{"id":"fe399ccf-5b73-4f91-b480-606fa1665c1f","arxiv_id":"2507.06212","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Fixed-count clustering, such as k-means, can force Mapper complexes to display arbitrary, misleading topological features.","lead":"A Topological Data Analysis researcher warns that using fixed-count clustering methods like k-means inside the Mapper algorithm can create misleading topological features, and calls these methods harmful for Mapper analysis. The paper gives a concrete example of holes being created and genuine structures being collapsed, and recommends adaptive clustering instead.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central claim of arbitrary topological distortion is asserted, not derived; k-means is not shown to realize the cited universality results.","rationale":"The paper is a short position piece, and its practical warning is plausible: global fixed-count clustering cannot adapt to each pullback, so it will merge and split clusters relative to any reasonable ground truth. The collection of tutorial snippets in Section 3 is genuine evidence that k-means and fixed-count agglomerative clustering are used in practice, and the cited prior work does show that Mapper outputs can be manipulated by choosing covers and clusterings. However, the strongest claim in Section 4 is specifically that fixed-count clustering produces 'potentially arbitrarily large changes to the topology of the Mapper complex.' That claim requires more than the existence of some bad covers; it requires showing that the specific class of clusterings produced by fixed-count methods can realize the arbitrary distortions, or at least that the distortions grow without bound in natural families. The paper does not provide such a demonstration. The single Figure 1 example illustrates two failure modes but does not quantify 'arbitrary,' and no code or data is supplied for reproduction. This is exactly the weakest assumption identified by the reader: the gap between guaranteed over/under-production and the full universality results. My recommended verdict is unchanged because the reader's CONDITIONAL already reflects this gap; the fix would be a formal or computational demonstration that k-means-induced Mapper graphs can realize unbounded topological variation.","tokens_in":1042,"tokens_out":1218,"duration_ms":64049,"concrete_test":"Build a family of examples from the Alvarado et al. construction: for a sequence of target graphs G_m with increasing first Betti number, choose a dataset, lens, and cover so that an ideal adaptive clustering realizes G_m. Then run Mapper with scikit-learn KMeans(k) for fixed k = 2 and k = 4 on the same inputs, using multiple random seeds, and compute the first Betti number (or graph edit distance to G_m) of the resulting Mapper graph. If the Betti number of the k-means output remains bounded for all m (e.g., bounded by a function of k and the number of cover elements), the 'arbitrary' claim fails; if it grows without bound or systematically realizes growing Betti numbers, the claim gains support. Also rerun Figure 1 with exact code and parameters to confirm the depicted failure modes.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is in Section 4: the paper infers that because nerve-lemma violations can change Mapper topology arbitrarily (Vejdemo-Johansson and Leshchenko 2020) and any graph is a Mapper graph (Alvarado et al. 2024), a fixed-count clustering method such as k-means is 'all but guaranteed' to produce arbitrarily large topological distortions. This inference does not follow. Both cited results are existence theorems: they show that some choice of lens, cover, and cluster assignment can realize a given output. They say nothing about the cluster assignments produced by k-means with a globally fixed k, nor about how often such assignments approximate the pathological cases. The paper offers no theorem connecting k-means outputs to the arbitrary nerve-lemma obstructions or to the Alvarado construction; the only evidence is Figure 1, a single illustrative example with no dataset description, no code, no seed, and no quantitative measure of distortion. The informal claim that over-production 'splits clusters' and under-production 'forms quotient spaces with arbitrary identification' is not enough: splitting and merging are real, but 'arbitrary' requires control over which topological changes can occur and how large they can be. It is possible that fixed-count clustering, for a fixed k and cover, produces only a bounded family of distortions, in which case the strongest claim would be false even though the practical warning remains sound. The reader's conditional verdict is therefore justified, but the 'arbitrary' part needs either a universality theorem for k-means-induced Mapper graphs or a systematic family of examples showing unbounded distortion.