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Cosmological information content of Betti curves and $k$-nearest neighbor distributions

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For halo clustering on nonlinear scales, nearly all Betti-curve information sits in beta0 and beta1, while kNNs match it.

desk verdict Careful Fisher comparison of Betti curves and kNNs with a useful convergence diagnostic; the beta2 claim is overstated and needs qualification, but the paper is worth refereeing. read the letter →

arxiv 2502.09709 v2 pith:RONNO6WH submitted 2025-02-13 astro-ph.CO

Aaron Ouellette , Gilbert Holder This is my paper · ORCID
classification astro-ph.CO MSC 85A4055N31
keywords Betticurvesk-nearestneighbordistributionspersistenthomologycosmologicalinformationFishermatrixconvergenceQuijotesimulationshaloclusteringnon-Gaussian
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Using Fisher matrices built from 5,000 Quijote simulations, this paper sets out to compare how much cosmological information two modern summary statistics extract from halo clustering on scales $2\,h^{-1}\mathrm{Mpc} < r < 50\,h^{-1}\mathrm{Mpc}$: Betti curves, a persistent-homology summary of topological features, and $k$-nearest-neighbor ($k$NN) distance distributions. The authors find that the first two Betti curves, $\beta_0$ and $\beta_1$, which track connected components and loops, carry essentially all of the constraining power in the well-converged parameter space; the void-counting curve $\beta_2$ contributes almost nothing to constraints on $\Omega_m$ and $\sigma_8$. They also find that $k$NNs, especially the data-data variant, give constraints competitive with the full Betti set, and that combining the two probes yields less information than independent statistics would. A secondary, methodological result is that only two parameter directions ($\Omega_m$ and $\sigma_8$) are reliably converged with the available simulations, which limits the dimensionality of the comparison.

What carries the argument

Betti curves are computed from an $\alpha$-complex filtration of the halo point cloud: at each filtration scale $r$, $\beta_d(r)$ counts the number of $d$-dimensional homology classes per halo (components for $d=0$, loops for $d=1$, voids for $d=2$). $k$NN CDFs record the fraction of query points whose distance to their $k$-th nearest halo is below $r$, with two variants: DR-$k$NNs (random volume-filling points to data) and DD-$k$NNs (data point to data point). The comparison instrument is the Fisher matrix from a Gaussian likelihood with constant covariance, with derivatives estimated by finite differences from 500 Quijote simulations; the paper's key methodological addition is an eigendecomposition-based convergence test that fits $F(N_{\mathrm{deriv}}) = F_{\infty} + F_{\mathrm{noise}}/N_{\mathrm{deriv}}$ for each eigenvalue and identifies which parameter directions are noise-dominated. This convergence test is what justifies restricting the headline comparison to the two well-constrained directions $\Omega_m$ and $\sigma_8$.

What would settle it

Recompute the Fisher comparison without the $\beta_d > 5\times10^{-4}$ truncation, or with a likelihood that allows non-Gaussian tails (for instance, simulation-based inference on the full data vectors), and see whether $\beta_2$ or the $k$NN tails add constraints that change the probe ranking. Alternatively, increase $N_{\mathrm{deriv}}$ well beyond 500 and check whether the third Fisher eigenvalue, aligned with $n_s$, converges; if it does, the reduced $\{\Omega_m, \sigma_8\}$ comparison may miss information that distinguishes the probes.

Watch

Extended reading notes

Core claim

The paper's central claim is that, for this halo sample, Betti curves and $k$NN distributions are measuring largely the same non-Gaussian clustering information, and that within the Betti curves the information is concentrated in the first two homology dimensions. Quantitatively, combining $\beta_0$ and $\beta_1$ gives 98% of the figure of merit of all three Betti curves in the $\{\Omega_m, \sigma_8\}$ plane; $\beta_2$, the void-counting curve, gives weak constraints on $\Omega_m$ and almost none on $\sigma_8$. The DD-$k$NNs (distances between halo pairs) achieve a figure of merit nearly equal to the Betti curves, while the DR-$k$NNs (distances from random volume-filling points to halos) perform worse in this setup. Combining Betti curves with $k$NNs improves constraints, but not by the factor $\sqrt{2}$ expected for independent probes, which the authors interpret as evidence that the two statistics are connected, possibly through their shared sensitivity to density regions.

