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once we extract subspaces(subset of features) for a dataset using unsupervised learning, can we use them for any task I mean for classification, clustering, or outlier detection?

Is the selection of subspaces for different task must me different ? that is based on task selection of subsets of features changes?

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Yes, Different tasks need different subspaces. So learning subspace depends on the type of task we perform. For Example, Consider both clustering and outlier detection tasks. To perform clustering we need dense subspaces because dense subspace(dense attributes) is useful to extract the local groups whereas to perform outlier detection we need sparse subspaces because sparse subspace is useful to extract the local outliers.

So from this, we can tell the selection of relevant subspaces depends on the type of task

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