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper argues that using fixed-count clustering algorithms such as k-means in the Mapper construction can arbitrarily distort the topology of the output. It reviews the Mapper algorithm and k-means, documents instances in open-source Mapper tutorials where k-means or fixed-count agglomerative clustering is used, and warns that fixed-count methods over-produce or under-produce clusters in the pullback cover elements, causing spurious holes, mergers, and other topological changes. It then suggests adaptive alternatives such as the largest gap heuristic, silhouette scoring, and DBSCAN. The supporting evidence is a single illustrative example (Figure 1) and appeals to prior results on nerve-lemma failures and universality of Mapper graphs.","tokens_in":3784,"tokens_out":3336,"duration_ms":40191,"significance":"The warning is practically important: if correct, it identifies a class of widely used Mapper implementations (Kepler-Mapper, tda-mapper, and others) that can be misleading, and it points to concrete tutorial code where fixed-count clustering is used. The paper also correctly emphasizes that the problem is the global fixed count itself, not uneven point density, so preprocessing or changing the fixed k cannot solve it. That said, the central claim is currently not derived: the paper asserts that fixed-count clustering is 'all but guaranteed' to cause 'arbitrarily large' topological distortions, but offers only one hand-illustrated example and no theorem or quantitative analysis connecting k-means outputs to the cited universality results.","major_comments":[{"comment":"The claim that a fixed-count clustering algorithm is 'all but guaranteed' to over-produce clusters in some cover elements and under-produce clusters in others is asserted without a formal model or evidence. No probability distribution over datasets, lenses, and covers is given, nor is there any argument about how often over/under-production occurs for k-means in realistic settings. As written, this is an informal plausibility statement, not a result.","section":"Section 4, paragraph 3"},{"comment":"The inference from the cited existence theorems to 'arbitrarily large changes' does not follow. Vejdemo-Johansson and Leshchenko (2020) show that breaking nerve-lemma assumptions can produce arbitrary topological changes, and Alvarado et al. (2024) show that any graph is realizable as a Mapper graph, but both are existential results: they do not state that k-means or any fixed-count clustering algorithm realizes those changes. The paper provides no construction or quantitative bound linking k-means outputs to these results. To sustain the word 'arbitrary', the paper would need either a theorem showing that for any target distortion there exists a dataset, lens, cover, and fixed k such that k-means produces that distortion, or a substantial empirical study; neither is present.","section":"Section 4, paragraphs 4-5"},{"comment":"Figure 1 is the only concrete evidence for the central claim, but it lacks the dataset description, the exact construction of the lens and cover, the k-means initialization parameters, and any quantitative measure of topological distortion. It also does not specify how the 'expected output' is defined (e.g., by single-linkage clustering or by connected components). Without these details, the figure is not reproducible and cannot support the strong 'arbitrary' claim.","section":"Figure 1"}],"minor_comments":[{"comment":"The phrase 'V oronoi cells' has a spacing typo and should read 'Voronoi cells'.","section":"Section 2"},{"comment":"The sentence 'one of the reviewers graciously suggested using the silhouette score' is an unusual meta-comment for a research paper; consider removing or rephrasing it.","section":"Section 5"},{"comment":"The Alvarado et al. reference is an arXiv preprint; please include the arXiv identifier version and, if available, a DOI or publication venue.","section":"References"},{"comment":"The caption should state explicitly how the bottom-row outputs are generated, including the clustering algorithm used for the 'expected output' and any parameters used for the 2-means and 4-means runs.","section":"Figure 1 caption"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is very short and reads more like a position/critique note than a full research paper. If the 'arbitrary' claim cannot be supported formally, the authors should either weaken the claim to 'unpredictable and uncontrolled' changes or provide the missing proof/experiments. The practical warning itself is sound and worth publishing in a suitable venue, but as it stands the title and abstract overstate what is established."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a short position paper, not a research paper. The new thing is the transfer of an existing critique—nerve lemma violations in Mapper can change topology arbitrarily—to the specific case of fixed-count clustering like k-means, plus a useful survey of how widespread this choice is in Kepler Mapper, tda-mapper, and scikit-TDA examples. The author also makes a good conceptual point: the problem is the global mandate on k, not uneven point density, so preprocessing or picking another fixed k won't help. That is worth saying clearly.\n\nThe paper does some things well. It separates discrimination from description honestly, and it points to adaptive alternatives (largest gap, silhouette, DBSCAN with care). Figure 1 gives an intuitive illustration of over- and under-production, and the writing is clear.\n\nThe soft spot is the load-bearing claim. Section 4 says a fixed-count algorithm is 'all but guaranteed' to produce 'arbitrarily large changes' to the Mapper topology. The cited results are existence theorems: Alvarado et al. show any graph can be realized by some lens/cover/clustering; Vejdemo-Johansson–Leshchenko show nerve lemma failure can cause arbitrary changes. But that does not imply k-means with fixed k realizes those arbitrary changes, or that the distortions are unbounded. The single hand-drawn example conveys the failure mode but is not evidence of arbitrariness. No code, data, or quantitative measure is provided. So the strongest claim is not derived.\n\nThat said, the practical warning survives even if 'arbitrary' is too strong. Even bounded distortions from forced over/under-clustering are enough to mislead a user. The paper would be more honest if it said 'can cause serious distortions' rather than 'arbitrary.' As a position statement it deserves attention; as a theorem it doesn't.