In jefferis/manifoldreduction: Manifold Dimension Reduction after Chigirev and Bialek

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manifoldreduction

An implementation of Chigirev and Bialek's algorithm described in "Optimal Manifold Representation of Data: An Information Theoretic Approach".

Quick Start

For the impatient ...

# install
if (!require("devtools")) install.packages("devtools")
devtools::install_github("jefferis/manifoldreduction")

# use
library(manifoldreduction)

# help for functions
?manifold_reduction

# run tests
library(testthat)
test_package("manifoldreduction")

Installation

Currently there isn't a released version on CRAN but you can use the remotes package to install the development version:

if (!require("remotes")) install.packages("remotes")
devtools::install_github("jefferis/manifoldreduction")

Note: Windows users may need Rtools and remotes to install this way.

Example

As an example, let's use a noisy 2D circle. First a helper function for the parametric form of the equation of a circle:

circle <- function(r = 1, x = 0, y = 0, n = 360) {
  theta = seq(from = 0,
              to = 2 * pi,
              length.out = n)
  cbind(X = x + r * cos(theta), Y = y + r * sin(theta))
}

Now we can add a bit of noise to the radius

noisy_circle=circle(r=rnorm(360, mean=1, sd=.1))
plot(noisy_circle, asp = 1)
lines(circle(), col='black')

Now let's try recovering a low dimensional manifold. The default parameters will essentially look for a line.

library(manifoldreduction)
# NB expects d x N points (not N x d)
q=manifold_reduction(t(noisy_circle), no_iterations = 10, Verbose=FALSE)

This toy example converges fast, so I have reduced the number of iterations. If you plot the results (red line), you can see that it does a good job of smoothly recovering the structure of the original input points.

plot(noisy_circle, asp=1)
lines(circle(), col='black')
lines(t(q$gamma), col='red')

However there is a clear bias in the position of the recovered circle. This is down to the structure in these data - distant points in the dataset still contribute to the reconstruction of the manifold which will therefore be biased towards a medial position. If one reduces the number of points that are used:

q2=manifold_reduction(t(noisy_circle), no_iterations = 10, Verbose=FALSE, knntouse = 40)
plot(noisy_circle, asp=1)
lines(circle(), col='black')
lines(t(q$gamma), col='red')
lines(t(q2$gamma), col='blue')

then one obtains a less biased but somewhat noisier estimate (blue) (compare with the previous estimate in red or the underlying circle in black).

Of course this toy example could be better solved if one used a knowledge of the expected distribution (a circle) as part of the fitting process. However the interesting point with this method is that it is a rather general procedure that can be used for arbitrary point distributions in higher dimensions.

jefferis/manifoldreduction documentation built on Oct. 25, 2019, 8:03 a.m.