In what ways is the term “topology” applied to Artificial Intelligence?

Question

I have a 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?

Answer 1

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:

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, Homology, Homotopy and Applications
• Persistence Images: A Stable Vector Representation of Persistent Homology
• Topological data analysis and machine learning theory
• Topological Kernel Learning: Discrete Morse Theory is used to connect kernel machine learning with topological data analysis.

Answer 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 for example.

The other major meaning is with reference to the structure (topology) of a combinatorial graph. Many modern machine learning algorithms are based in the idea of combinatorial graphs, including Bayesian Networks, Sum/Product Networks, and Deep Neural Networks. Here, topology refers to the topological ordering 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.