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Paying attention to feature distribution alignment (pun intended)
We discuss the meaning of weighted-orthogonality of function bases in feature engineering, and the relationship between the weight function and the feature distribution.
The differential equations community has been designing orthogonal bases w.r.t various weights for a long time, and have come up with plenty of methods and an enormous body of literature. This idea aligns with our intuition at the intro - mapping using a “uniformizing” transformation before computing Legendre polynomials produces an orthogonal basis w.r.t the original raw feature. Now it appears clear why the provocative title fits this post - we indeed paid close attention to the alignment between our non-linear features and the data distribution.
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