Intrinsic Dimension Estimation based on Manifold Assumption and Graph Distance

As the name suggests, this function assumes that the data is sampled from the manifold in that graph representing the underlying manifold is first estimated via \(k\)-nn. Then graph distance is employed as an approximation of geodesic distance to locally estimate intrinsic dimension.

est.gdistnn(X, k = 5, k1 = 3, k2 = 10)

Arguments

X

an \((n\times p)\) matrix or data frame whose rows are observations.

k

the neighborhood size used for constructing a graph. We suggest it to be large enough to build a connected graph.

k1

local neighborhood parameter (smaller radius) for graph distance.

k2

local neighborhood parameter (larger radius) for graph distance.

Value

a named list containing containing

estdim

the global estimated dimension, which is averaged local dimension.

estloc

a length-\(n\) vector of locally estimated dimension at each point.

References

He J, Ding L, Jiang L, Li Z, Hu Q (2014). “Intrinsic Dimensionality Estimation Based on Manifold Assumption.” Journal of Visual Communication and Image Representation, 25(5), 740--747.

Author

Kisung You

Examples

# \donttest{
## create 3 datasets of intrinsic dimension 2.
X1 = aux.gensamples(dname="swiss")
X2 = aux.gensamples(dname="ribbon")
X3 = aux.gensamples(dname="saddle")

## acquire an estimate for intrinsic dimension
out1 = est.gdistnn(X1, k=10)
out2 = est.gdistnn(X2, k=10)
out3 = est.gdistnn(X3, k=10)

## print the results
sprintf("* est.gdistnn : estimated dimension for 'swiss'  data is %.2f.",out1$estdim)
#> [1] "* est.gdistnn : estimated dimension for 'swiss'  data is 2.04."
sprintf("* est.gdistnn : estimated dimension for 'ribbon' data is %.2f.",out2$estdim)
#> [1] "* est.gdistnn : estimated dimension for 'ribbon' data is 2.34."
sprintf("* est.gdistnn : estimated dimension for 'saddle' data is %.2f.",out3$estdim)
#> [1] "* est.gdistnn : estimated dimension for 'saddle' data is 2.46."

line1 = paste0("* est.gdistnn : 'swiss'  estiamte is ",round(out1$estdim,2))
line2 = paste0("* est.gdistnn : 'ribbon' estiamte is ",round(out2$estdim,2))
line3 = paste0("* est.gdistnn : 'saddle' estiamte is ",round(out3$estdim,2))
cat(paste0(line1,"\n",line2,"\n",line3))
#> * est.gdistnn : 'swiss'  estiamte is 2.04
#> * est.gdistnn : 'ribbon' estiamte is 2.34
#> * est.gdistnn : 'saddle' estiamte is 2.46

## compare with local-dimension estimate
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3))
hist(out1$estloc, main="Result-'Swiss'", xlab="local dimension")
abline(v=out1$estdim, lwd=3, col="red")
hist(out2$estloc, main="Result-'Ribbon'", xlab="local dimension")
abline(v=out2$estdim, lwd=3, col="red")
hist(out3$estloc, main="Result-'Saddle'", xlab="local dimension")
abline(v=out2$estdim, lwd=3, col="red")

par(opar)
# }