Optimal Recovery and Statistical Estimation in Lp Sobolev Classes
Technical report
"Optimal recovery and statistical estimation in Lp Sobolev classes"
is available for downloading.
Abstract
We present algorithms for the optimal recovery of a partial derivative
of a function at a point
when we have approximate measurements
of the Riesz transform of the function,
and this function belongs to a $L_p$ Sobolev class.
The algorithms are exactly optimal among linear algorithms
for the cases $p=1$ and
$p=2$ and we give tight bounds for the performance of the
algorithms when $p>1$, $p \neq 2$.
Previously only the case $p=2$ has been studied.
Algorithms for the optimal recovery problem provide optimal
estimators for several statistical problems,
when we calibrate algorithms suitably.
As examples we construct a nearly minimax estimator in
the Gaussian white noise model and an asymptotically nearly minimax
estimator for the problem of regression function estimation
with iid data.
Reproducing the figures
To reproduce and modify the figures of the article, apply
R functions
sobolp.R
and
sobolp.dim.R
(Information on R is available in
R archive network.)
source("~/public_html/Art/sobolp/sobolp.R")
source("~/public_html/Art/sobolp/sobolp.dim.R")
Figure1: d=1, g=0, 0.5, 0.9
d<-1
r<-0
postscript(file="/home/jsk/Arti/sobolp/sobolp-kuva1.ps",horizontal=FALSE)
par(mfrow=c(3,2))
g<-0
pala<-1.1
pyla<-2
pstep<-0.1
pvec<-seq(pala,pyla,pstep)
sala<-1
syla<-5
sstep<-0.25
svec<-seq(sala,syla,sstep)
kv<-sobolp(svec,pvec,g,d,r)
contour(kv$x,kv$y,kv$z,xlab=expression(beta),ylab="p",nlevels=25)
title(sub=expression(paste("a.1) ", 1<=p<=2, " , " , gamma==0)))
pala<-2
pyla<-50
pstep<-1
pvec<-seq(pala,pyla,pstep)
sala<-0.5
syla<-10
sstep<-0.5
svec<-seq(sala,syla,sstep)
kv<-sobolp(svec,pvec,g,d,epsi=0.5)
contour(kv$x,kv$y,kv$z,xlab=expression(beta),ylab="p",nlevels=12)
title(sub=expression(paste("a.2) ", p>=2, " , " , gamma==0)))
g<-0.5
pala<-1.1
pyla<-2
pstep<-0.1
pvec<-seq(pala,pyla,pstep)
sala<-0.6
syla<-4
sstep<-0.25
svec<-seq(sala,syla,sstep)
kv<-sobolp(svec,pvec,g,d,r)
contour(kv$x,kv$y,kv$z,xlab=expression(beta),ylab="p",nlevels=20)
title(sub=expression(paste("b.1) ", 1<=p<=2, " , " , gamma==0.5)))
pala<-2
pyla<-4.8
pstep<-0.2
pvec<-seq(pala,pyla,pstep)
sala<-2
syla<-10
sstep<-0.1
svec<-seq(sala,syla,sstep)
kv<-sobolp(svec,pvec,g,d,epsi=0)
contour(kv$x,kv$y,kv$z,xlab=expression(beta),ylab="p",nlevels=20)
title(sub=expression(paste("b.2) ", p>=2, " , " , gamma==0.5)))
g<-0.9
pala<-1.1
pyla<-2
pstep<-0.1
pvec<-seq(pala,pyla,pstep)
sala<-0.6
syla<-4
sstep<-0.25
svec<-seq(sala,syla,sstep)
kv<-sobolp(svec,pvec,g,d,r)
