RandPro (version 0.1.0)

find_dim_JL: Johnson - Lindenstrauss Function

Description

Johnson Lindenstrauss Transform[JLT] is the heart of random projection. The lemma states that a small set of points in a high-dimensional space can be embedded into a space of much lower dimension in such a way that distances between the points are nearly preserved. The lemma has used in dimensionality reduction, compressed sensing, manifold learning and graph embedding. \((1 - epsilon) ||x - y||^2 < ||RP(x) - RP(y)||^2 < (1 + epsilon) ||x - y||^2\) where x and y are number of rows and columns respectively

Usage

find_dim_JL(sample, epsilon = 0.1)

Arguments

sample

- number of samples

epsilon

- error tolerance level with default value 0.1

Value

minimum number of dimension required to maintain the pai wise distance with the controlled amount of error

Details

The function find_dim_JL() is used to find the minimum dimension required to project the data from high dimensional space to low dimensional space. The number of sample and error tolerant level was passed as an input argument to the function find_dim_JL() . It will return the minimal size of the random subspace to guarantee a bounded distortion introduced by the random projection.

References

[1] William B.Johnson, Joram Lindenstrauss, "Extension of Lipschitz mappings into a Hilbert space (1984)"

[2] Sanjoy Dasgupta , Anupam Gupta "An elementary proof of a theorem of Johnson and Lindenstrauss (2003)"

See Also

Johnson-Lindenstrauss Elementary Proof

Examples

# NOT RUN {
#load library
library(RandPro)

#Calculate minimum dimension using eps =0.5 for 1000000 sample
y <- find_dim_JL(1000000,0.5)

#Calculating minimum dimension using different epsilon value for 1000000 sample
d <-  c(0.5,0.1)
x<- find_dim_JL(103260,d)

# }