Load-bearing premise

The headline comparison assumes each summary statistic follows a multivariate Gaussian distribution with a constant, parameter-independent covariance; the paper enforces this by truncating Betti curves at $\beta_d > 5\times10^{-4}$ and checks it qualitatively, but if the excluded tails carry non-Gaussian information, the ranking of probes could change.

Editorial extensions

If this is right

  • If the result holds, future Betti-curve analyses on similar halo samples can drop $\beta_2$ in the $\Omega_m$-$\sigma_8$ plane and lose little: $\beta_0$ plus $\beta_1$ recovers 98% of the combined figure of merit.
  • $k$-nearest-neighbor distributions, especially DD-$k$NNs, can serve as a computationally cheaper stand-in for Betti curves at these scales, with comparable constraining power and no need for $\alpha$-complex triangulation.
  • Combining Betti curves with $k$NNs will not double the information; the realized gain is closer to the gain from adding $k$NNs to the two-point function, consistent with a shared non-Gaussian signal.
  • The 2-point correlation function adds almost no new information once Betti curves and both $k$NN variants are included, so the topological and neighbor statistics are absorbing the small-scale non-Gaussian information that the power spectrum misses.
  • Forecasts in higher-dimensional parameter spaces from Quijote-style derivative sets should first run the eigenvalue convergence test; the paper finds only two directions are reliably converged, with the $h$ and $M_\nu$ directions fully noise-dominated and the $n_s$ direction still converging.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The overlap between Betti curves and $k$NNs hints at a mathematical correspondence: the $\alpha$-complex filtration threshold is itself a distance criterion, so the homology classes counted by $\beta_d$ are functions of the same pairwise-distance information that $k$NN CDFs compress; one could test this by asking whether the Euler characteristic ($\beta_0 - \beta_1 + \beta_2$) fully predicts the $
  • $\beta_2$'s near-zero constraining power in the two-parameter plane may depend on the number-density cut at 150,000 halos; on sparser or denser samples, or with a void-oriented filtration, void counting could become informative, so the 'drop $\beta_2$' advice should be re-checked per survey selection.
  • The DR-$k$NN versus DD-$k$NN split maps onto low-density versus high-density environments, which suggests the Betti curves' sensitivity could be decomposed the same way; a joint analysis in redshift space, where $k$NNs have a natural decomposition, could clarify which physical regions carry the shared signal.
  • A direct observational test would be to apply the same Fisher comparison to a real galaxy catalog in redshift space, where $k$NNs have known advantages; if the overlap persists, pipeline designers could choose $k$NNs on cost grounds alone.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper compares the cosmological information content of Betti curves and k-nearest neighbor (kNN) distributions as summary statistics for halo clustering, using Fisher matrices estimated from the Quijote simulations on scales 2–50 h^-1 Mpc. The authors pay careful attention to convergence: they use 5,000 fiducial simulations for covariance, apply the Hartlap correction, and diagnose derivative-noise convergence through an eigenvalue decomposition, concluding that only two parameter directions are reliably constrained. Restricting to {Omega_m, sigma8}, they find that beta0 and beta1 contain nearly all the Betti-curve information while beta2 contributes little, that DD-kNNs are highly competitive while DR-kNNs are less so, and that the kNN and Betti probes are complementary but not fully independent.

Significance. If the main results hold, the paper provides practical guidance for survey analysis: Betti-curve analyses could focus on beta0/beta1, and kNNs could serve as a cheaper substitute or complement. The Fisher convergence methodology is a genuine strength: the use of 5,000 simulations, Hartlap correction, eigenvalue-based noise fractions, and the explicit restriction to converged parameter directions is careful and reproducible with public Quijote data. The work also highlights a possible connection between topological statistics and kNNs. However, the headline claims rest on several choices—the Betti tail cut, the unspecified treatment of kNN tails, and the way probe combinations are constructed—that are not yet fully validated, so the quantitative conclusions should be treated with caution until those choices are tested.

