Answers.Body: <p>I spent some time thinking about it, but I'm aware of only two main meanings. There might be more that aren't coming to me right now though...</p>
  
  <p>In <strong>local search</strong> problems or sometimes in optimization for machine learning, the "topology" of a problem corresponds to the change in the function you're optimizing as you move between adjacent states. If the change is sharp, you have a "rugged topology". If it's gentle and continuous, you have a "smooth topology". See on page 2 of <a href="https://pdfs.semanticscholar.org/f79e/15480c283700317b85e931da748b3ca56a23.pdf" rel="nofollow noreferrer">An Introduction to Fitness Landscape Analysis
  and Cost Models for Local Search</a> for example. </p>
  
  <p>The other major meaning is with reference to the structure (topology) of a <a href="https://en.wikipedia.org/wiki/Combinatorics#Graph_theory" rel="nofollow noreferrer">combinatorial graph</a>. Many modern <strong>machine learning</strong> algorithms are based in the idea of combinatorial graphs, including <a href="https://en.wikipedia.org/wiki/Bayesian_network" rel="nofollow noreferrer">Bayesian Networks</a>, <a href="https://arxiv.org/abs/1202.3732" rel="nofollow noreferrer">Sum/Product Networks</a>, and Deep Neural Networks. Here, topology refers to the <a href="https://en.wikipedia.org/wiki/Topological_sorting" rel="nofollow noreferrer">topological ordering</a> of a directed graph, or more informally, to "how the graph is structured". For example, in a neural network, the depth and width of the network's layers, and the nature of the connections between layers, define the topology of the network.</p>
  
  <p>Additionally, it gets used a lot in the sense of the second meaning in other areas of AI, just because those areas also use graphs. For example, in automated planning, or in probabilistic reasoning, it is also common to represent your problem as a combinatorial graph. You could then talk about the "topology" of the problem. </p>
  
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Answers.UserDisplayName: John Doucette
Answers.Body: <p>In addition to the ways the term topology is itself used generically to describe the "shape" of various aspects of Machine Learning, the term appears in the field <a href="https://en.wikipedia.org/wiki/Topological_data_analysis" rel="nofollow noreferrer">Topological Data Analysis</a>:</p>
  
  <blockquote>
    <p>In applied mathematics, topological data analysis (TDA) is an approach to the analysis of datasets using techniques from topology. Extraction of information from datasets that are high-dimensional, incomplete and noisy is generally challenging. TDA provides a general framework to analyze such data in a manner that is insensitive to the particular metric chosen and provides dimensionality reduction and robustness to noise. Beyond this, it inherits functoriality, a fundamental concept of modern mathematics, from its topological nature, which allows it to adapt to new mathematical tools.</p>
  </blockquote>
  
  <p>Some examples of its use in ML:</p>
  
  <ul>
  <li><a href="https://www.intlpress.com/site/pub/files/_fulltext/journals/hha/2016/0018/0001/HHA-2016-0018-0001-a021.pdf" rel="nofollow noreferrer">The ring of algebraic functions on persistence bar codes</a>, Homology, Homotopy and Applications</li>
  <li><a href="https://arxiv.org/abs/1507.06217" rel="nofollow noreferrer">Persistence Images: A Stable Vector Representation of Persistent Homology</a></li>
  <li><a href="http://www.birs.ca/events/2012/5-day-workshops/12w5081" rel="nofollow noreferrer">Topological data analysis and machine learning theory</a></li>
  <li><a href="https://www.researchgate.net/publication/327427685_Topological_Kernel_Learning" rel="nofollow noreferrer">Topological Kernel Learning: Discrete Morse Theory is used to connect kernel machine learning with topological data analysis.</a></li>
  </ul>
  
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Answers.UserDisplayName: brazofuerte
Body: <p>I have only a general understanding of General Topology, and want to understand the scope of the term &quot;topology&quot; in relation to the field of Artificial Intelligence.</p>
  <p>In what ways are topological structure and analysis applied in Artificial Intelligence?</p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2021-12-13T14:24:07.340Z
Comments.Text: [Here](https://ai.stackexchange.com/q/20879/2444) is a related question, which focuses on the applications of topology (as a field) to deep learning.
Comments.UserDisplayName: nbro
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2024-05-10T19:41:02.990Z
Comments.Text: So, besides TDA (Topological Data Analysis), there does not seem to be any definable locus were topology, stricto sensu, is utilized by AI, if we define topology as: "the study of the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself." A further question might be: are any/all forms of graphs (particularly data graphs) definable as "geometric objects"?
Comments.UserDisplayName: Raphael Pallais
Tags: <machine-learning><terminology><applications><topology>
Title: In what ways is the term "topology" applied to Artificial Intelligence?
UserDisplayName: DukeZhou
ThreadText: # In what ways is the term "topology" applied to Artificial Intelligence?
  