\n\nFor peer review: I'd send it out, but with a request to temper the universal claim or support it with experiments showing unbounded distortion across datasets. The citations are appropriate and the self-citation is fair. This is a reading-group paper for TDA methodologists and a caution flag for practitioners.\n\nRecommendation: conditional accept as a position/critique piece, not as a formal result.","headline":"Useful practitioner warning about fixed-count clustering in Mapper, but the 'arbitrary distortion' claim is asserted rather than proven.","tokens_in":4385,"tokens_out":3108,"would_cite":true,"duration_ms":32080,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55N31","62H30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Fixed-count clustering such as k-means can change the topology of a Mapper complex arbitrarily, making Mapper output unreliable as a description of data shape.","keywords":["Mapper","k-means","topological data analysis","nerve lemma","fixed-count clustering","clustering","Mapper complex","lens function"],"falsifier":"Take a dataset whose ground-truth shape is two parallel strands, choose a lens such that every cover element contains exactly two well-separated components, and run Mapper with $k=2$ on many such datasets; if the resulting complexes remain within a bounded homotopy type of the ground truth even when clusters are forced to be over- or under-produced, the claim that fixed-count clustering produces arbitrary distortion in realistic settings would be weakened.","tokens_in":3335,"feed_emoji":"🗺️","tokens_out":7589,"duration_ms":75096,"temperature":0.7,"pith_summary":"The paper argues that using clustering algorithms with a fixed, globally chosen number of clusters—most prominently $k$-means—inside the Mapper construction can change the output Mapper complex in arbitrarily large ways, not merely add small noise. The mechanism is that each preimage of a lens function has its own natural number of clusters; a single global $k$ is almost guaranteed to split clusters in some preimages and merge them in others. Splitting can create spurious holes or components, and merging can collapse genuine topological structure, so the final complex no longer describes the shape of the data. This matters because Mapper is widely used to visualize and explore data shape, and many popular Mapper tutorials and libraries currently recommend $k$-means or fixed-count agglomerative clustering. The paper's central recommendation is that fixed-count clustering should be avoided in Mapper, in favor of adaptive methods such as the original largest-gap heuristic.","feed_headline":"k-means can distort Mapper topology arbitrarily, paper warns","feed_subtitle":"Fixed-count clustering appears in Mapper tutorials, but it can both invent and erase shape features.","key_machinery":"The load-bearing object is the refined covering step of the Mapper algorithm: each preimage $L^{-1}Y_i$ is handed to a clustering algorithm, and the clusters become the vertex sets of the nerve complex. What carries the argument is the fixed-count constraint of the clustering method, combined with the nerve lemma, whose assumptions require each cover element to be contractible. The paper's mechanism is the unavoidable over- and under-production of clusters that results when one global $k$ is applied to cover elements with different natural component counts. This mechanism, together with the universality result that Mapper complexes can realize arbitrary complexes, is what turns a routine clustering mismatch into potentially arbitrary topological distortion.","core_discovery":"The central claim is that a fixed-count clustering algorithm is structurally incompatible with the Mapper refinement step, because it forces exactly $k$ clusters onto every pullback cover element regardless of how many connected components the data actually has there. Over-production of clusters splits single components and can create holes or extra branches when the pieces reconnect in neighboring cover elements; under-production identifies distinct strands and quotients out real topological features. Building on the earlier observation that breaking nerve-lemma assumptions can produce arbitrary topological changes, and on the recent universality result that any sufficiently small simplicial complex can be realized as a Mapper complex over a large dataset, the paper concludes that the merging and splitting induced by fixed-count clustering can result in arbitrarily large distortions of the Mapper complex. The paper emphasizes that the problem is the mandated global count itself, not uneven point density, so preprocessing or switching to another fixed-count method does not fix it.","pith_inferences":["A natural extension, not stated in the paper: the same mechanism should affect any clustering parameter that is fixed globally, including scale parameters such as a fixed distance threshold, not just a fixed cluster count.","One practical diagnostic suggested by the argument: compare the Mapper output from $k$-means against the output of an adaptive clusterer on the same data; large differences concentrated in cover elements where the natural cluster count is far from $k$ would corroborate the mechanism.","If the claim is right, Mapper-based discriminative statements that two datasets differ because their Mapper complexes differ become unsafe under fixed-count clustering, since the global $k$ alone could generate the difference."],"forward_implications":["Mapper complexes built with $k$-means or fixed-count agglomerative clustering cannot