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arxiv: 2605.25541 · v1 · pith:XQUQ5OAMnew · submitted 2026-05-25 · 💻 cs.CG · cs.AI· cs.HC· cs.LG

TopoAlign: Topology-Aware Visual Representation Alignment

Xinyuan Yan , Rita Sevastjanova , Mennatallah El-Assady , Bei Wang This is my paper

Pith reviewed 2026-06-29 19:37 UTC · model grok-4.3

classification 💻 cs.CG cs.AIcs.HCcs.LG
keywords representation alignmenttopological data analysismapper graphsvisualizationneural networksmodel comparison
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The pith

TopoAlign aligns model representations by jointly optimizing mapper graphs from topological data analysis.

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

The paper presents TopoAlign to compare neural network representations using topology instead of geometry. It builds mapper graphs for different models on the same inputs and aligns them globally with force-directed optimization before finding local matches. This offers a structural view that can aid model interpretation and selection. Readers would care because current alignment measures miss global organization. The framework is demonstrated on language and multimodal models.

Core claim

The central discovery is that joint force-directed optimization on mapper graphs from representations of shared inputs produces coordinated layouts that enable detection of structurally matching regions across models or layers, providing topological insights into representation alignment.

What carries the argument

Mapper graphs from topological data analysis that are jointly optimized with force-directed layout to align global structures and reveal correspondences.

Load-bearing premise

The mapper graphs preserve enough of the true global structure in the representations to make the joint optimization meaningful for finding real correspondences.

What would settle it

A test where models with known alignment levels from other metrics are compared and the TopoAlign correspondences either match or fail to match those levels.

Figures

Figures reproduced from arXiv: 2605.25541 by Bei Wang, Mennatallah El-Assady, Rita Sevastjanova, Xinyuan Yan.

Figure 1
Figure 1. Figure 1: Topology-aware visual representation alignment during fine-tuning with [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: An illustration of mapper graph construction. Left: a point cloud [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Mapper graphs of the final layer of BERT-base after fine-tuning [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: An illustration of five local alignment patterns. [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: A local alignment between structures 1 and 2 and their corre [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: TopoAlign workspace comparing fine-tuned BERT-Base representations at layer 3 (left) and layer 12 (right), organized into three coordinated panels: (A) Overview Panel: a 2×2 layout showing two model representations, each consisting of a mapper graph (top) and a projection view (bottom). (B) Control Panel: controls for coloring strategies (B1), global alignment strength (B2), local alignment discovery (B3),… view at source ↗
Figure 7
Figure 7. Figure 7: Mapper graphs of the final layer of fine-tuned BERT-base at [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: Mapper graphs of the final layers of BERT-base (left) and BERT [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Mapper graphs of the CLIP image modality (left) and caption [PITH_FULL_IMAGE:figures/full_fig_p008_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Details of local alignments A, B, and C. For each alignment, [PITH_FULL_IMAGE:figures/full_fig_p009_11.png] view at source ↗
Figure 13
Figure 13. Figure 13: Mapper graphs of layers 7 (left) and 12 (right) of the fine-tuned [PITH_FULL_IMAGE:figures/full_fig_p013_13.png] view at source ↗
read the original abstract

Neural networks encode inputs as high-dimensional vectors, known as representations, that capture how models process data by encoding task-relevant structure and semantics. Representation alignment refers to the degree to which different models, layers, or training conditions produce similar representations for the same inputs, with important implications for model interpretation, selection, and robustness analysis. Existing approaches to measure alignment primarily rely on geometric properties, such as neighborhood and cluster similarity, offering limited insight into the global organization of representations. In this work, we present TopoAlign, a topology-aware framework for visually comparing model representations from a structural perspective. Leveraging mapper graphs from topological data analysis, TopoAlign jointly analyzes graphs constructed from representations of shared inputs across different models or layers. The framework supports a top-down comparative workflow: it first performs global structure alignment via joint force-directed optimization to produce coordinated graph layouts; it then identifies local correspondences through automated detection of structurally matching regions, visualized with Bubble Sets; and finally it enables fine-grained pattern inspection through motif-based queries and membrane-inspired visualizations. We demonstrate TopoAlign through case studies on language and multimodal models, complemented by expert feedback. Our results show that TopoAlign provides meaningful insights into representation structure and alignment from a topological perspective.

