Can any function be decomposed as sum of Gaussians?

Question

In Fourier series, any function can be decomposed as sum of sine and cosine
In neural networks, any function can be decomposed as weighted sum over logistic functions. (A one layer neural network)
In wavelet transforms, any function can be decomposed as weighted sum of Haar functions

Is there also such property for decomposition into mixture of Gaussians? If so, is there a proof?

You may also want to check out https://math.stackexchange.com/ and https://mathoverflow.net/. I tried to edit my answer to give more context, but it is hard to do here because there's no latex/mathjax in the markdown. If you're interested, you can ask a similar question there and I'll try to answer? — Riley, Jul 06 '20 at 19:26

score 4 · Answer 1 · answered Apr 08 '19 at 19:33

There's a theorem, the Stone-Weierstrass theorem, which gives conditions for when a family of functions can approximate any continuous function. You need

an algebra of functions (closed under addition, subtraction, and multiplication)
the constant functions
and you need the functions to separate points:
- (for any two distinct points you can find a a function that assigns them different values)

You can approximate a constant function with increasingly wide gaussians. You can time-shift gaussians to separate points. So if you form an algebra out of gaussians, you can approximate any continuous function with them.

+1 This answer is significantly more thorough than the top-voted answer. I've referenced your answer [here](https://mattermodeling.stackexchange.com/questions/9681/do-the-cc-pc-def2-basis-sets-mathematically-converge-to-the-cbs-limit-assuming/9682?noredirect=1#comment20415_9682) in case you're interested! Perhaps you can write a better answer for that user who seems to be much more interested in the mathematical convergence properties than the typical user of that Stack Exchange site? — Nike, Sep 23 '22 at 18:56

score 3 · Answer 2 · answered May 08 '17 at 14:43

3

If the sum allows to be infinite, then the answer is Yes. Please refer to Yves Meyer's book of "Wavelet and Operators", section 6.6, lemma 10.

answered May 08 '17 at 14:43

user7981320

31
2

Note that this lemma is for complex functions expressed as sum of positive Gaussians with complex coefficients, so it might take a bit more work to show (if true) that any positive function can be written as a sum of positive Gaussians with positive coeffcients. – Jess Riedel Sep 13 '20 at 18:00

score 1 · Answer 3 · edited Sep 26 '22 at 08:25

Yes. Decomposing any function to a sum of any kind of Gaussians is possible, since it can be decomposed to a sum of Dirac functions :) (and Dirac is a Gaussian where the variance approaches zero).

Some more interesting questions would be:

Can any function be decomposed to a sum of non-zero variance Gaussians, with a given, constant variance, that are defined around varying centers?
Can any function be be decomposed to a sum of non-zero variance Gaussians, all having 0 as the center, but defined with alternating variances?

The Mathematics Stack Exchange might be a better place to answer these questions though.

Can any function be decomposed as sum of Gaussians?

3 Answers3