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"MultidimensionalScaling" (Machine Learning Method)

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 d_(ij) 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||≈ d_(ij). The lower-dimensional embeddings are computed by minimizing the embedding cost: ∑i,j [||yi-yj||- d_(ij)]2.
  • The following suboptions can be given:
  • MaxIterations Automaticmaximum number of optimization steps
    "MinRelativeChange"Automaticminimum relative change of the cost value to continue the optimization process

Examples

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Basic Examples  (2)Summary of the most common use cases

Reduce the dimension of random vectors using the "MultidimensionalScaling" method:

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https://wolfram.com/xid/0fcu9cribt9h07oxaro-b9tmml
Out[1]=1

Create and visualize a "Swiss roll" dataset:

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https://wolfram.com/xid/0fcu9cribt9h07oxaro-fqjqny
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https://wolfram.com/xid/0fcu9cribt9h07oxaro-7bpljt
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Train a nonlinear dimension reducer using "MultidimensionalScaling" on the dataset to map to two-dimensional space:

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https://wolfram.com/xid/0fcu9cribt9h07oxaro-4b74ci
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Find and visualize the data coordinates in the reduced space:

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https://wolfram.com/xid/0fcu9cribt9h07oxaro-hxupzm
Out[4]=4

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

In[5]:=5
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https://wolfram.com/xid/0fcu9cribt9h07oxaro-2lgnr6
Out[5]=5

Scope  (1)Survey of the scope of standard use cases

Dataset Visualization  (1)

Load the Fisher Iris dataset from ExampleData:

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https://wolfram.com/xid/0fcu9cribt9h07oxaro-i0qojo
In[2]:=2
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https://wolfram.com/xid/0fcu9cribt9h07oxaro-ljl8f4
Out[2]=2

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

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https://wolfram.com/xid/0fcu9cribt9h07oxaro-hw72bx
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Group the examples by their species:

In[4]:=4
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https://wolfram.com/xid/0fcu9cribt9h07oxaro-u41y3o

Reduce the dimension of the features:

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https://wolfram.com/xid/0fcu9cribt9h07oxaro-2geip6

Visualize the reduced dataset:

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https://wolfram.com/xid/0fcu9cribt9h07oxaro-ck8dgi
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Options  (1)Common values & functionality for each option

MaxIterations  (1)

Load the "MNIST" dataset:

In[1]:=1
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https://wolfram.com/xid/0fcu9cribt9h07oxaro-tcwdw7

Reduce the dimension of images using "MultidimensionalScaling":

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https://wolfram.com/xid/0fcu9cribt9h07oxaro-51nrd6

Find the reduced features using a different MaxIterations option:

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https://wolfram.com/xid/0fcu9cribt9h07oxaro-o6o2zn

Visualize the obtained features and compare the results:

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