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
Or if we look at a measure theoretic view, certain polynomial families are orthogonal with respect to a probability measure.
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Replying to @EmilyGorcenski
Norbert Wiener studied this in the 30s and 40s and called it "Polynomial Chaos." It's one of the most powerful unheard of techniques around.
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Replying to @EmilyGorcenski
It's basically a tool that allows us to write arbitrary (finite variance) distributions with respect to chosen, happy distributions.
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Replying to @EmilyGorcenski
The math looks a lot different, but it's basically PCA with an (infinite) orthogonal polynomial basis.
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Replying to @EmilyGorcenski
It's used in CFD somewhat, that's about it. I've never met a data scientist or even a math professor who's heard of it.
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But it is incredibly powerful and parallelizable, particularly for simulation. Maybe I should write an open source python library someday..
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