AI and Topology

: [Stanford University - Andrew Brodhead]

- Overview

Topology studies spatial properties that remain unchanged under any continuous deformation. It is sometimes called "rubber sheet geometry" because objects can stretch and contract like rubber, but cannot break.

For example, a simple loop in a plane and the boundary edge of a square in a plane are topologically equivalent, as may be observed by imagining the loop as a rubber band that can be stretched to fit tightly around the square.

For example, a square can be transformed into a circle without destroying it, but the number 8 cannot. Therefore, the square is topologically equivalent to the circle, but not the same as the number 8.

Please refer to the following for more information:

Wikipedia: Topology

- Topological Properties

Topology is the study of shapes that can be stretched and moved while points on the shape continue to stay close to each other. Topology is the part of mathematics concerned with 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 topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of topological spaces, as any distance or metric defines a topology.

The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property.

The following are basic examples of topological properties: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connectedness, which allows distinguishing a circle from two non-intersecting circles.

Topology is a relatively new branch of mathematics. Most studies of topology date from 1900. The following are some of the subfields of topology.

- General Topology

General topology (or point set topology) usually considers the local properties of space and is closely related to analysis. It generalizes the concept of continuity to define a topological space where sequence constraints can be taken into account.

Sometimes distances can be defined in these spaces, in which case they are called metric spaces; sometimes the concept of distance does not make sense.

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- Combinatorial Topology

Combinatorial topology considers the global properties of the space and is constructed from a network of vertices, edges, and faces. This is the oldest branch of topology, dating back to Euler. It has been shown that topologically equivalent spaces have the same numerical invariants, which we now call the Euler property. This is the number (V - E + F), where V, E, and F are the number of vertices, edges, and faces of the object.

For example, tetrahedrons and cubes are topologically equivalent to spheres, and any "triangulation" of a sphere has an Euler eigenvalue of 2.

- Algebraic Topology

Algebraic topology also considers the global properties of space and uses algebraic objects such as groups and rings to answer topological questions. Algebraic topology transforms topological problems into algebraic problems that are easier to solve.

For example, a group called a homology group can be associated with each space, and a torus and a Klein bottle can be distinguished from each other because they have different homology groups.

- Differential Topology

Differential topology considers a space with some degree of smoothness associated with each point. In this case, the square and circle are not smoothly (or differentiable) equivalent to each other.

Differential topology is useful for studying the properties of vector fields, such as magnetic or electric fields.

- Topological Deep Learning

The study of the form and structure of objects, focusing on features that support continuous transformation, is called topology.

In recent years, topology has become an effective set of tools for machine learning to analyze complex data. Topology can provide insight into underlying relationships between variables that may be difficult to obtain using other techniques because it focuses on the overall structure of the data rather than specific aspects.

Topological deep learning (TDL) represents a field at the intersection of topology and deep learning, offering approaches to analyze and learn from data structured in topological spaces. By leveraging the principles of topology, TDL offers an approach to understanding and processing data supported on topological spaces.

Traditional deep learning (DL) typically assumes that the data being examined lies in a linear vector space and can be efficiently characterized using feature vectors. However, there is a growing recognition that this traditional view may not be adequate for describing a variety of real-world data sets.

For example, molecules are better represented as graphs rather than eigenvectors. Likewise, three-dimensional objects, such as those encountered in computer graphics and geometry processing, are best represented by meshes.

Furthermore, data from social networks cannot be represented by simple vector-based descriptions, and participants in social networks are interconnected in complex ways.

Therefore, there has been a surge of interest in integrating topological concepts into traditional DL frameworks to obtain more accurate structural representations of the underlying data.