contour(kv$x,kv$y,kv$z,xlab=expression(beta),ylab="p",nlevels=15)
title(sub=expression(paste("c.1) ", 1<=p<=2, " , " , gamma==0.9)))
pala<-2
pyla<-3.5
pstep<-0.1
pvec<-seq(pala,pyla,pstep)
sala<-2
syla<-10
sstep<-0.1
svec<-seq(sala,syla,sstep)
kv<-sobolp(svec,pvec,g,d,epsi=0.5)
contour(kv$x,kv$y,kv$z,xlab=expression(beta),ylab="p",nlevels=12)
title(sub=expression(paste("c.2) ", p>=2, " , " , gamma==0.9)))
dev.off()
Figure 2: d=2, g=0, 0.5, 0.9
d<-2
r<-0
postscript(file="/home/jsk/Arti/sobolp/sobolp-kuva2.ps",horizontal=FALSE)
par(mfrow=c(3,2))
g<-0
pala<-1.1
pyla<-2
pstep<-0.1
pvec<-seq(pala,pyla,pstep)
sala<-2
syla<-4
sstep<-0.25
svec<-seq(sala,syla,sstep)
kv<-sobolp(svec,pvec,g,d,r)
contour(kv$x,kv$y,kv$z,xlab=expression(beta),ylab="p",nlevels=25)
title(sub=expression(paste("a.1) ", 1<=p<=2, " , " , gamma==0)))
pala<-2
pyla<-50
pstep<-1
pvec<-seq(pala,pyla,pstep)
sala<-1 #1.1
syla<-10
sstep<-0.5
svec<-seq(sala,syla,sstep)
kv<-sobolp(svec,pvec,g,d,epsi=0.5)
contour(kv$x,kv$y,kv$z,xlab=expression(beta),ylab="p",nlevels=15)
title(sub=expression(paste("a.2) ", p>=2, " , " , gamma==0)))
g<-0.5
pala<-1.1
pyla<-2
pstep<-0.1
pvec<-seq(pala,pyla,pstep)
sala<-1.5
syla<-4
sstep<-0.25
svec<-seq(sala,syla,sstep)
kv<-sobolp(svec,pvec,g,d,r)
contour(kv$x,kv$y,kv$z,xlab=expression(beta),ylab="p",nlevels=20)
title(sub=expression(paste("b.1) ", 1<=p<=2, " , " , gamma==0.5)))
pala<-2
pyla<-3.5
pstep<-0.1
pvec<-seq(pala,pyla,pstep)
sala<-2
syla<-8
sstep<-0.1
svec<-seq(sala,syla,sstep)
kv<-sobolp(svec,pvec,g,d,epsi=0.5)
contour(kv$x,kv$y,kv$z,xlab=expression(beta),ylab="p",nlevels=25)
title(sub=expression(paste("b.2) ", p>=2, " , " , gamma==0.5)))
g<-0.9
pala<-1.1
pyla<-2
pstep<-0.1
pvec<-seq(pala,pyla,pstep)
sala<-1
syla<-3.5
sstep<-0.2
svec<-seq(sala,syla,sstep)
kv<-sobolp(svec,pvec,g,d,r)
contour(kv$x,kv$y,kv$z,xlab=expression(beta),ylab="p",nlevels=12)
title(sub=expression(paste("c.1) ", 1<=p<=2, " , " , gamma==0.9)))
pala<-2
pyla<-3
pstep<-0.05
pvec<-seq(pala,pyla,pstep)
sala<-2.5
syla<-15
sstep<-0.2
svec<-seq(sala,syla,sstep)
kv<-sobolp(svec,pvec,g,d,epsi=0.5)
contour(kv$x,kv$y,kv$z,xlab=expression(beta),ylab="p",nlevels=20)
title(sub=expression(paste("c.2) ", p>=2, " , " , gamma==0.9)))
dev.off()
Figure 3: Constants and the dimensionality
dims<-c(1,2,3,4,5,6,7,8,9,10)
bets<-2+dims/2 #rep(6,length(dims))
p<-1
sd1<-sobolp.dim(dims,p,bets)
p<-2
sd2<-sobolp.dim(dims,p,bets)
p<-4
sd4<-sobolp.dim(dims,p,bets)
tyypit1<-c(rep(19,length(dims)),rep(21,length(dims)),rep(22,2*length(dims)))
tyypit2<-c(rep(19,length(dims)),rep(21,length(dims)))
postscript(file="/home/jsk/Arti/sobolp/sobolp-kuva3.eps",
