Pretty much anything in classical linear algebra can be abstracted to something more powerful by taking a swig of the Axiom of Choice.
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Replying to @EmilyGorcenski
A great example is PCA. It's often viewed as a dimensionality reduction technique, where we construct an orthonormal basis from data.
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Replying to @EmilyGorcenski
But orthonormalization exists in infinite dimensional spaces, too, and so with the AoC we could choose an infinite basis instead.
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Replying to @EmilyGorcenski
This is often called a spectral expansion, and one common form is the Karhunen-Loève Transform.
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Replying to @EmilyGorcenski
In KLT, we compute eigenfunctions, which are equivalent to eigenvectors in a discrete basis over ℝ^n. In fact, PCA is just the discrete KLT.
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Replying to @EmilyGorcenski
But we can restrict the KLT in other useful ways. Polynomials form a linear vector space, so choosing orthogonal polynomials makes sense.
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Replying to @EmilyGorcenski
As it turns out, the weighting functions in the inner product space for orthogonal polynomials are often closely related to probability fxns
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Replying to @EmilyGorcenski
shouldnt this only appky to orthonormal polynomials if you want the total space to sum to 1 (
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So as it turns out you can just scale the polynomial to get the desired behavior.
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Replying to @EmilyGorcenski
ofc i just want to make sure my understanding is correct
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