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arxiv: 2312.03925 · v3 · submitted 2023-12-06 · 🧮 math.AT · cs.CG

Pruning vineyards: updating barcodes and representative cycles by removing simplices

Barbara Giunti , J\=anis Lazovskis This is my paper

Pith reviewed 2026-05-24 05:00 UTC · model grok-4.3

classification 🧮 math.AT cs.CG
keywords persistent homologybarcoderepresentative cyclesboundary matrixsimplex removalupdate algorithmfiltrationalgebraic topology
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The pith

SiRUP updates the reduced boundary matrix and representative cycles after simplex removal using fewer column operations than recomputing the barcode from scratch.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents an algorithm for updating persistent homology barcodes and their representative cycles when simplices are removed from a filtration. Existing methods handle additions and order changes but not removals, which often force a full recomputation. SiRUP works directly on an already-reduced boundary matrix, performs the necessary updates locally, and remains compatible with clearing optimizations. Theoretical bounds and experiments show the number of matrix column additions is minimal and the overall cost stays below that of restarting the reduction.

Core claim

The Simplicial Removal Update Procedure (SiRUP) updates a reduced boundary matrix when simplices are removed from the filtration; the same procedure automatically maintains the associated representative cycles, requires only a minimal number of column additions, and has strictly lower complexity than recomputing the barcode and cycles on the pruned filtration from scratch.

What carries the argument

SiRUP, a procedure that identifies the affected columns in the reduced boundary matrix and performs targeted additions to restore reduction after simplex removal.

If this is right

  • Barcodes remain correct under simplex removal without restarting the reduction algorithm.
  • Representative cycles are maintained automatically as part of the matrix update.
  • The method composes with clearing optimizations already used in standard persistence computations.
  • Total column operations stay below those of a fresh reduction on the updated filtration.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Dynamic data sets that lose points over time could maintain topological summaries incrementally.
  • Combined with existing addition and swap updates, general small edits to a filtration become tractable.
  • The minimal-operation guarantee suggests the procedure could serve as a building block for more complex dynamic persistence pipelines.

Load-bearing premise

A local update to an already-reduced boundary matrix after simplex removal can preserve the correct barcode and cycles while executing strictly fewer column operations than a complete recomputation.

What would settle it

A filtration and removal sequence where running SiRUP yields a different barcode or different representative cycles than those obtained by reducing the boundary matrix of the pruned filtration from scratch, or where the column-addition count exceeds the full-recomputation count.

Figures

Figures reproduced from arXiv: 2312.03925 by Barbara Giunti, J\=anis Lazovskis.

Figure 1
Figure 1. 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. Simplicial complexes and filtrations. Given a finite set V , an k-simplex (over V ) is a subset of V of size k + 1. Whenever there is a relationship τ… view at source ↗
Figure 2
Figure 2. Figure 2: A simplicial complex (left), the sub-matrix of its boundary matrix given by the edges and [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A simplicial complex K and its unreduced boundary matrix D (left), factored as R = DU by 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. Thi… view at source ↗
Figure 4
Figure 4. Figure 4: The penultimate step (left, Fn−1) and final step (middle, Fn) of a filtration of a simplicial complex, and the filtration after removing a simplex (right, F \ st (τ )). The reduced boundary matrix corresponding to each step is also given. In this example, the last step of the presented filtration ( [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Several examples of how the barcode of a simplicial complex may change when the star [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A simplicial complex without the 1-simplex [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: A visual overview of our main testing results. The plots are colored semi-transparently to [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Steps and outputs of full testing environment. The values reported in Table 4 are from [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
read the original 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 clearing 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript introduces the Simplicial Removal Update Procedure (SiRUP) to update a reduced boundary matrix (and its representative cycles) when simplices are removed from a filtration. The central claims are that SiRUP has lower complexity than recomputing the barcode from scratch, executes a minimal number of matrix column additions, and remains compatible with clearing optimizations; both theoretical arguments and experiments are invoked to support the complexity and minimality statements.

Significance. If the correctness, complexity, and minimality claims hold, the work fills a documented gap in dynamic persistent-homology algorithms by handling removals (as opposed to additions or order swaps). The provision of an implementation together with both theoretical and experimental validation of the operation counts is a concrete strength.

minor comments (2)
  1. The abstract states that the number of column additions is minimal 'with both theoretical and experimental methods,' but the precise statement of the minimality theorem (including any assumptions on the filtration or the reduction strategy) is not visible in the provided excerpt; a short clarifying sentence would help readers locate the claim.
  2. Notation for the reduced boundary matrix and the representative cycles should be introduced once, early, with a consistent symbol (e.g., R or D) rather than alternating between descriptive phrases.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their review and for recognizing the significance of filling the gap in dynamic persistent-homology algorithms for simplex removals, as well as the value of the implementation and validation. The report does not list any specific major comments.

Circularity Check

0 steps flagged

No circularity: algorithmic update procedure is independently specified

full rationale

The paper presents the SiRUP algorithm as a direct constructive procedure for updating a reduced boundary matrix and its cycles under simplex removal. Correctness, minimality of column operations, and complexity bounds are argued via explicit algorithmic steps, theoretical comparison to full recomputation, and experimental counts; none of these steps reduce by definition or self-citation to the target quantities themselves. No fitted parameters are relabeled as predictions, no uniqueness theorems are imported from the authors' prior work, and the central claims rest on standard persistent-homology matrix reduction rather than any self-referential loop.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review is based solely on the abstract; no free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.0 · 5661 in / 1040 out tokens · 30828 ms · 2026-05-24T05:00:49.963161+00:00 · methodology

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