horizontal=FALSE, onefile=FALSE, paper="special",
width=9, height=4)
par(mfrow=c(1,2))
plot(c(dims,dims,dims,dims),c(sd1$lowe,sd2$lowe,sd4$lowe,sd4$over),
xlab="d",ylab="",pch=tyypit1)
title(sub="a) constants")
plot(c(dims,dims),
c(log(sd2$lowe/sd1$lowe,base=10),log(sd4$over/sd2$over,base=10)),
xlab="d",ylab="",pch=tyypit2)
title(sub="b) log-ratios of the constants")
dev.off()
gv /home/jsk/Arti/sobolp/sobolp-kuva3.eps &
Find the bound for beta
g<-0
d<-10
r<-0
pala<-1.1
pyla<-4
pstep<-0.1
pvec<-seq(pala,pyla,pstep)
postscript(file="/home/jsk/Arti/sobolp/sobolpbound.ps") #,horizontal=FALSE)
b<-matrix(0,length(pvec),1)
for (i in 1:length(pvec)){
b[i]<-max(r/pvec[i]+d/2,
(r-g)/pvec[i]+d*(3/(2*pvec[i])-1/2),
d/2+(pvec[i]-2)*g+(pvec[i]-1)*r ,
d*(3/(2*pvec[i])-1/2) + r/pvec[i] - g/(pvec[i]*(pvec[i]-1)),
d/2+2*r*(pvec[i]-1)/pvec[i]+g*(pvec[i]-2)/pvec[i])
}
plot(pvec,b)
dev.off()
postscript(file="/home/jsk/Arti/sobolp/sobolp-kuva1.ps",horizontal=FALSE)
par(mfrow=c(3,2))
g<-0
pala<-1.1
pyla<-2
pstep<-0.1
pvec<-seq(pala,pyla,pstep)
sala<-1
syla<-5
sstep<-0.25
svec<-seq(sala,syla,sstep)
kv<-sobolp(svec,pvec,g,d,r)
contour(kv$x,kv$y,kv$z,xlab=expression(beta),ylab="p",nlevels=25)
title(sub=expression(paste("a.1) ", 1<=p<=2, " , " , gamma==0)))
pala<-2
pyla<-50
pstep<-1
pvec<-seq(pala,pyla,pstep)
sala<-0.5
syla<-10
sstep<-0.5
svec<-seq(sala,syla,sstep)
kv<-sobolp(svec,pvec,g,d,epsi=0.5)
contour(kv$x,kv$y,kv$z,xlab=expression(beta),ylab="p",nlevels=12)
title(sub=expression(paste("a.2) ", p>=2, " , " , gamma==0)))
g<-0.5
pala<-1.1
pyla<-2
pstep<-0.1
pvec<-seq(pala,pyla,pstep)
sala<-0.6
syla<-4
sstep<-0.25
svec<-seq(sala,syla,sstep)
kv<-sobolp(svec,pvec,g,d,r)
contour(kv$x,kv$y,kv$z,xlab=expression(beta),ylab="p",nlevels=20)
title(sub=expression(paste("b.1) ", 1<=p<=2, " , " , gamma==0.5)))
pala<-2
pyla<-4.8
pstep<-0.2
pvec<-seq(pala,pyla,pstep)
sala<-2
syla<-10
sstep<-0.1
svec<-seq(sala,syla,sstep)
kv<-sobolp(svec,pvec,g,d,epsi=0)
contour(kv$x,kv$y,kv$z,xlab=expression(beta),ylab="p",nlevels=20)
title(sub=expression(paste("b.2) ", p>=2, " , " , gamma==0.5)))
g<-0.9
pala<-1.1
pyla<-2
pstep<-0.1
pvec<-seq(pala,pyla,pstep)
sala<-0.6
syla<-4
sstep<-0.25
svec<-seq(sala,syla,sstep)
kv<-sobolp(svec,pvec,g,d,r)
contour(kv$x,kv$y,kv$z,xlab=expression(beta),ylab="p",nlevels=15)
title(sub=expression(paste("c.1) ", 1<=p<=2, " , " , gamma==0.9)))
pala<-2
pyla<-3.5
pstep<-0.1
pvec<-seq(pala,pyla,pstep)
sala<-2
syla<-10
sstep<-0.1
svec<-seq(sala,syla,sstep)
kv<-sobolp(svec,pvec,g,d,epsi=0.5)
contour(kv$x,kv$y,kv$z,xlab=expression(beta),ylab="p",nlevels=12)
title(sub=expression(paste("c.2) ", p>=2, " , " , gamma==0.9)))
dev.off()
