The function that returns the right scalar multiplication neural network
srm(nu, a)nu %srm% a
Returns a neural network that is \(\nu \triangleleft a\). This instantiates as \(f(a \cdot x)\).under continuous function activation. More specifically we will define this operation as:
Let \(\lambda \in \mathbb{R}\). We will denote by \((\cdot) \triangleleft (\cdot): \mathsf{NN} \times \mathbb{R} \rightarrow \mathsf{NN}\) the function satisfying for all \(\nu \in \mathsf{NN}\) and \(\lambda \in \mathbb{R}\) that \(\nu \triangleleft \lambda = \nu \bullet \mathsf{Aff}_{\lambda \mathbb{I}_{\mathsf{I}(\nu)},0}\).
A neural network
A real number.
Definition 2.3.4. Jentzen, A., Kuckuck, B., and von Wurstemberger, P. (2023). Mathematical introduction to deep learning: Methods, implementations, and theory. https://arxiv.org/abs/2310.20360.
Note: We will have two versions of this operation, a prefix and an infix version.