MC {nnR}

The MC neural network

Description

This function implements the 1-D approximation scheme outlined in the References.

Note: Only 1-D interpolation is implemented.

Usage

MC(X, y, L)

Arguments

X

a list of samples from the functions domain.

y

the function applied componentwise to each point in the domain.

L

the Lipschitz constant for the function. Not necessarily global, but could be an absolute upper limit of slope, over the domain.

Value

A neural network that gives the maximum convolution approximation of a function whose outputs is $y$ at $n$ sample points given by each row of $X$, when instantiated with ReLU.

References

Lemma 4.2.9. Jentzen, A., Kuckuck, B., and von Wurstemberger, P. (2023). Mathematical introduction to deep learning: Methods, implementations, and theory. https://arxiv.org/abs/2310.20360.

Examples

seq(0, 3.1416, length.out = 200) -> X
sin(X) -> y
MC(X, y, 1) |> inst(ReLU, 0.25) # compare to sin(0.25)