cems (version 0.0)

cem: Conditional Expectation Manifolds

Description

This package computes principal surfaces based on the approach described in Gerber et. al. 2009. Prinicpal surfaces are typically found by minimizing $E[ || Y -g(\lambda(Y))|| ]$ over the functions $g:R^m \mapsto R^n$ with $m < n$ {m < n} and $\lambda_g:R^n \mapsto R^m$ defined as an orthogonal projection onto $g$. In Gerber et. al. 2009 the oppoiste approach is described; fixing $g_{\lambda}(x) = E[Y | \lambda(Y) = x]$ and minimzing over $\lambda$, i.e. optimizing the conditonal expectation manifold (CEM) given $\lambda$. Gerber et. al. 2009 called this approach kernel map manifolds (KMM) since both mappings where defined by kernel regression.

Usage

cem(y, z, knn = 15, iter = 100, step = 0.1, fudge = 2.5, verbose=1)
cem.optimize(object, iter = 50, step=0.1, verbose=1)
## S3 method for class 'cem':
predict(object, newdata, ...)

Arguments

y

$n$-dimensional data to compute conditional expectation manifold for.

z

Initalization for low dimensional mapping $\lambda$. For example an isomap or lle or PCA embedding of $y$.

knn

Number of nearest neighbors for graph computation.

iter

Number of optimization iterations, i.e. number of gradient desecent with line search setps.

step

Gradient descent step size

fudge

Fudge factor, amount of over-smoothing of inital CEM.

verbose

Report gradient descent information. 1 reports iteration number and mean squared projetcion distances. 2 has additonal information on step size and line search.

object

CEM object to do prediction for

newdata

Data to do prediction for. If ncol(newdata) == m for each point x in newdata g(x) is computed. If col{newdata} == n for each point y in newdata $\lambda(y)$ is computed.

...

Additional arguments have no effect.

Value

• An object of class "cem".

References

Samuel Gerber, Tolga Tasdizen, Ross Whitaker, Dimensionality Reduction and Principal Surfaces via Kernel Map Manifolds, In Proceedings of the 2009 International Conference on Computer Vison (ICCV 2009).

Examples

library(vegan)
library(rgl)
library(cems)

###Swissroll data set example

#Create swissroll toy data set
data(swissroll)
d <- swissroll(500, nstd = 0.5)
plot3d(d$Xn)
#d$Xn contains noisy (orthogonal Normal distirbuted to the swissroll) of d$X

#Compute isomap
iso <- isomap(dist(d$Xn), k=15)

#Create and optimize CEM
ps <- cem(y = d$Xn, z = iso$points[, 1:2], knn=20, iter=5)

#Create test set
dtest <- swissroll(500, nstd = 0.25)

#parametrize manifold coordinates from test set
coords <- predict(ps, dtest$Xn)

#reconstruct points on manifold from coordinates
p <- predict(ps, coords)

#mean squared test error
mean( rowSums((dtest$X - p)^2) )

#optimize some more with samller step size
ps <- cem.optimize(ps, iter=5, step = 0.05)

#mean sqyuared error
coords <- predict(ps, dtest$Xn)
p <- predict(ps, coords)
mean( rowSums((dtest$X - p)^2) )



###Frey faces example
#Do not run
data("frey_faces")

#create isomap
iso <- isomap(dist(faces), k=25)

#Create conditional expectation surface (no optimization here, 20 iteration take
#roughly 2h on a 2Ghz laptop)
ps <- cem(y=faces, z = iso$points[, 1:3], knn=45, iter=0)

#parametrize manifold coordinates
coords <- predict(ps)

#reconstruct points on manifold from coordinates
p <- predict(ps, coords)

#mean squared test error
mean( rowSums((faces - p)^2) )

#plot images
split.screen(c(2,3))


ind <- c(1, 120, 333)
for(i in 1:length(ind)){
 screen(i)
 im <- matrix(faces[ind[i], 560:1], 20, 28)
 image(1:nrow(im), 1:ncol(im), im, xlab="", ylab="")
 screen(i+3)
 im <- matrix(p[ind[i], 560:1], 20, 28)
 image(1:nrow(im), 1:ncol(im), im, xlab="", ylab="")
}