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Feature selection in a gist from what i understand is reducing the variables but retaining the labels as much as possible, from that pov this seems correct but i haven't found anything on this. Any help is appreciated.

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In your definition, feature selection (FS) is done in a supervised manner, but FS can also be unsupervised.

From a practitioner's point of view, your pipeline is reasonable. For instance, you can cluster your data points, then perform supervised FS (on the clustered points), and finally, if wanted, do classification, but you can also perform unsupervised FS, then clustering (on the featured data points), and finally classification.

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