{"id":"00ed23d3-307c-4490-b5c2-f6efe17f33c3","arxiv_id":"2509.03681","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":8,"one_line_summary":"A stochastic subsampling protocol scales topological data analysis to brain-scale matrices, but its geometric findings are generated by fitting then testing on the same data.","lead":"This paper presents ALBATROSS, a stochastic subsampling protocol that computes topological summaries of large similarity matrices with far less memory, and applies it to human brain scans to map each region's geometric shape. It matters because it promises to make topological data analysis practical for big biological datasets, although the statistical evidence for the brain map has a self-fitting problem.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The p-values supporting the hyperbolic-geometry claims are computed after choosing the model to minimize the same l1/ibv statistics, so they are in-sample fitting statistics, not valid hypothesis tests.","rationale":"The reader's weakest assumption identified the exact issue I consider most load-bearing: post-selection validity. The protocol's model selection and p-value computation use the same l1 and ibv statistics on the same empirical data, so the p-values cannot support the claim of geometric consistency. This is not a matter of consensus; it is an internal flaw in the statistical pipeline that directly undermines the headline result. Even if the CLT variance formula (Eq 3) and hyperbolic radius density (Eq 7) were corrected, this circularity would remain. The concern is concrete and testable via a simple split-sample or null-simulation experiment. I do not find grounds to move away from the reader's REJECT verdict; hence 'UNCHANGED' is appropriate. The paper's memory-reduction contribution is credible, but the statistical geometry inference, which is the central advertised outcome, is not supported without post-selection correction.","tokens_in":21514,"tokens_out":6056,"duration_ms":66440,"concrete_test":"For the HCP global 392x392 rsFC matrices (or the strawberry data), split the I=150 subsampled Betti-curve iterations into a model-selection set (e.g., 100 iterations) and a held-out test set (50 iterations). Run the full ALBATROSS selection procedure (Eqs 10-12) on the selection set only, fix the best-fit model parameters, and then compute the six Z statistics and Fisher's chi-square (Eqs 3-5) using only the held-out iterations. If the held-out p-values no longer classify the data as 'consistent with hyperbolic' (or the reported chi-square values change substantially), the original p-values are an artifact of post-selection bias. A complementary check: simulate many synthetic datasets from the best-fit hyperbolic model, run the entire selection-plus-testing pipeline on each, and compare the observed test statistic to the distribution of the minimum statistic.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that ALBATROSS enables valid statistical geometry inference rests on the p-values produced in Online Methods Eqs 3-5. These p-values compare empirical mean Betti curves to the best-fit model using the same normalized l1 and ibv statistics that were minimized during model selection (Eqs 10-12, grid search and Bayesian optimization). Because the selected model is chosen to minimize these exact statistics on the same data, the observed l1/ibv values are systematically lower than typical draws from the fitted model; the null distribution estimated from resamplings of that fitted model is not the distribution of the selection-minimized statistic. This is a double-dipping / selective-inference problem. Consequently, the reported p-values in Supplementary Table 5 and the headline 'all ten individuals consistent with Hyperbolic geometry' are in-sample goodness-of-fit results, not statistical evidence for the geometric model. The validation results on strawberry odors and hippocampal place cells (Figure 2) are subject to the same circular comparison across candidate spaces. No split-sample, hold-out, or post-selection-adjusted inference is provided to address this, so the geometric conclusions are currently unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper introduces ALBATROSS, a protocol for computing Betti curves of filtered simplicial complexes of large adjacency matrices by stochastic sub-sampling. The protocol computes average empirical Betti curves from random subsamples, searches over Euclidean, hyperbolic, and shuffled candidate models using integrated Betti value (ibv) and L1-distance statistics, and