Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The minimal width of universal $p$-adic ReLU neural networks

Sándor Z. Kiss, Ambrus Pál

Abstract

We determine the minimal width of $p$-adic neural networks with the universal approximation property for continuous $\mathbb Q_p$-valued functions on compact open subsets with respect to the $L_q$ norms and the $C_1$ norm, where the activation function is a natural $p$-adic analogue of the ReLU function.

The minimal width of universal $p$-adic ReLU neural networks

Abstract

We determine the minimal width of -adic neural networks with the universal approximation property for continuous -valued functions on compact open subsets with respect to the norms and the norm, where the activation function is a natural -adic analogue of the ReLU function.
Paper Structure (3 sections, 25 theorems, 53 equations)

This paper contains 3 sections, 25 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Lower bound
  3. Upper bound

Key Result

Theorem 1.2

For every $q\in [1,\infty]$ the $\mathrm{pReLU}$-networks of width $w$ have the universal approximation property for continuous functions $f:\mathbb Z_p^{d_x} \to\mathbb Q_p^{d_y}$ in the $L_q$ norm if and only if $w\geq\max(d_x+1,d_y)$.

Theorems & Definitions (62)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • ...and 52 more