Object-oriented Persistent Homology

Abstract

Persistent homology provides a new approach for the topological simplification of big data via measuring the life time of intrinsic topological features in a filtration process and has found its success in scientific and engineering applications. However, such a success is essentially limited to qualitative data classification and analysis. Indeed, persistent homology has rarely been employed for quantitative modeling and prediction. Additionally, the present persistent homology is a passive tool, rather than a proactive technique, for classification and analysis. In this work, we outline a general protocol to construct object-oriented persistent homology methods. By means of differential geometry theory of surfaces, we construct an objective functional, namely, a surface free energy defined on the data of interest. The minimization of the objective functional leads to a Laplace-Beltrami operator which generates a multiscale representation of the initial data and offers an objective oriented filtration process. The resulting differential geometry based object-oriented persistent homology is able to preserve desirable geometric features in the evolutionary filtration and enhances the corresponding topological persistence. The cubical complex based homology algorithm is employed in the present work to be compatible with the Cartesian representation of the Laplace-Beltrami flow. The proposed Laplace-Beltrami flow based persistent homology method is extensively validated. The consistence between Laplace-Beltrami flow based filtration and Euclidean distance based filtration is confirmed on the Vietoris-Rips complex for a large amount of numerical tests. The convergence and reliability of the present Laplace-Beltrami flow based cubical complex filtration approach are analyzed over various spatial and temporal mesh sizes. The Laplace-Beltrami flow based persistent homology approach is utilized to study the intrinsic topology of proteins and fullerene molecules. Based on a quantitative model which correlates the topological persistence of fullerene central cavity with the total curvature energy of the fullerene structure, the proposed method is used for the prediction of fullerene isomer stability. The efficiency and robustness of the present method are verified by more than 500 fullerene molecules. It is shown that the proposed persistent homology based quantitative model offers good predictions of total curvature energies for ten types of fullerene isomers. The present work offers the first example to design object-oriented persistent homology to enhance or preserve desirable features in the original data during the filtration process and then automatically detect or extract the corresponding topological traits from the data.

Keywords: Computational topology; Fullerene; Laplace-Beltrami flow; Protein; Total curvature energy; Variation.

Figure 1

A flow chart for the construction of object-oriented persistent homology.

Figure 2

Selected frames of fullerene C₆₀ generated from the time evolution of the Laplace-Beltrami flow. Charts from left to right and from top to bottom are frames 1 to 6, respectively. According to Table 1, the second last frame has a central cavity while the last frame has no void.

Figure 3

Comparison of the topological evolution and persistence of the C₆₀ molecule. Top row: Barcodes obtained from the proposed Laplace-Beltrami flow based filtration; Bottom row: Barcodes obtained from the Rips complex filtration.

Figure 4

Comparison of the persistence of β₁ barcodes obtained from the growth of atomic radius filtration and from the geometric flow based filtration for fullerene C₃₆. Top left: Atomic radius filtration; Top right: Geometric flow filtration, h = 0.5Å; Bottom left: Geometric flow filtration, h = 0.25Å; Bottom right: Geometric flow filtration, h = 0.125Å.

Figure 5

Comparison of the persistence of β₁ barcodes obtained from the growth of atomic radius filtration and from the geometric flow based filtration for fullerene C₁₀₀. Top left: Atomic radius filtration; Top right: Geometric flow filtration, h = 0.5Å; Bottom left: Geometric flow filtration, h = 0.25Å; Bottom right: Geometric flow filtration, h = 0.125Å.

Figure 6

The initial structure of protein 2GR8. Left chart: Secondary structure representation; Right chart: atomic representation. Colors indicate different types of atoms.

Figure 7

Geometric evolution of protein 2GR8 under the Laplace-Beltrami flow. Charts from left to right and from top to bottom are frames 1 to 6, respectively.

Figure 8

The time evolution of the topological invariants of protein 2GR8 under the Laplace-Beltrami flow.

Figure 9

The initial structure of a beta barrel. Left chart: Secondary structure representation; Right chart: atomic representation. Colors indicate different types of atoms.

Figure 10

The geometric evolution of a beta barrel under the Laplace-Beltrami flow. Charts from left to right and from top to bottom are frames 1 to 6, respectively.

Figure 11

The evolution of the topological invariants of the beta barrel under the Laplace-Beltrami flow.

Figure 12

The comparison of fullerene isomer total curvature energies and persistent homology theory predictions.

Figure 13

The comparison of fullerene isomer total curvature energies and persistent homology theory predictions