then uses a CLT-based z-test combined by Fisher's method to produce p-values. The authors validate the approach on strawberry odor and hippocampal place-cell datasets and apply it to resting-state fMRI functional connectivity matrices from 392 cortical parcels in 10 HCP subjects, reporting that all ten global matrices and 353/392 local parcels are consistent with hyperbolic geometry, and that local hyperbolic radius correlates with the global radial coordinate (Spearman rs = 0.52).","tokens_in":21869,"tokens_out":9882,"duration_ms":110954,"significance":"If the statistical inference were valid, this paper would make a useful practical contribution: the memory benchmarks (e.g., roughly 71 GB for a direct n=300 computation versus 1.5 GB for ALBATROSS with n=30 on a laptop) and the public code would substantially lower the barrier to TDA on large biological datasets. The HCP application is also novel in scale. However, the statistical engine of the paper is not valid as implemented: the hypothesis tests use the same statistics that were minimized during model selection, and the CLT variance is mis-scaled. The headline geometric conclusions are therefore unsupported, and the central claim that ALBATROSS provides valid statistical geometry inference is not established.","major_comments":[{"comment":"Circular model selection and testing. The best-fit model is chosen by minimizing the objective Σ_{k=1}^3 (l1^k + ibv^k) in Eq. 12, where l1^k and ibv^k are exactly the normalized L1-distance and integrated Betti value statistics defined in Eqs. 10–11. The subsequent hypothesis test in Eqs. 4–5 evaluates the same empirical Betti curves against that selected model using the same L1 and ibv statistics. No split-sample, hold-out, or post-selection adjustment is provided. Consequently, the observed statistics are minimizers of the same functional used to build the null distribution, and the p-values in Supplementary Table 5 and the 'consistent with hyperbolic' statements in Figure 3 are in-sample fit statistics rather than valid hypothesis tests. The same issue affects the strawberry and hippocampal validation results.","section":"Online Methods, Eqs. 10–12 and Eqs. 4–5"},{"comment":"Incorrect CLT variance scaling. For a mean of r independent resamplings, the variance of the mean is Var(β_r)/r, not Var(β_r)·r. As written, Eq. 3 inflates the variance by a factor of r^2 relative to the correct expression and deflates every Z statistic in Eq. 4 by a factor of r. For r=100, this is a two-order-of-magnitude change in the test statistic, so all reported p-values would change after correction.","section":"Online Methods, Eq. 3"},{"comment":"Hyperbolic radial density and unspecified dimension. The radial density p(r) ∼ sinh((d−1)r) is not the radial marginal of the uniform measure in d-dimensional hyperbolic space; the correct density is proportional to sinh^{d−1}(r). For d>2, the simulated 'hyperbolic' model is a different, non-uniform ensemble, so consistency with that model does not directly imply consistency with hyperbolic geometry. The manuscript applies 3-D hyperbolic MDS but does not report the d used in the Betti-curve fits; d is absent from the Bayesian-optimization parameter list and from Supplementary Tables 3–5. The sampler should be corrected or the model explicitly re-described as a non-uniform radial model.","section":"Online Methods, Eq. 7"},{"comment":"The 'shuffled' candidate space is mentioned repeatedly in the main text, Figure 3, and Supplementary Table 5, but the manuscript never specifies how the shuffled adjacency matrices are generated. Without this definition, the claim that zero parcels are consistent with the a-geometric shuffled model is not reproducible or interpretable.","section":"Online Methods, Statistical Inference"}],"minor_comments":[{"comment":"The denominator in the cosine formula reads ||xi||||xi||; it should be ||xi|| ||xj||.","section":"Online Methods, Eq. 8"},{"comment":"Figure 3a states that all parcels had topology consistent with a hyperbolic shell, but the main text says 353 of 392 regions are consistent and 39 are inconsistent with any candidate space. This contradiction needs to be resolved.","section":"Figure 3 and main text"},{"comment":"The listed n values contain duplicates and omissions: n ∈ [10,15,20,25,40,45,50,45,50,55] should be cleaned. Also, the table reports only rmax and rmin; the best-fit d and noise level ε are described as varying but are not tabulated, which hampers reproducibility.","section":"Supplementary Table 3"},{"comment":"There are repeated