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

2 major / 2 minor

Summary. The paper introduces TopoAlign, a topology-aware visual framework for comparing neural network representations. It constructs mapper graphs from representations of shared inputs across models or layers, applies joint force-directed optimization to produce coordinated layouts, detects matching regions via Bubble Sets, and supports motif queries with membrane visualizations. The approach is demonstrated through qualitative case studies on language and multimodal models plus expert feedback, with the central claim that it yields meaningful insights into representation structure and alignment from a topological perspective.

Significance. If the layouts and detected correspondences can be shown to recover genuine cross-model topological structure rather than optimization artifacts, the work would offer a useful structural complement to existing geometric alignment measures (e.g., CKA, RSA) for interpretability and model selection tasks. The mapper-graph plus coordinated-layout pipeline is a plausible extension of TDA tools, but the current qualitative-only evidence limits assessed significance.

major comments (2)
  1. [Abstract] Abstract (top-down workflow paragraph): the claim that joint force-directed optimization on mapper graphs produces coordinated layouts reflecting true cross-model correspondences is load-bearing for all subsequent insights, yet the manuscript supplies no quantitative validation (e.g., correlation with known layer-wise similarities, transfer-task alignment scores, or comparison to geometric baselines) that the optimized node positions or Bubble-Set overlaps recover ground-truth structure.
  2. [Case studies] Case studies section: expert feedback and visual inspection are presented as evidence of 'meaningful insights,' but without reported metrics such as region-overlap precision, motif-recovery rates against synthetic correspondences, or inter-rater agreement on alignment quality, it remains unclear whether the detected matches exceed what random or purely geometric layouts would produce.
minor comments (2)
  1. [Methods] Notation for mapper-graph parameters (e.g., cover resolution, clustering linkage) should be defined explicitly in the methods section to allow reproducibility.
  2. [Figures] Figure captions for the coordinated layouts and Bubble Sets should state the exact force-directed parameters and overlap thresholds used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (top-down workflow paragraph): the claim that joint force-directed optimization on mapper graphs produces coordinated layouts reflecting true cross-model correspondences is load-bearing for all subsequent insights, yet the manuscript supplies no quantitative validation (e.g., correlation with known layer-wise similarities, transfer-task alignment scores, or comparison to geometric baselines) that the optimized node positions or Bubble-Set overlaps recover ground-truth structure.

    Authors: TopoAlign is presented as an exploratory visual analytics framework rather than a quantitative alignment metric. The joint force-directed layout is a heuristic to support visual coordination of mapper graphs, and the detected correspondences are based on structural similarity within those graphs. We will revise the abstract to clarify the interpretive and visual nature of the claims and add an explicit limitations paragraph noting the absence of quantitative validation against geometric baselines such as CKA. This is a partial revision that preserves the qualitative focus while addressing the concern directly. revision: partial

  2. Referee: [Case studies] Case studies section: expert feedback and visual inspection are presented as evidence of 'meaningful insights,' but without reported metrics such as region-overlap precision, motif-recovery rates against synthetic correspondences, or inter-rater agreement on alignment quality, it remains unclear whether the detected matches exceed what random or purely geometric layouts would produce.

    Authors: We agree that the evaluation is qualitative. In visualization and TDA research, case studies combined with expert feedback constitute standard evidence of utility when the goal is to demonstrate novel visual workflows. We will expand the case-studies section with additional detail on the feedback protocol and further examples. Introducing quantitative metrics would require defining synthetic ground truth or new user studies, which we will acknowledge as future work rather than add to the current manuscript. revision: partial

Circularity Check

0 steps flagged

No circularity: standard TDA and layout methods applied to new domain

full rationale

The paper introduces TopoAlign as a workflow applying mapper graphs (standard TDA), joint force-directed optimization (standard graph layout), Bubble Sets, and motif queries to compare neural representations. No equations, fitted parameters, or derived predictions appear that reduce by construction to the inputs or to self-citations. The central claims rest on qualitative case studies using these off-the-shelf techniques rather than any self-referential derivation or uniqueness theorem imported from the authors' prior work. Self-citations, if present, are not load-bearing for the method's validity. This is a methodological application paper whose derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The approach rests on the domain assumption that mapper graphs capture global organization of representations; no free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption Mapper graphs from topological data analysis capture the global organization of high-dimensional representations
    Invoked to justify the structural comparison workflow.

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