Harmonic Chain Barcode and Stability

Abstract

The persistence barcode is a topological descriptor of data that plays a fundamental role in topological data analysis. Given a filtration of the space of data, a persistence barcode tracks the evolution of its homological features. In this paper, we introduce a novel type of barcode, referred to as the canonical barcode of harmonic chains, or harmonic chain barcode for short, which tracks the evolution of harmonic chains. As our main result, we show that the harmonic chain barcode is stable and it captures both geometric and topological information of data. Moreover, given a filtration of a simplicial complex of size $n$ with $m$ time steps, we can compute its harmonic chain barcode in $O(m^2n^ω + mn^3)$ time, where $n^ω$ is the matrix multiplication time. Consequently, a harmonic chain barcode can be utilized in applications in which a persistence barcode is applicable, such as feature vectorization and machine learning. Our work provides strong evidence in a growing list of literature that geometric (not just topological) information can be recovered from a persistence filtration.

Figures (8)

• Figure 1: (a) A filtration of a simplicial complex, the time that a simplex is inserted is written next to the simplex. For the four triangles we have times $t_1 < t_2 < t_3 < t_4$. We also use these time stamps as names of the simplices. (b) A part of the boundary matrix of the complex which is relevant to 1-dimensional homology. The boundary matrix is already reduced, therefore, the generators of persistent homology computed by the standard matrix reduction algorithm are given by columns of the matrix. (c) The 1-dimensional ordinary persistence barcode. (d) The 1-dimensional harmonic chain barcode based on the cycles computed by the matrix reduction algorithm.
• Figure 2: Changing the order of two simplices might effect an arbitrary large change in the harmonic barcode. The filtration differ by exchange of two edges. The ordinary persistence barcode is in black, whereas the harmonic chain barcode is in purple (constructed using the basis given by the standard matrix reduction algorithm).
• Figure 3: The canonical harmonic chain barcode (in blue) for the filtration in \ref{['fig:first-attempt-example']}.
• Figure 4: The ordinary persistence barcode and the canonical barcode of harmonic chains for the filtration $F_1$. (a) The insertion times of of edges and triangles. (b) The 2-dimensional boundary matrix, the matrices of different times steps are marked. (c) The ordinary persistence barcode (in black). (d) The canonical barcode of harmonic chains (in blue).
• Figure 5: The ordinary persistence barcode and the canonical barcode of harmonic chains. (a) The insertion times of edges and triangles for the filtration $F_2$. (b) The 2-dimensional boundary matrix, the matrices of different times steps are marked. (c) The ordinary persistence barcode (in black). (d) The canonical barcode of harmonic chains (in blue). The persistence reduction of the boundary matrix remains the same since the matrix is unchanged. The basis for ordinary persistence barcode therefore remains the same chains as our first ordering, however, the insertion times of simplices are changed, resulting in different bars.

Theorems & Definitions (19)

• Lemma 1: eckmann1944harmonische
• Proposition 1
• Definition 1
• Theorem 1
• Definition 2
• Definition 3
• Theorem 2
• proof
• Definition 4
• Theorem 3