major comments (4)
  1. [§5.2, Fig. 2, App. A] The claim that beta2 has almost no constraining power is not robust to the ad hoc tail cut beta_d > 5e-4. Because beta2 decays toward zero at large r (Fig. 2), this cut removes exactly the large-scale tail of the beta2 curve; the surviving bins are low-amplitude and potentially noise-dominated. The Gaussianity check in Appendix A is qualitative (a chi-square histogram with roughly 1,500 simulations and 60-150 degrees of freedom), so it has limited power to detect non-Gaussianity in these low-count bins. The paper should show that the beta2 conclusion is stable to the threshold (for example, by repeating the Fisher calculation for several values of the cut) or quantify the information contained in the discarded tail; as written, the abstract claim that 'almost no constraining power comes from beta2' may be an artifact of the cut. The parameter dependence is also visible in §7 and Fig. 9, where beta2 does constrain n_s, so the statement should be qualified.
  2. [§7, Table 1] The method used to build Fisher matrices for probe combinations is not stated. If each combination is computed as the sum of the individual Fisher matrices, then independence is assumed by construction and the conclusion that the probes are 'not fully independent' is circular. To test independence, the authors need to compute the joint Fisher matrix from the concatenated data vector (including the cross-covariance between probes) and compare it with the sum of the individual Fisher matrices. In addition, the baseline stated in the text is incorrect: with FoM = sqrt(det F) in two parameters, combining two independent probes with equal Fisher matrices doubles the FoM, not multiplies it by sqrt(2). The quantitative independence claim therefore needs to be rederived or softened.
  3. [§5.3 vs §7] The treatment of kNN tails is inconsistent. Section 5.3 defines the kNN CDFs and specifies the k values but no scale or tail cut, while Section 7 states that 'we have explicitly removed the tails of the kNN distributions in order to ensure that our summary statistics are Gaussian.' The threshold and the number of retained bins must be specified, and the impact of this cut on the Fisher results should be quantified; as written, the comparison protocol between kNNs and Betti curves is ambiguous.
  4. [§6, Eq. (3)] The entire comparison rests on the assumption that each summary statistic is multivariate Gaussian with parameter-independent covariance, but the evidence for this is only the qualitative chi-square test in Appendix A. Given that the Betti curves are truncated specifically to enforce Gaussianity, and that the tails that are removed may contain non-Gaussian information, the paper should either provide a more quantitative Gaussianity assessment (for example, goodness-of-fit statistics or a comparison of Fisher information with and without tail bins) or explicitly frame the results as conditional on this assumption. The ranking of probes in Table 1 could change if the excluded tails carry information.
minor comments (5)
  1. [Abstract] The abstract's statement that 'the kNNs provide very competitive constraints' overstates the results: in Table 1, DR-kNN has FoM 0.42 versus 0.52 for the 2-point function in the Omega_m-sigma8 space, so only DD-kNN is clearly competitive.
  2. [§7, Fig. 9] The finding that beta2 does constrain n_s in the three-parameter space should be echoed in the abstract or conclusions to avoid overgeneralizing the 'beta2 has almost no constraining power' statement, which is specific to the reduced two-parameter space.
  3. [§6.2, Table 2] Please clarify whether the two-parameter Fisher matrices are recomputed from derivatives and covariances restricted to {Omega_m, sigma8} or derived by projecting the five-dimensional matrices; the eigenvectors in Table 2 are mixtures involving h and n_s, so the relationship between the 5D and 2D convergence statements is not immediate.
  4. [§5.2] Please state the number of bins retained for each Betti curve after the beta_d > 5e-4 cut; Figure 3 suggests roughly 60 total bins, but the exact numbers matter for reproducibility and for interpreting the Appendix A chi-square degrees of freedom.
  5. [Fig. 10] Please label the number of degrees of freedom and the number of simulations used for each chi-square distribution in Figure 10, since the power of the Gaussianity test depends on both.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Fisher constraints are computed from external Quijote simulations with numerical derivatives, and the probe-comparison conclusions are inferred from those Fisher matrices rather than imposed by construction.

