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In eigenfaces, one used the inverse transformation PCA is capable of doing to reconstruct the low dimensional face image. In tsne one may not reconstruct the original dataset to produce something akin to eigenfaces as a result of the non invertibility of the transformation.

What dimensionality reduction methods besides PCA allows reconstruction of the original image in lower dimensional space, .I.e what eigenfaces does?

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Of the top of my head, some dimensionality reduction methods to consider are:

  • Non-negative matrix factorisation (NNMF). We approximate a non-negative matrix as the product of two other (hopefully simpler) non-negative matrices. Effectively each component is guaranteed to be positive and therefore have an clear additive representation.
  • Independent component analysis (ICA). We solve the blind source signal separation problem; effectively each component is a statistically independent source of variation.
  • CUR decomposition (CUR). We try to approximate our matrix with rows and columns come from the original matrix; effectively each component is a pre-existing row. As such all our components have a natural interpretation as they exactly represent single units from our existing sample.
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