```
tags:
- algorithm
- application
- transport
```

Nongaussian KL Expansion

Suppose we have a scalar-valued stochastic process

where we can define

Then, by construction,

which simply comes from the orthonormality of

It suffices to show that

The problem arises when *not* a GP; we of course maintain zero mean and identity covariance, but the higher moments will become nontrivial (and thus the joint distribution of

and consider

Suppose we, for the time being, can perform the integrals in ^992e28,^b94337 analytically (or, at the very worst, "close enough"), and thus can find the eigenfunctions and -values. Therefore, for a given set of samples

If we could sample

Suppose, though, we only have access to samples of the solution (and no knowledge/desire for any stochastic inputs). Then, we create an *invertible* map

From there, we can sample new approximate solutions

If we truly have

What to do? A few options, I suppose.

- Find a map
such that . Note that this is ill-posed; the*measures*can be equivalent even when the*samples*get transported poorly. For example, suppose the input is a coin-flip on 0 1 and the output is a coin flip on 1 2. There's two ways of transporting these distributions, but one of them will guarantee you're always wrong. - Find the map
as above such that . This totally disregards interpolation and just matches points (though I suppose, with enough parameters, you could pass overfitting?) - Use something like entropically-regularized OT? Idk enough to say much. Would I need the input/output to match?
- Suppose that
is in fact still stochastic for fixed , i.e., doesn't capture all of the randomness.*Then*, the approach described by is perfectly well-posed, I think?

We observe that the high order stochastic modes of the KLE are increasingly Gaussian (or at least log-convex looking). Why is that? Is there any reason the bimodal behavior shouldn't get pushed out to parameter number 28? The spectral decay is particularly fast, so this isn't altogether unreasonable to assume that, since the KLE hinges on everything being L2, and the Gaussian is the pretty reasonable choice of measure on L2, it tends to look log-convex or something like that?

The case studied in the siam UQ presentation is a reaction with bimodal output concentration. It seems entirely reasonable to then consider a stochastic process source term, for example, which can be represented via the KLE, and we use the diffusion equation, e.g.

Here,

Then, consider

Where

Suppose now that

Now we examine the inner product to note that

Here, though, we can precisely recover that this has a strong relationship to the Fourier series!

- Problem doesn't actually reduce here? The fact this is spectral doesn't make a huge difference
- See traditional spectral methods: Learning intrusively via
would be the goal. - I don't personally like this-- training
and simultaneously seems... dissatisfying

**Further Thoughts**

- What's the problem with spectral UQ? Number of basis functions to solve for gets prohibitively large in high dimension

Suppose we just do a spectral expansion of with linear differential operator with random coefficients

Practically, we end up solvingIf all are uncorrelated or something, then we can solve each equation separately via inner product with and find the accordingly. The problem, however, is

- What if we don't know the distribution of the input
(i.e. we don't know what basis functions to use) - Even if we did know
, if it's high-dimensional, we're out of luck (it needs to be a projection in many dimensions and the coefficients explode exponentially in the dimension)