full rationale

The paper's central claims—that beta0 and beta1 contain almost all of the Betti-curve information, that kNNs give competitive constraints, and that kNNs and Betti curves are not fully independent—are outputs of Fisher matrices built entirely from external simulation data. Section 6.1 states that covariance matrices use 5,000 fiducial Quijote simulations and that derivatives are estimated by finite differences from 500 perturbed realizations; no parameter is fitted to the target conclusions. The 'not fully independent' statement is inferred from the FoM combination values in Table 1 using Eq. (5), not encoded in the summary statistics themselves. The beta2 result is a measured Fisher constraint rather than a definitional artifact, and the paper itself notes that beta2 does add useful information in the three-parameter space, so the abstract's blanket 'almost no constraining power' claim is an overstatement but not circular. The only self-citation, Ouellette et al. (2023), supplies the alpha-complex Betti-curve pipeline; it is prior methodological support, not a load-bearing citation that defines the present result. Potential robustness concerns—such as the beta_d > 5e-4 Gaussianity cut in Section 5.2, the qualitative chi-square test in Appendix A, and the finite-derivative convergence treatment—are correctness or assumption risks, not circularity, because they do not make the Fisher output equivalent to the input by construction.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted constants; the listed free parameters are hand-chosen analysis settings and domain assumptions that the central comparison depends on. The axioms are standard Fisher and persistent-homology background plus the Gaussianity and convergence assumptions stated in Sections 5, 6, and Appendix A.

free parameters (5)
  • Scale range for all summary statistics = 2 to 50 h^-1 Mpc
    Chosen by hand in Sections 5.1-5.3; all Fisher information comparisons depend on this range.
  • Betti curve tail threshold = beta_d > 5e-4 per halo
    Section 5.2; imposed to keep the statistics Gaussian, but it removes the tails that may carry information.
  • kNN order set = k = 1, 2, 4, 8, 16
    Section 5.3; a finite set is selected, so the full kNN hierarchy is not used.
  • Random point count for DR-kNN = 1e6
    Section 5.3; chosen to fill the volume and affects the noise in DR-kNN CDFs.
  • Halo number density cut = 150,000 halos; nbar = 1.5e-4 h^3 Mpc^-3
    Section 4; top-mass selection keeps number density constant but changes the mass threshold between cosmologies.
assumptions (5)
  • domain assumption Each summary statistic is drawn from a multivariate Gaussian distribution with parameter-independent covariance.
    Assumed in Section 6 (Fisher formalism) and checked only qualitatively in Appendix A; the tail cut in Section 5.2 is imposed to make this plausible.
  • domain assumption Quijote simulations and FoF halo catalogs at z=0 faithfully represent nonlinear halo clustering.
    Section 4; all forecasts are simulation-based and no comparison to observations is made.
  • domain assumption The eigendecomposition noise-fraction test identifies all unconverged Fisher directions, so restricting to Omega_m and sigma_8 removes derivative noise bias.
    Section 6.2 and Table 2; the method is new to this paper and is validated only by the observed 1/Nderiv scaling.
  • ad hoc to paper The chosen scale cuts and the beta_d > 5e-4 threshold do not remove significant cosmological information.
    Section 5.2; these cuts are justified by Gaussianity, but the Fisher ranking depends on them.
  • standard math Persistent homology with alpha-complex filtration and kNN CDFs are computed as described by the cited libraries.
    Sections 2, 3, and 5; relies on Gudhi, Corrfunc, scipy k-d tree, and the cited theory.

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Cite this review

Pith. "Pith review of Cosmological information content of Betti curves and $k$-nearest neighbor distributions." pith.science (2026). https://pith.science/paper/RONNO6WH

@misc{pith2026250209709,
  author       = {Pith},
  title        = {Pith review of: Cosmological information content of Betti curves and $k$-nearest neighbor distributions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RONNO6WH}},
  note         = {Machine review of arXiv:2502.09709}
}
abstract

