"Isomap" (Machine Learning Method)

Details & Suboptions

  • "Isomap", which stands for isometric mapping, is a nonlinear neighbor-based dimensionality reduction method. The method attempts to find a low-dimensional embedding of data via a transformation that preserves geodesic distances.
  • "Isomap" is able to learn nonlinear manifolds; however, it gives poor results on boundaries, can fail if data has high-density variations, and is computationally expensive.
  • The following plots show two-dimensional embeddings learned by the "Isomap" method applied to the benchmark datasets Fisher's Irises, MNIST and FashionMNIST:
  • "Isomap" constructs a neighborhood graph on N data points given their nearest neighbors. The geodesic distances between all pairs of points on the manifold are estimated by the shortest path in the nearest neighbors graph, , which results in the matrix of graph distances . The method attempts to find the embedding for which the Euclidean distance (shortest distance) is equal to the geodesic distance. The lower-dimensional embeddings are computed by minimizing the embedding cost:                                     min ∑ Ni=1 [||yi-yj||- dijG]2
  • The method is equivalent to performing the classical "MultidimensionalScaling" on the matrix of graph distances . Consequently, the Euclidean distances in the embedding space match the graph distances: ||yi-yj||≈ dijG.
  • The following suboption can be given:
  • "NeighborsNumber" Automaticnumber of neighbors k

Examples

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Basic Examples  (1)

Create and visualize a "Swiss roll" dataset:

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

Find and visualize the data coordinates in the reduced space:

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

Scope  (1)

Dataset Visualization  (1)

Load the Fisher Iris dataset from ExampleData:

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

Group the examples by their species:

Reduce the dimension of the features:

Visualize the reduced dataset:

Options  (1)

"NeighborsNumber"  (1)

Generate a dataset of different head poses from 3D geometry data with random viewpoints:

Visualize different head poses:

Reduce the dataset to a two-dimensional representation by specifying the "NeighborsNumber" in the neighborhood graph for performing the isometric mapping:

Visualize the original images in the reduced space, in which the up-down and front-side poses are disentangled: