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A. Cole and G. Shiu
9.1 Introduction
Physics and information theory have long had a rich exchange. Many concepts in
information theory admit an understanding via statistical physics. Moreover, information theory contributes a great deal to both experimental and theoretical physics.
The experimental physicist seeks to understand the information content of various
experimental observations and the constraining power of future experiments, making extensive use of the Fisher information. On the theoretical side, the “it from
qubit” program has revealed that many aspects of gravity seem to emerge from
information-theoretic quantities such as entanglement between different statistical
systems. A similar relationship exists more broadly between machine learning/data
science and physics, with physically-motivated architectures and algorithms such as
the Boltzmann machine [3] and importance sampling and a growing presence of deep
learning in both theoretical and experimental physics. This symbiotic relationship
between fields shows no signs of slowing down.
In this chapter, we review the application of persistent homology to several physical data sets. Persistent homology forms the main pillar of Topological Data Analysis
(TDA), and formalizes a notion of “shape” for discrete data sets. As such, we can
regard it as a powerful pattern recognition tool with potential to identify signatures of
subtle physical phenomena. In particular, we are motivated by problems in cosmology and string theory. In cosmology, we would like to understand how fundamental
physics affects the complex structure (such as the spatial distribution of galaxies)
we observe in our universe. For string theory, we are interested in the problem of
navigating the so-called string landscape, the vast set of ground states of the theory.
The organization of this chapter is as follows. In Sect. 9.2 we briefly review the
basics of persistent homology as a general pattern recognition tool. We emphasize
its stability theorems and describe how one may derive coarser statistics than the
persistence diagram. In Sect. 9.3 we note how topological statistics have previously
appeared in the cosmology literature, and describe how persistent homology techniques make contact with these statistics. We describe how persistent homology
provides novel statistical descriptions of the topology of cosmological observables
including the Cosmic Microwave Background and Large-Scale Structure. Section 9.4
motivates the use of data-centric approaches to the string theory landscape. After
reviewing relevant aspects of a construction of string vacua supported by fluxes,
we discuss how computational topology allows one to characterize these data sets,
recovering previously known features and discovering new structure.
9.2 A Short Review of Persistent Homology
In this section, we review persistent homology as a tool allowing one to formalize a
notion of “shape” for discrete data sets. We begin by defining simplicial complexes,
the discrete mathematical structures in which we embed our data. We describe how
A. Cole and G. Shiu
9.1 Introduction
Physics and information theory have long had a rich exchange. Many concepts in
information theory admit an understanding via statistical physics. Moreover, information theory contributes a great deal to both experimental and theoretical physics.
The experimental physicist seeks to understand the information content of various
experimental observations and the constraining power of future experiments, making extensive use of the Fisher information. On the theoretical side, the “it from
qubit” program has revealed that many aspects of gravity seem to emerge from
information-theoretic quantities such as entanglement between different statistical
systems. A similar relationship exists more broadly between machine learning/data
science and physics, with physically-motivated architectures and algorithms such as
the Boltzmann machine [3] and importance sampling and a growing presence of deep
learning in both theoretical and experimental physics. This symbiotic relationship
between fields shows no signs of slowing down.
In this chapter, we review the application of persistent homology to several physical data sets. Persistent homology forms the main pillar of Topological Data Analysis
(TDA), and formalizes a notion of “shape” for discrete data sets. As such, we can
regard it as a powerful pattern recognition tool with potential to identify signatures of
subtle physical phenomena. In particular, we are motivated by problems in cosmology and string theory. In cosmology, we would like to understand how fundamental
physics affects the complex structure (such as the spatial distribution of galaxies)
we observe in our universe. For string theory, we are interested in the problem of
navigating the so-called string landscape, the vast set of ground states of the theory.
The organization of this chapter is as follows. In Sect. 9.2 we briefly review the
basics of persistent homology as a general pattern recognition tool. We emphasize
its stability theorems and describe how one may derive coarser statistics than the
persistence diagram. In Sect. 9.3 we note how topological statistics have previously
appeared in the cosmology literature, and describe how persistent homology techniques make contact with these statistics. We describe how persistent homology
provides novel statistical descriptions of the topology of cosmological observables
including the Cosmic Microwave Background and Large-Scale Structure. Section 9.4
motivates the use of data-centric approaches to the string theory landscape. After
reviewing relevant aspects of a construction of string vacua supported by fluxes,
we discuss how computational topology allows one to characterize these data sets,
recovering previously known features and discovering new structure.
9.2 A Short Review of Persistent Homology
In this section, we review persistent homology as a tool allowing one to formalize a
notion of “shape” for discrete data sets. We begin by defining simplicial complexes,
the discrete mathematical structures in which we embed our data. We describe how