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Universal approximation theorem: History, Arbitrary-depth case & Arbitrary-width case

In the field of machine learning, the universal approximation theorems (UATs) state that neural networks with a certain structure can, in principle, approximate any continuous function to any desired degree of accuracy. These theorems provide a mathematical justification for using neural networks, assuring researchers that a sufficiently large or deep…

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Universal approximation theorem topic overview

The analysis highlights History, Arbitrary-depth case and Arbitrary-width case as prominent areas in the source structure around Universal approximation theorem.

Related topics
56
Source areas
6
Connected nodes
62
Extracted relationships
50
Concept neighborhoods
31
Bridge connections
62

What this topic covers Research coverage

Source areas are shown by the number of related topics found in each part of the analysis. Use smaller areas too: they can reveal specialized angles and content gaps.

History · 19 topics
Arbitrary-depth case · 13 topics
Overview · 11 topics
Arbitrary-width case · 10 topics
Setup · 2 topics
Bounded depth and bounded width case · 1 topics

Smaller areas are not necessarily less important. They contain fewer connections in this analysis and can be useful for finding specialized angles or coverage gaps.

Explore all related topics Closing gaps

Browse the complete topic structure, not only the most central items. Less prominent entities and concepts can reveal missing angles, specialized context and useful research gaps. Each item opens a new analysis centered on that subject.

Overview

Setup

History

Arbitrary-width case

Arbitrary-depth case

Bounded depth and bounded width case

Advanced semantic analysis

Deeper signals for content research, entity SEO and topical coverage. The plain-language headings explain what each technical view is useful for.

How Universal approximation theorem connects Entity context

The extracted context around Universal approximation theorem shows recurring relationship patterns in the source. For example, Universal approximation theorem → Bochner, For, In, It, L1, Lebesgue, Lebesgue-integrable, Moreover, ReLU, The, They, Universal, Zhou Lu Another extracted example is Universal approximation theorem → Arnold, In, Indeed, Kolmogorov, Robert Hecht-Nielsen, The Kolmogorov, This, Vugar Ismailov, Ziming Liu. Use these groups to spot repeated connection types before inspecting the individual relationships.

Universal approximation theorem

Top relations

related to Arbitrary-depth case · 13
Universal approximation theorem → Bochner, For, In, It, L1, Lebesgue, Lebesgue-integrable, Moreover, ReLU, The, They, Universal, Zhou Lu
related to Kolmogorov network · 9
Universal approximation theorem → Arnold, In, Indeed, Kolmogorov, Robert Hecht-Nielsen, The Kolmogorov, This, Vugar Ismailov, Ziming Liu
related to Bounded depth and bounded width case · 7
Universal approximation theorem → For, Maiorov, Pinkus, The, Their, There, Universal
related to Arbitrary-width case · 5
Universal approximation theorem → George Cybenko, In, Kurt Hornik, See, The
related to Setup · 4
Universal approximation theorem → Artificial, In, Most, The

Important terminology

Use these terms to understand the vocabulary surrounding the topic, not as a checklist for keyword stuffing.

Important terminology

function displaystyle networks approximation universal neural activation width network functions depth theorem arbitrary also mathbb approximate continuous relu hidden layer

Universal approximation theorem relationships Subject–Predicate–Object triples

TTTA extracted 50 structured relationships around Universal approximation theorem. Examples in this analysis include Gustaf Gripenberg in 2003 → instance of → Moshe Leshno et al in 1993 and later Allan Pinkus in 1999 showed that the universal approximation property is equivalent to having a nonpolynomial activation function.Arbitrary… and Gustaf Gripenberg in 2003 → instance of → Arbitrary depthThe arbitrary depth case was also studied by a number of authors. The table shows each extracted connection, where it came from and its confidence.

