
A One Parameter Equation That Can Exactly Fit Any Scatter Plot - richardhod
https://marginalrevolution.com/marginalrevolution/2018/05/one-parameter-equation-can-exactly-fit-scatter-plot.html
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woopwoop
Their formula is neat, but the fact that you can fit any finite set of data to
arbitrary precision with a smoothly varying one parameter family of functions
is easy. Here is another construction. Let P_1, P_2, ... be an enumeration of
the polynomials with rational coefficients, let phi_i be a smooth function
supported on [i-1/2,i+1/2] with phi_i(i)=1, and let f_theta(x) = sum
phi_i(theta) P_i(x) = phi_round(theta)(theta) P_theta(x), where round(theta)
is the closest integer to theta. Whatever you mean by smoothly varying one-
parameter family of functions, this is surely it, since on any open interval
the image is contained in a finite dimensional vector space and the curve of
functions is smooth. Since we can hit any polynomial with rational
coefficients by picking the right (integer) theta, and for any finite set E, f
: E to R, and epsilon>0 there exists a polynomial P with rational coefficients
such that |P(x) - f(x)|<epsilon for all x in E, we are done.

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RobertoG
"[..] the paper also tells us that Occam’s Razor is wrong [..]"

I doubt the paper say that, and I don't think it's true.

It seems to me that Occam's Razor is about the quantity of information, not
about how many parameters you use to store the information.

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mlazos
This paper was really interesting and it's awesome, but I'm tired of everyone
saying that this means parameter counting is a useless heuristic for the
complexity of a model. The more parameters you add to some ML models (for
instance NN and SVM) the higher the VC dimension, and that's that. Many models
are simply not constructed in a way that a single parameter can encode all of
the necessary information. (And still be a continuous function of that
parameter)

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sgentle
Heh, neat. Now all you need is to implement that logistic map solver in
hardware and write a function that maps other functions into their equivalent
"decimal code" and you've got a pretty fun computer going.

The paper notes that these functions are infinitely continuously
differentiable. I wonder if they'd have any applications in machine learning?

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Waterluvian
If I created a formula that randomly plotted each "pixel" on or off, there's a
seed out there that will plot my name, no?

Is this what the paper is about, or is it something different?

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viraptor
Doesn't Gödel numbering already guarantee that kind of function?

