Linear and non-linear algorithms

Dimensionality reduction algorithms differ in the constraints they impose on the new variables and how they aim to minimize the loss of information:

  • Linear algorithms such as PCA and ICA constrain the new variables to be linear combinations of the original features; that is, hyperplanes in a lower-dimensional space. Whereas PCA requires the new features to be uncorrelated, ICA goes further and imposes statistical independencethe absence of both linear and non-linear relationships. The following screenshot illustrates how PCA projects three-dimensional features into a two-dimensional space:
  • Non-linear algorithms are not restricted to hyperplanes and can capture more complex structure in the data. However, given the infinite number of options, the algorithms still need to make assumptions to arrive at a solution. In this section, we show how t-distributed Stochastic Neighbor Embedding (t-SNE) and Uniform Manifold Approximation and Projection (UMAP) are very useful for visualizing higher-dimensional data. The following screenshot illustrates how manifold learning identifies a two-dimensional sub-space in the three-dimensional feature space (the manifold_learning notebook illustrates the use of additional algorithms, including local linear embedding):
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