typos: 'stoachstic' (main text), 'itegrated' (Online Methods), and 'the first n Betti numbers' should be 'the first three Betti numbers' in context.","section":"Main text and Online Methods"},{"comment":"p-values are printed as 0.0. After Bonferroni correction and Fisher's method, p-values cannot be exactly zero; they should be reported with inequalities such as p < 1e-10.","section":"Supplementary Table 5"}],"recommendation":"reject","confidential_remarks":"The computational side of this paper is potentially useful, and the authors have made their code available. My concern is solely with the validity of the statistical conclusions. If the authors resubmit after (i) correcting Eq. 3, (ii) replacing the in-sample p-values with hold-out or post-selection inference, and (iii) specifying the shuffled null and the hyperbolic dimension used in fitting, I would be willing to reconsider. As it stands, the headline geometric claims are not supported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The memory reduction is genuine: stochastic subsampling lets you compute Betti curves on hundreds-by-hundreds weighted networks on a laptop, and they reproduce prior odor and place-cell results with a fraction of the peak memory. Code is posted. That is a real, citable contribution to the toolkit. But the statistical inference that carries the headline findings is circular. The model is chosen by minimizing normalized l1 plus ibv distances (Eqs 10–12), and the p-values are then computed from the same l1 and ibv measures (Eqs 4–5) against that chosen model. So the small l1/ibv values that drive 'consistent with hyperbolic' are the ones the optimizer worked to produce; the null distribution is the distribution of the statistic without selection, not the distribution of the minimized statistic. There is no hold-out, no split-sample, no selection-adjusted inference. The cortical hyperbolic map, the headline result, is in-sample goodness of fit.\n\nWhat is actually new: applying already-known subsampling theory to clique complexes of weighted networks, and the ambitious 392-parcel cortical atlas. The text properly acknowledges that the subsampling theorems were proved for point clouds and that the clique-complex case is new. The validation on two independent datasets is a nice sanity check that the subsampling itself doesn't destroy the topological signal.\n\nSoft spots, in descending severity. Equation 3 looks wrong: the variance of a sample mean should shrink with the number of resamples, but they multiply by r. As written that would make the test conservative, but it is still an error. The radial density in Eq 7 is sinh((d−1)r), not the standard hyperbolic volume element; this needs a fix or an explicit source. Fisher's method on six dependent p-values is not valid without a correction for dependence. Minor but still worth saying: treating a non-significant p-value as 'consistent with' the model is a weak inference even in a proper test; here it is a byproduct of the circularity.\n\nWho gets value from this? Anyone building large-scale TDA pipelines will want the protocol and benchmarks. The geometry claims should be treated as hypotheses until the inference is redone. The paper deserves a serious referee, but the current central claims do not hold as stated.\n\nRecommendation: send to peer review with a request for major revision — redo the statistical inference with post-selection control or independent test data, correct the CLT formula and the radial density, and report correlations among the six statistics. With that, the cortical map might be worth believing.","headline":"Real memory win; circular geometry testing.","tokens_in":22310,"tokens_out":4851,"would_cite":false,"duration_ms":50627,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper's central claim is that stochastic sub-sampling makes topological data analysis (TDA) cheap enough to map the human cortex's functional geometry, and the resulting map is mostly hyperbolic.","keywords":["topological data analysis","Betti curves","stochastic sub-sampling","hyperbolic geometry","persistent homology","resting-state fMRI","functional connectivity","cortical parcellation"],"falsifier":"Split the sub-samples into two halves: use one half to select each model's parameters (minimizing normalized l1 plus integrated-Betti-value distance) and the other half to compute the reported p-values. If ALBATROSS's geometry claims are genuine, 'consistent with hyperbolic' should survive the split for most parcels and for the global matrices; if the p-values collapse, the