be trusted as descriptions of data shape; conclusions about loops, components, and flares may be clustering artifacts.","Choosing a different fixed $k$ does not fix the problem, since the distortion comes from the global mandate rather than the specific value; tuning may move features but not remove the mechanism.","Preprocessing the data will not resolve the issue, because the mismatch is between a global parameter and cover-wise natural cluster counts.","Adaptive clustering—for example the largest-gap hierarchical heuristic from the original Mapper paper, or per-preimage cluster-count selection via a quality score such as silhouette—avoids the specific failure mode and should be the recommended practice.","Open-source Mapper tutorials and documentation that currently suggest KMeans or fixed-count agglomerative clustering should be revised, since following them can silently corrupt downstream analyses."],"supporting_citations":[{"why":"Establishes that breaking nerve-lemma assumptions can create or remove homological features, grounding the criterion for a bad cover.","marker":"Vejdemo-Johansson and Leshchenko [2020]"},{"why":"Proves that any sufficiently small simplicial complex can be realized as a Mapper complex over a large dataset, supplying the universality behind arbitrarily large distortion.","marker":"Alvarado et al. [2024]"},{"why":"Introduces the Mapper algorithm and the largest-gap heuristic, the recommended adaptive alternative that does not fix the cluster count beforehand.","marker":"Singh et al. [2007]"},{"why":"Kepler-Mapper documentation; cited as one of the open-source tutorials that showcase fixed-count clustering choices.","marker":"van Veen et al. [2019]"},{"why":"Giotto-TDA documentation; another open-source Mapper implementation whose examples use fixed-count clustering.","marker":"Tauzin et al. [2020]"},{"why":"Python Mapper implementation, part of the open-source ecosystem where fixed-count choices appear.","marker":"Müllner and Babu [2013]"},{"why":"Provides the silhouette score, which the paper suggests as one adaptive way to select cluster numbers per cover element.","marker":"Rousseeuw [1987]"},{"why":"DBSCAN is offered as an alternative clustering approach, with the caveat that it requires handling unclustered points.","marker":"El-Sonbaty et al. [2004]"}],"fun_headline_variants":["Fixed-count clustering distorts Mapper topology arbitrarily","k-means can invent or erase Mapper features","Mapper arbitrary topological changes from k-means","k-means forces false clusters, damaging Mapper results","Why fixed-count clustering fails for Mapper"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes the generic over- and under-production of clusters by fixed-count methods actually drives the Mapper complex through the full range of arbitrary topological distortions, rather than only through the limited set observed in the single illustrated example.","fun_headline_variants_meta":{"raw":{"variants":["Fixed-count clustering distorts Mapper topology arbitrarily","k-means can invent or erase Mapper features","Mapper arbitrary topological changes from k-means","k-means forces false clusters, damaging Mapper results","Why fixed-count clustering fails for Mapper"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00031,"raw_usage":{"total_tokens":1681,"prompt_tokens":771,"completion_tokens":910,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":387,"completion_tokens_details":{"reasoning_tokens":838}},"tokens_in":387,"tokens_out":910,"duration_ms":7338,"temperature":1.0,"reasoning_tokens":838,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:07:55.446096+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a dataset whose ground-truth shape is two parallel strands, choose a lens such that every cover element contains exactly two well-separated components, and run Mapper with $k=2$ on many such datasets; if the resulting complexes remain within a bounded homotopy type of the ground truth even when clusters are forced to be over- or under-produced, the claim that fixed-count clustering produces arbitrary distortion in realistic settings would be weakened.","supporting_citations":[{"cited_title":"Topological methods for the analysis of high dimensional data sets and 3d object recognition","cited_arxiv_id":null,"evidence_quote":"Introduces the Mapper algorithm and the largest-gap heuristic, the recommended adaptive alternative that does not fix the cluster count beforehand."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Kepler-Mapper documentation; cited as one of the open-source tutorials that showcase fixed-count clustering choices."},{"cited_title":"giotto-tda: A topological data analysis toolkit for machine learning and data exploration, 2020","cited_arxiv_id":null,"evidence_quote":"Giotto-TDA documentation; another open-source Mapper implementation whose examples use fixed-count clustering."},{"cited_title":"Python mapper: An open-source toolchain for data exploration, analysis and visualization","cited_arxiv_id":null,"evidence_quote":"Python Mapper implementation, part of the open-source ecosystem where fixed-count choices appear."},{"cited_title":"Silhouettes: a graphical aid to the interpretation and validation of cluster analysis","cited_arxiv_id":null,"evidence_quote":"Provides the silhouette score, which the paper suggests as one adaptive way to select cluster numbers per cover element."},{"cited_title":"An efficient density based clustering algorithm for large databases","cited_arxiv_id":null,"evidence_quote":"DBSCAN is offered as an alternative clustering approach, with the caveat that it requires handling unclustered points."}],"review_version":1}