Qy <- Q %*% y K <- sqrt(diag(crossprod(Q))) * sigma * Kval f <- DSelector(Q, Qy, K) # Reprojection Step s <- abs(f\$coef) > sigma e <- rep(0,nrow(A)) e[s] <- lm(Qy ~ Q[,s] -1)\$coef xhat <- lm((y-e) ~ A - 1)\$coef return(xhat) } # Test Problem as in Section 4 of Candes (2007) n <- 256 m <- 2 * n A <- matrix(rnorm(m*n),m,n) rho <- 0.10 k <- round(rho * m) x <- rnorm(n) y <- A %*% x yt <- y sigma <- median(abs(y))/16 s <- sample(1:m,k) yt[s] <- -yt[s] # Original Candes example u <- rnorm(m)*sigma yt <- yt + u xhat <- LPDecoderRing(A, yt, sigma) pdf("Eg3.pdf",width=8,height=3) par(mfrow=c(1,3)) g <- lm(yt ~ A -1)\$coef rmseG <- round(sqrt(var(x-g)),3) plot(x,g,ylab=expression(hat(x)),main="Gauss") text(1,-2,paste("RMSE = ",rmseG)) plot(x,xhat,ylab=expression(hat(x)),main="Dantzig") rmseD <- round(sqrt(var(x-xhat)),3) text(1,-2,paste("RMSE = ",rmseD)) h <- lm((y + u) ~ A - 1)\$coef plot(x,h,ylab=expression(hat(x)),main="Oracle") rmseH <- round(sqrt(var(x-h)),3) text(1,-2,paste("RMSE = ",rmseH)) dev.off()