geometry finding is an artifact of using the same data to fit and test.","tokens_in":21438,"feed_emoji":"🧠","tokens_out":6201,"duration_ms":61684,"temperature":0.7,"pith_summary":"Topological data analysis can reveal whether a dataset lives on a curved space, but computing filtered simplicial complexes on large matrices once required hundreds of gigabytes of memory. The paper introduces ALBATROSS, a protocol that computes Betti curves on many small random sub-samples of an adjacency matrix and averages them, so memory use depends on the sub-sample size rather than the full dataset. It validates the protocol by reproducing earlier findings on strawberry odor chemistry and rat hippocampal place-cell firing. It then runs TDA where it was previously infeasible: voxel-wise resting-state fMRI correlation matrices for 392 cortical parcels in ten people. The headline finding is that all ten global 392-by-392 matrices, and 353 of 392 local parcels, are statistically consistent with hyperbolic geometry.","feed_headline":"Brain's resting-state geometry comes out hyperbolic in cheap TDA","feed_subtitle":"Memory drops from ~71 GB to ~1.5 GB; 392 cortical parcels and all 10 global maps fit hyperbolic geometry.","key_machinery":"The object that carries the argument is the Betti curve of a filtered simplicial complex, which counts n-dimensional holes in the clique complex of a weighted adjacency matrix as the threshold varies. ALBATROSS replaces the full complex with an average over many random sub-samples of size n, a stochastic sub-sampling estimate; it then searches candidate geometries (Euclidean, hyperbolic shell or full space, or a shuffled null) by minimizing a normalized l1-plus-integrated-Betti-value objective, and converts the fit into p-values through a central-limit-theorem z-statistic with Bonferroni and Fisher corrections.","core_discovery":"On the paper's own terms, the discovery is that ALBATROSS turns filtered-simplicial-complex TDA from a supercomputer-scale computation into a laptop-scale one, and that doing so exposes a coherent geometric organization in the human cortex: the resting-state functional connectivity of every individual studied is consistent with a hyperbolic space at the global level, and most parcels show local hyperbolic geometry. The authors are careful to frame this as statistical consistency with candidate model spaces rather than direct proof that the cortex 'is' hyperbolic. What would follow is that cortical functional organization has a negatively curved, hierarchical structure, with local topology va","pith_inferences":["The candidate-geometry family is limited to Euclidean, hyperbolic, and shuffled null spaces; including spherical or product geometries could reclassify the 39 parcels currently inconsistent with all three, so the 'hyperbolic' label is relative to that family.","Because the same data choose and then test the model, the reported p-values are likely optimistic; a held-out split of sub-samples would turn ALBATROSS into a general model-selection tool whose geometry calls are confirmatory.","The correlation between local hyperbolic radius and global radial coordinate predicts a testable signature: interventions that shift a region's global hierarchical position should change its local Betti curves even when local signal quality is unchanged.","Replacing the radial sampling density in the hyperbolic model with the exact hyperbolic volume element could change the best-fit radii and noise levels, though probably not the coarse geometry family."],"forward_implications":["TDA on matrices with hundreds of rows and columns moves from a 475 GB compute node to a 24 GB laptop, so researchers without institutional clusters can run geometry checks.","The same protocol applies to any weighted adjacency matrix, not just neuroimaging, and the two validation datasets show it generalizes across data types.","Because classic network-science measures are a subset of filtered-complex information, ALBATROSS also provides a route to large-scale versions of measures such as rich-club structure.","If the cortical finding is correct, most of the brain's functional parcels sit in a negatively curved space, linking local connectivity topology to the brain's global hierarchical organization.","Memory use stays roughly constant for a chosen sub-sample size, so the approach scales to datasets that would otherwise be untouchable."],"supporting_citations":[{"why":"Supplies the theoretical result that Betti-curve measures on filtered complexes are stably estimated by stochastic sub-sampling, the