Topology in Physics and Computing

Topology has been shown to reveal important information about geometry and shape from data, [Carlsson 2015][Carlsson 2009] as I have talked about in various TDA blog entries. I have also demonstrated how to describe the topology if discrete data points by constructing simplicial complexes, and then calculated the homology and Betti numbers. (I will talk about persistent homology in the future.) Dealing with discrete data points in our digital storage devices, homology is the best way to describe it.

But if you are from a physics background, you may be familiar with the concept of homotopy and fundamental group. Some physicists deal with topology without digging into advanced mathematical tools but simply through solitons. There is a well-written introduction in this blog. In the physical world, an object is said to be topological if:

• there is a singular point that cannot be removed by a continuous deformation of field; [Mermin 1979]
• it has a saddle-point equation of the model that is different from another object of another topology, [Rajaraman 1987] inducing different kinds of physical dynamics; [Bray 1994]
• it can only be removed by crossing an energy barrier, which can be described by an instanton; [Calzetta, Ho, Hu 2010]
• it can proliferate through Kosterlitz-Thouless (BKT) phase transition; [Kosterliz, Thouless 1973]
• it can form in a system through a second-order phase transition at a finite rate, a process known as Kibble-Zurek mechanism (KZM); [Kibble 1976] [Zurek 1985] and
• its topology can be described by a winding number. (c.f. Betti numbers in homology)

Topological objects include vortices in magnets, superfluids, superconductors, or Skyrmions in helimagnets. [Mühlbauer et. al. 2009] [Ho et. al. 2010] They may come in honeycomb order, like Abrikosov vortices in type-II superconductors, [Abrikosov 1957] and helical nanofilaments in smectics. [Matsumoto et. al. 2009] It is widely used in fractional quantum Hall effect [Tsui et. al. 1982] and topological insulators (a lot of references can be found…). They can all be described using homotopy and winding numbers. We can see that topology is useful to describe the physical world for the complexities and patterns. There are ideas in string-net theory to use topology to describe the emergence of patterns and new phases of quantum matter. [Zeng et. al. 2015] Of course, I must not omit topological quantum computing that makes the qubits immune to environmental noise. [Das Sarma, Freedman, Nayak 2005]

A-phase of helimagnet (MnSi) (taken from [Ho et. al. 2010]

However in data analytics, we do not use homotopy, albeit its beauty and usefulness in the physical world. Here are some of the reasons:

• In using homotopy, sometimes it takes decades for a lot of brains to figure out which homotopy groups to use. But in data analysis, we want to grasp the topology simply from data.
• Homotopy deals with continuous mappings, but data are discrete. Simplicial homology captures it more easily.
• In a physical system, we deal with usually one type of homotopy groups. But in data, we often deal with various topologies which we are not aware of in advance. Betti numbers can describe the topology easily by looking at data.
• Of course, homotopy is difficult to compute numerically.

Afra Zomorodian argued the use of homology over homotopy in his book as well. [Zomorodian 2009]

Starting the Journey of Topological Data Analysis (TDA)

Topology has been around for centuries, but it did not catch the attention of many data analysts until recently. In an article published in Nature Scientific Reports, the authors demonstrated the power of topology in data analysis through examples including gene expression from breast rumors, voting data in the United States, and player performance data from the NBA. [Lum et. al. 2013]

As an introduction, they described topological methods “as a geometric approach to pattern or shape recognition within data.” It is true that in machine learning, we never care enough pattern recognition, but topology adds insights regarding the shapes of data that do not change with continuous deformation. For example, a circle and an ellipse have “the same topology.” The distances between data points are not as important as the shape. Traditional machine learning methods deal with feature vectors, distances, or classifications, but the topology of the data is usually discarded. Gunnar Carlsson demonstrated in a blog that a thin ellipse of data may be misrepresented as two straight parallel lines or one straight lines. [Carlsson 2015] Dimensionality reduction algorithms such as principal component analysis (PCA) often disregard the topology as well. (I heard that Kohenen’s self-organizing maps (SOM) [Kohonen 2000] retain the topology of higher dimensional data during the dimensionality reduction, but I am not confident enough to say that.)

Euler introduced the concept of topology in the 18th century. Topology has been a big subject in physics since 1950s. The string theory, as one of the many efforts in unifying gravity and other three fundamental forces, employs topological dimensions. In condensed matter physics, the fractional quantum Hall effect is a topological quantum effect. There are topological solitons [Rajaraman 1987] such as quantum vortices in superfluids, [Simula, Blakie 2006; Calzetta, Ho, Hu 2010] columns of topological solitons (believed to be Skyrmions) in helical magnets, [Mühlbauer et. al. 2009; Ho et. al. 2010; Ho 2012] hexagonal solitonic objects in smectic liquid crystals [Matsumoto et. al. 2009]… When a field becomes sophisticated, it becomes quantitative; when a quantitative field becomes sophisticated, it requires abstract mathematics such as topology for a general description. I believe analysis on any kinds of data is no exception.

There are some good reviews and readings about topological data analysis (TDA) out there, for example, the ones by Gunnar Carlsson [Carlsson 2009] and Afra Zomorodian [Zomorodian 2011]. While physicists talk about homotopy, data analysts talk about persistent homology as it is easier to compute. Data have to be described in a simplicial complex or a graph/network. Then the homology can be computed and represented in various ways such as barcodes. [Ghrist 2008] Then we extract insights about the data from it.

Topology has a steep learning curve. I am also a starter learning about this. This blog entry will not be the last talking about TDA. Therefore, I opened a new session called TDA for all of my blog entries about it. Let’s start the journey!

There is an R package called “TDA” that facilitates topological data analysis. [Fasy et. al. 2014] A taste of homology of a simplicial complex is also demonstrated in a Wolfram demo.

(Taken from TheGuardian)