Pruning vineyards: updating barcodes and representative cycles by removing simplices

TL;DR

This work addresses updating persistent barcode and representative cycles when simplices are removed from a filtration. It introduces SiRUP, an algorithm that updates the reduced boundary matrix factorization $D=RU$ to produce $R',U'$ for the filtration with the removed simplices, achieving fewer column additions than recomputing from scratch and preserving informative representative cycles. Theoretical guarantees show asymptotic improvements over naïve removal and comparisons with existing dynamic methods (zigzag, vine swaps) demonstrate practical efficiency gains, supported by extensive experiments. The approach enables efficient dynamic topology in applications such as neuroscience, manifold learning, and streaming data, where local deletions frequently occur and rapid updates are essential.

Abstract

The barcode of a filtration and its representative cycles encode rich information often useful in data analysis. However, obtaining them can be computationally expensive. Therefore, it is useful to have methods that update them if the associated filtration undergoes small changes. There are already efficient algorithms updating a barcode if simplices exchange entrance order or are added, but not if simplices are removed. We provide an implementation to update a reduced boundary matrix when simplices in the filtration are removed. Our algorithm, the Simplicial Removal Update Procedure (SiRUP), intrinsically updates also the representative cycles, and is compatible with the twist optimizations. We show that the complexity of our algorithm is lower than recomputing the barcode from scratch and that the number of executed matrix column additions is minimal, with both theoretical and experimental methods.

Figures (8)

• Figure 1: A simplicial complex $K$ (left) with its simplices indexed, its (unreduced) boundary matrix (middle) corresponding to the simplex-wise filtrations, and its barcode (right), with dimension 0 bars below the dashed line and dimension 1 bars above the dashed line.
• Figure 2: A simplicial complex (left), the sub-matrix of its boundary matrix given by the edges and their boundaries (a), the formal sums $e_2+e_3$ (dashed) and $e_1+e_2+e_3$ (dashed and solid), (b) and (c) respectively.
• Figure 3: A simplicial complex $K$ and its unreduced boundary matrix $D$ (left), factored as $R=DU$ by \ref{['alg_sba']}, and the outputs with clearing (right) and without (center). Clearing puts column $i=5$ of the boundary matrix $R'$ to zero if and only if there is a pivot pairing $(i,j)=(5,6)$. That is, if the formal sum corresponding to column $j=6$ kills the homological class generated by simplex corresponding to row $i=5$.
• Figure 4: The penultimate step (left, $\mathcal{F}_{n-1}$) and final step (middle, $\mathcal{F}_n$) of a filtration of a simplicial complex, and the filtration after removing a simplex (right, $\mathcal{F}\setminus\mathrm{st}\left(\tau\right)$). The reduced boundary matrix corresponding to each step is also given.
• Figure 5: Several examples of how the barcode of a simplicial complex may change when the star of a simplex (highlighted in light red) is removed, as observed in the simplicial complex (top row) and the reduced boundary matrix (bottom row). Finite bars (one in dimension $0$ and several in dimension $1$) disappear and an infinite bar in dimension $1$ appears in (a), a finite bar becomes shorter and another finite bar disappears in (b), $4$ infinite bars in dimension $1$ appear in (c), and an infinite bar disappears in (d).

Theorems & Definitions (12)

• Example 3.1
• Remark 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Theorem 3.5
• proof
• Proposition 3.6
• proof