SubjectPredicateObjectConfidenceSrc
Gustaf Gripenberg in 2003instance ofMoshe Leshno et al in 1993 and later Allan Pinkus in 1999 showed that the universal approximation property is equivalent to having a nonpolynomial activation function.Arbitrary…0.80text
Dmitry Yarotskyinstance ofMoshe Leshno et al in 1993 and later Allan Pinkus in 1999 showed that the universal approximation property is equivalent to having a nonpolynomial activation function.Arbitrary…0.80text
Zhou Lu et al in 2017instance ofMoshe Leshno et al in 1993 and later Allan Pinkus in 1999 showed that the universal approximation property is equivalent to having a nonpolynomial activation function.Arbitrary…0.80text
Boris Hanininstance ofMoshe Leshno et al in 1993 and later Allan Pinkus in 1999 showed that the universal approximation property is equivalent to having a nonpolynomial activation function.Arbitrary…0.80text
Mark Sellke in 2018 who focused on neural networks with ReLU activation functioninstance ofMoshe Leshno et al in 1993 and later Allan Pinkus in 1999 showed that the universal approximation property is equivalent to having a nonpolynomial activation function.Arbitrary…0.80text
Gustaf Gripenberg in 2003instance ofArbitrary depthThe arbitrary depth case was also studied by a number of authors0.80text
Dmitry Yarotskyinstance ofArbitrary depthThe arbitrary depth case was also studied by a number of authors0.80text
Zhou Lu et al in 2017instance ofArbitrary depthThe arbitrary depth case was also studied by a number of authors0.80text
Boris Hanininstance ofArbitrary depthThe arbitrary depth case was also studied by a number of authors0.80text
Mark Sellke in 2018 who focused on neural networks with ReLU activation functioninstance ofArbitrary depthThe arbitrary depth case was also studied by a number of authors0.80text
the step function can be approximated by continuous activation functionsinstance ofcertain non-continuous activation functions0.80text
which then allows the approximation result to apply to those functionsinstance ofcertain non-continuous activation functions0.80text

Related concept clusters Concept neighborhoods

The concept neighborhoods around Universal approximation theorem bring nearby vocabulary together. In this analysis, examples include Universal, Networks and Theorem. Use the clusters to find adjacent concepts and terminology that may deserve separate research.

  • Universal approximation theorem
    • Universal
    • Networks
    • Theorem
    • Width
    • Activation
    • Neural
    • Bounded
    • Property
    • Function
    • Two
    • Functions
    • Depth
  • universal approximation theorem
    • Universal
    • Networks
    • Theorem
    • Width
    • Activation
    • Neural
    • Bounded
    • Property
    • Function
    • Displaystyle
    • Two
    • Depth
  • neural networks
    • Networks
    • Neural
    • Universal
    • Hidden
    • Network
    • Activation
    • Layers
    • Width
    • Approximate
    • Bounded
    • Also
    • Functions
  • continuous function
    • Displaystyle
    • Activation
    • Mappings
    • Sigma
    • Relu
    • Mathbb
    • Also
    • Universal
    • Function
    • Networks
    • Neural
    • Network
  • feedforward networks
    • Neural
    • Universal
    • Hidden
    • Activation
    • Layers
    • Width
    • Bounded
    • Also
    • Functions
    • Function
    • Depth
    • Layer
  • activation function
    • Functions
    • Displaystyle
    • Activation
    • Function
    • Universal
    • Relu
    • Networks
    • Sigma
    • Approximation
    • Mathbb
    • Also
    • Neural
  • sigmoid function
    • Displaystyle
    • Activation
    • Sigma
    • Relu
    • Mathbb
    • Also
    • Universal
    • Networks
    • Neural
    • Network
    • Exists
    • Width
  • existence theorems
    • Neurons
    • Two
    • Universal
    • Number
    • One
    • Bounded
    • Layers
    • Hidden
    • Layer
    • Width
    • Depth
    • Neural

Connections between topic areas Semantic bridges

For Universal approximation theorem, one of the stronger structural bridges in this analysis connects Universal approximation theorem with History. Bridges highlight paths between different parts of the map and can reveal research angles that are easy to miss in a flat list.

Min side: 3
Universal approximation theorem — History · splits 43 ⟂ 20
Universal approximation theorem — Arbitrary-depth case · splits 49 ⟂ 14
Universal approximation theorem — Overview · splits 51 ⟂ 12
Universal approximation theorem — Arbitrary-width case · splits 52 ⟂ 11
Universal approximation theorem — Setup · splits 60 ⟂ 3

Map overview Semantic statistics

Universal approximation theorem

Nodes63
Edges62
Triples50
Avg. degree1.97
Density0.031746
Components1

Source & methodology

TTTA analyzes the structure around Universal approximation theorem to surface related topics, entities, relationships, concept neighborhoods and bridge connections. Use the map to explore areas such as History, Arbitrary-depth case & Arbitrary-width case, including less central topics that may reveal useful research gaps. Automatically extracted connections are research leads rather than rewritten encyclopedia content.

Source: Wikipedia — Universal approximation theorem · EN edition · Analysis: TopicsToTalkAbout

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