We compare the cosmological constraints that can be obtained from halo clustering on non-linear scales ($2 h^{-1}$ Mpc < $r$ < $50 h^{-1}$ Mpc) using Betti curves, a topological summary statistic, and $k$-th nearest neighbor ($k$NN) distributions. We quantify the information content of each summary statistic through Fisher matrices computed from the Quijote simulations. Due to the use of simulation-based Fisher forecasts, we pay careful attention to the convergence of the Fisher matrices by looking at their eigendecompositions. We find that, in general, only two directions in the parameter space have constraints that are well converged given the number of Quijote simulations available. We then compare the information content of each summary statistic in the reduced parameter space $\{\Omega_m, \sigma_8\}$. We find that almost all of the information present in the Betti curves comes from the first two, $\beta_0$ and $\beta_1$, which track the number of connected components and one-dimensional loops respectively, and almost no constraining power comes from $\beta_2$ which tracks the number of topological voids. In comparison, we find that the $k$NNs provide very competitive constraints along with several potential advantages in regards to real data. Finally, we find that while the $k$NNs and Betti curves provide some complementary constraints, they are not fully independent, potentially indicating a connection between the two statistics.

Figures

Figures reproduced from arXiv: 2502.09709 by the authors.

Figure 1
Figure 1. — An example of an 𝛼-complex filtration using random points in a 2D periodic box. As the filtration parameter 𝛼 increases, larger simplices are added to the complex and more neighboring points get connected. Topological features such as loops appear in the complex at some value of 𝛼 and then disappear at a larger value as more simplices are added. Betti curves (𝛽0 and 𝛽1 in 2D) track the numbers of these features as… view at source ↗
Figure 2
Figure 2. — Top panel: Average halo Betti curves computed from 5,000 sim￾ulations at the fiducial cosmology. The Betti curves are normalized by the number of halos in the catalog (𝑁ℎ = 150, 000). Bottom panel: Fractional variations in the Betti curves when changing Ω𝑚 / 𝜎8 by 5%. The faint shaded regions indicate the range of 1𝜎 variations at the fiducial cosmology. 0 20 40 60 bin number 0 10 20 30 40 50 60 bin number −0.4 −0… view at source ↗
Figure 3
Figure 3. — Betti curve correlation matrix computed from 5,000 simulations at the fiducial cosmology. The three sub-matrices along the diagonal represent the correlation matrices for 𝛽0, 𝛽1, and 𝛽2, in order of increasing bin number. 6. FISHER MATRIX FORMALISM The Fisher matrix is used ubiquitously in cosmology to es￾timate or forecast the constraints on cosmological parameters given some summary statistic and the expected me… view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: — Top panel: Average DR-𝑘NNs for the fiducial cosmology. For visualization purposes, we plot the peaked CDF. From left to right, the curves represent the 𝑘NN CDFs for 𝑘 = 1, 2, 4, 8, 16. Bottom panel: Fractional variations in the 𝑘 = 1, 𝑘 = 4, and 𝑘 = 16 curves when ch…
Figure 7
Figure 7. Figure 7: — The DD-𝑘NN correlation matrix. The sub-blocks along the diag￾onal correspond to 𝑘 = 1, 2, 4, 8, 16. parameters, is then given by the Cramér-Rao bound: 𝜎𝛼 ≥ √︁ (𝐹−1 )𝛼𝛼. (4) We also define a figure-of-merit (FoM) to quantify how infor￾mative a given summary statistic …
Figure 8
Figure 8. Figure 8: — Constraints in the Ω𝑚-𝜎8 parameter space computed from the top 1.5 × 105 halos in real space. On the left we show the individual Betti curves and on the right we compare the combined Betti curves with the 𝑘NNs. 0.30 0.35 Ωm 0.9 1.0 ns 0.82 0.83 0.84 0.85 σ8 0.82 0.83…
Figure 9
Figure 9. Figure 9: — Same as [PITH_FULL_IMAGE:figures/full_fig_p007_9.png]
Figure 10
Figure 10. Figure 10: — The distribution of 𝜒 2 values for each of the considered summary statistics. The blue histograms show the distributions of 𝜒 2 values calculated from the data vectors. The orange curves show the expected distributions from theory, and the green histograms show real…
Figure 11
Figure 11. Figure 11: — Convergence of Fisher matrix eigenvalues with respect to the number of derivative simulations. The dotted lines indicate the estimated noise level for each eigenvalue [PITH_FULL_IMAGE:figures/full_fig_p009_11.png]
Figure 12
Figure 12. Figure 12: — Convergence of the Fisher matrix eigenvalues with respect to the number of covariance simulations [PITH_FULL_IMAGE:figures/full_fig_p010_12.png]

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