"MultidimensionalScaling" (Machine Learning Method)
- Method for DimensionReduction, DimensionReduce, FeatureSpacePlot and FeatureSpacePlot3D.
- Reduce the dimension of data using a metric multidimensional scaling.
Details & Suboptions
- "MultidimensionalScaling" is a nonlinear distance-based dimensionality reduction method. The method attempts to find a low-dimensional embedding of data using a transformation that preserves the pairwise distances.
- "MultidimensionalScaling" is able to learn nonlinear manifolds; however, it can be slow when the number of examples is large.
- The following plots show two-dimensional embeddings learned by the "MultidimensionalScaling" method applied to the benchmarking datasets Fisher's Irises, MNIST and FashionMNIST:
- Given the distance matrix
of data points in the original space, "MultidimensionalScaling" attempts to find the lower-dimensional embeddings
, such that distances in the lower-dimensional space match the distances between data points in the original space,
yi-yj
≈
. The lower-dimensional embeddings
are computed by minimizing the embedding cost: ∑i,j [
yi-yj
-
]2.
- The following suboptions can be given:
-
MaxIterations Automatic maximum number of optimization steps "MinRelativeChange" Automatic minimum relative change of the cost value to continue the optimization process

Examples
open allclose allBasic Examples (2)Summary of the most common use cases
Reduce the dimension of random vectors using the "MultidimensionalScaling" method:

https://wolfram.com/xid/0fcu9cribt9h07oxaro-b9tmml

Create and visualize a "Swiss roll" dataset:

https://wolfram.com/xid/0fcu9cribt9h07oxaro-fqjqny

https://wolfram.com/xid/0fcu9cribt9h07oxaro-7bpljt

Train a nonlinear dimension reducer using "MultidimensionalScaling" on the dataset to map to two-dimensional space:

https://wolfram.com/xid/0fcu9cribt9h07oxaro-4b74ci

Find and visualize the data coordinates in the reduced space:

https://wolfram.com/xid/0fcu9cribt9h07oxaro-hxupzm

Visualize the dataset in the original space, with each point colored according to its reduced variable:

https://wolfram.com/xid/0fcu9cribt9h07oxaro-2lgnr6

Scope (1)Survey of the scope of standard use cases
Dataset Visualization (1)
Load the Fisher Iris dataset from ExampleData:

https://wolfram.com/xid/0fcu9cribt9h07oxaro-i0qojo

https://wolfram.com/xid/0fcu9cribt9h07oxaro-ljl8f4

Generate a reducer function using "MultidimensionalScaling" with the features of each example:

https://wolfram.com/xid/0fcu9cribt9h07oxaro-hw72bx

Group the examples by their species:

https://wolfram.com/xid/0fcu9cribt9h07oxaro-u41y3o
Reduce the dimension of the features:

https://wolfram.com/xid/0fcu9cribt9h07oxaro-2geip6
Visualize the reduced dataset:

https://wolfram.com/xid/0fcu9cribt9h07oxaro-ck8dgi

Options (1)Common values & functionality for each option
MaxIterations (1)

https://wolfram.com/xid/0fcu9cribt9h07oxaro-tcwdw7
Reduce the dimension of images using "MultidimensionalScaling":

https://wolfram.com/xid/0fcu9cribt9h07oxaro-51nrd6
Find the reduced features using a different MaxIterations option:

https://wolfram.com/xid/0fcu9cribt9h07oxaro-o6o2zn
Visualize the obtained features and compare the results:

https://wolfram.com/xid/0fcu9cribt9h07oxaro-5vrs5c