basis of ALBATROSS.","marker":"[24]"},{"why":"Compares sub-sampling and clustering-based reduction for TDA, supporting the claim that stochastic sub-sampling performs well outside point-cloud settings.","marker":"[23]"},{"why":"The software used to compute Betti curves on each sub-sample.","marker":"[17]"},{"why":"Provides the strawberry-odor hyperbolic-shell finding and the model-comparison framework ALBATROSS must replicate.","marker":"[11]"},{"why":"Establishes the Betti-curve statistics and supplies the hippocampal place-cell dataset used for validation.","marker":"[10]"},{"why":"Gives the previous hyperbolic-geometry inference for hippocampal place cells that ALBATROSS reproduces.","marker":"[27]"},{"why":"Defines the 392 cortical parcels whose voxel-wise resting-state connectivity matrices are the target of the high-resolution demonstration.","marker":"[29]"},{"why":"Documents the acquisition and preprocessing pipeline for the resting-state fMRI data.","marker":"[44]"},{"why":"Code base for Bayesian hyperbolic multidimensional scaling used to place the 392 parcels in hyperbolic space.","marker":"[40]"}],"fun_headline_variants":["Hyperbolic brain maps now fit on a laptop","TDA memory cut 98%: brain geometry revealed","Cheap TDA: brain's global geometry is hyperbolic","From 71GB to 1.5GB: TDA uncovers brain curvature","Laptop-scale TDA shows brain's hyperbolic structure"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The protocol assumes that p-values obtained by comparing the empirical Betti curves to the model that was chosen to best fit those same curves are valid evidence of geometric consistency; if that post-model-selection comparison is not controlled, the reported 'consistent with hyperbolic' results are in-sample fit statistics rather than confirmatory tests.","fun_headline_variants_meta":{"raw":{"variants":["Hyperbolic brain maps now fit on a laptop","TDA memory cut 98%: brain geometry revealed","Cheap TDA: brain's global geometry is hyperbolic","From 71GB to 1.5GB: TDA uncovers brain curvature","Laptop-scale TDA shows brain's hyperbolic structure"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000168,"raw_usage":{"total_tokens":1028,"prompt_tokens":602,"completion_tokens":426,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":346,"completion_tokens_details":{"reasoning_tokens":356}},"tokens_in":346,"tokens_out":426,"duration_ms":4439,"temperature":1.0,"reasoning_tokens":356,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T10:45:22.217632+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Split the sub-samples into two halves: use one half to select each model's parameters (minimizing normalized l1 plus integrated-Betti-value distance) and the other half to compute the reported p-values. If ALBATROSS's geometry claims are genuine, 'consistent with hyperbolic' should survive the split for most parcels and for the global matrices; if the p-values collapse, the geometry finding is an artifact of using the same data to fit and test.","supporting_citations":[{"cited_title":"From Geometry to Topology: Inverse Theorems for Distributed Persistence","cited_arxiv_id":"2101.12288","evidence_quote":"Supplies the theoretical result that Betti-curve measures on filtered complexes are stably estimated by stochastic sub-sampling, the basis of ALBATROSS."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Compares sub-sampling and clustering-based reduction for TDA, supporting the claim that stochastic sub-sampling performs well outside point-cloud settings."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the strawberry-odor hyperbolic-shell finding and the model-comparison framework ALBATROSS must replicate."},{"cited_title":"Giusti, E","cited_arxiv_id":null,"evidence_quote":"Establishes the Betti-curve statistics and supplies the hippocampal place-cell dataset used for validation."},{"cited_title":"Zhang, P","cited_arxiv_id":null,"evidence_quote":"Gives the previous hyperbolic-geometry inference for hippocampal place cells that ALBATROSS reproduces."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the 392 cortical parcels whose voxel-wise resting-state connectivity matrices are the target of the high-resolution demonstration."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Documents the acquisition and preprocessing pipeline for the resting-state fMRI data."},{"cited_title":"Praturu, T","cited_arxiv_id":null,"evidence_quote":"Code base for Bayesian hyperbolic multidimensional scaling used to place the 392 parcels in hyperbolic space."}],"review_version":1}