  (Asked by: username_0 on 2018-08-08)
  
  I have only a general understanding of General Topology, and want to understand the scope of the term "topology" in relation to the field of Artificial Intelligence.
  
  In what ways are topological structure and analysis applied in Artificial Intelligence?
  
  **Tags:** machine-learning, terminology, applications, topology
  
  > **Comments:**
  > **username_1**: [Here](https://ai.stackexchange.com/q/20879/2444) is a related question, which focuses on the applications of topology (as a field) to deep learning.  
  > **username_2**: So, besides TDA (Topological Data Analysis), there does not seem to be any definable locus were topology, stricto sensu, is utilized by AI, if we define topology as: "the study of the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself." A further question might be: are any/all forms of graphs (particularly data graphs) definable as "geometric objects"?  
  
  ---
  
  ## 2 answers
  
  ### Answer by username_3 on 2018-08-08. Score: 2
  
  I spent some time thinking about it, but I'm aware of only two main meanings. There might be more that aren't coming to me right now though...
  
  In **local search** problems or sometimes in optimization for machine learning, the "topology" of a problem corresponds to the change in the function you're optimizing as you move between adjacent states. If the change is sharp, you have a "rugged topology". If it's gentle and continuous, you have a "smooth topology". See on page 2 of [An Introduction to Fitness Landscape Analysis
  and Cost Models for Local Search](https://pdfs.semanticscholar.org/f79e/15480c283700317b85e931da748b3ca56a23.pdf) for example.
  
  The other major meaning is with reference to the structure (topology) of a [combinatorial graph](https://en.wikipedia.org/wiki/Combinatorics#Graph_theory). Many modern **machine learning** algorithms are based in the idea of combinatorial graphs, including [Bayesian Networks](https://en.wikipedia.org/wiki/Bayesian_network), [Sum/Product Networks](https://arxiv.org/abs/1202.3732), and Deep Neural Networks. Here, topology refers to the [topological ordering](https://en.wikipedia.org/wiki/Topological_sorting) of a directed graph, or more informally, to "how the graph is structured". For example, in a neural network, the depth and width of the network's layers, and the nature of the connections between layers, define the topology of the network.
  
  Additionally, it gets used a lot in the sense of the second meaning in other areas of AI, just because those areas also use graphs. For example, in automated planning, or in probabilistic reasoning, it is also common to represent your problem as a combinatorial graph. You could then talk about the "topology" of the problem.
  
  
  ---
  
  ### Answer by username_4 on 2020-05-25. Score: 2
  
  In addition to the ways the term topology is itself used generically to describe the "shape" of various aspects of Machine Learning, the term appears in the field [Topological Data Analysis](https://en.wikipedia.org/wiki/Topological_data_analysis):
  
  > In applied mathematics, topological data analysis (TDA) is an approach to the analysis of datasets using techniques from topology. Extraction of information from datasets that are high-dimensional, incomplete and noisy is generally challenging. TDA provides a general framework to analyze such data in a manner that is insensitive to the particular metric chosen and provides dimensionality reduction and robustness to noise. Beyond this, it inherits functoriality, a fundamental concept of modern mathematics, from its topological nature, which allows it to adapt to new mathematical tools.
  
  Some examples of its use in ML:
  
  * [The ring of algebraic functions on persistence bar codes](https://www.intlpress.com/site/pub/files/_fulltext/journals/hha/2016/0018/0001/HHA-2016-0018-0001-a021.pdf), Homology, Homotopy and Applications
  * [Persistence Images: A Stable Vector Representation of Persistent Homology](https://arxiv.org/abs/1507.06217)
  * [Topological data analysis and machine learning theory](http://www.birs.ca/events/2012/5-day-workshops/12w5081)
  * [Topological Kernel Learning: Discrete Morse Theory is used to connect kernel machine learning with topological data analysis.](https://www.researchgate.net/publication/327427685_Topological_Kernel_Learning)
  
  
  ---
  
  