Backwards Principal Component Analysis and Principal Nested Relations

In non-Euclidean data spaces represented by manifolds (or more generally stratified spaces), analogs of principal component analysis can be more easily developed using a backwards approach. There has been a gradual evolution in the application of this idea from using increasing geodesic subspaces of submanifolds in analogy with PCA to using a “backward sequence” of a decreasing family of subspaces. We provide a version of the backwards approach by using a “nested sequence of relations” which define the decreasing sequences of subspaces which need not be geodesic. Because these are naturally inductively added in a backward sequence, they are frequently more tractable and overcome difficulties with using geodesics.

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