Pith. sign in

REVIEW 2 major objections 5 minor 1 cited by

Towards characterizing dark matter subhalo perturbations in stellar streams with graph neural networks

T0 review · 2 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A graph neural network on a stellar stream's full phase space tightens dark subhalo mass constraints by up to an order of magnitude.

desk verdict The GCNN+SBI pipeline is a genuine new tool for stream-subhalo inference, but the headline [11,7,3] gains are measured against an unvalidated in-house power-spectrum baseline, so treat them as upper-end, not settled. read the letter →

arxiv 2502.03522 v2 pith:46EA6NW7 submitted 2025-02-05 astro-ph.GA astro-ph.IM

Peter Xiangyuan Ma , Keir K. Rogers , Ting S. Li , Renée Hložek , Jeremy Webb , Ruth Huang , Julian Meunier This is my paper · ORCID
classification astro-ph.GAastro-ph.IM
keywords darkmattersubhalosstellarstreamsgraphconvolutionalneuralnetworkssimulation-basedinferenceGD-1posteriorcalibrationMilkyWaysubstructure
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

This paper argues that a graph convolutional neural network (GCNN) can compress the full six-dimensional phase space of a stellar stream into summary statistics that retain far more information about a passing dark matter subhalo than the standard one-dimensional density power spectrum. On simulated streams modelled on the Milky Way stream GD-1, the method improves the constraint on the perturbing subhalo's mass by a factor of 11 at $10^{8}$ solar masses, 7 at $10^{7}$, and 3 at $10^{6}$, compared with the power-spectrum analysis. The paper also shows that the resulting posteriors are better calibrated, meaning the uncertainty estimates are trustworthy. A practical consequence is that 300 stars with complete phase-space data constrain subhalo mass about as well as 3000 stars with only sky positions and line-of-sight velocities. The authors present this as a proof of principle for future photometric, spectroscopic, and astrometric surveys.

What carries the argument

The machinery is a graph convolutional neural network used as a learned data compressor, followed by simulation-based inference with normalizing flows. Each star is a graph node with a six-dimensional phase-space feature vector; edges connect each node to its five nearest neighbours with weight equal to the inverse squared distance, so the graph encodes local stream geometry and pairwise interactions. Graph-convolution layers perform message passing over this adjacency matrix, and a global-average pooling layer reduces the node features to a fixed-length vector that is mapped to two summary statistics estimating subhalo mass and velocity. A masked autoregressive flow, trained with neural posterior estimation, then learns the posterior distribution of mass and velocity from simulated pairs of parameters and compressed summaries. The paper compares this encoder against the current state-of-the-art power spectrum encoder on the same simulation-based inference pipeline.

What would settle it

Run the same GCNN-plus-SBI pipeline on training simulations that include the realistic population of many subhalos (the paper notes a real stream appreciably interacts with order 100 subhalos) and nonzero impact parameters; if the mass-constraint gains of 3 to 11 over the power spectrum do not survive, the central claim is refuted.

Watch

Extended reading notes

Core claim

The central claim is that replacing the lossy one-dimensional angular power spectrum with a graph neural network applied directly to the stream's phase-space point cloud yields both tighter and more accurate constraints on the mass and relative velocity of a perturbing subhalo. In a suite of streamspraydf simulations of a GD-1-like stream with a single Hernquist-profile subhalo, the GCNN encoder reduces the fractional mean absolute error on mass and velocity relative to the power-spectrum encoder, and the posterior mass constraints improve by factors of 11, 7, and 3 for subhalo masses of $10^{8}$, $10^{7}$, and $10^{6}$ solar masses, respectively. The GCNN posteriors pass a simulation-based calibration test across most of the prior volume, while the power-spectrum posteriors do not. The authors attribute the gain to the network using all six phase-space dimensions, to the graph structure capturing local geometric distortions such as gaps and spurs, and to the velocity information helping to break the mass-velocity degeneracy.

Load-bearing premise

The whole comparison rests on the assumption that a simulated stream perturbed by a single subhalo with zero impact parameter, a fixed Hernquist scale radius of 10 parsecs, and an interaction time fixed at -200 Myr captures how real subhalos perturb GD-1.

Editorial extensions

If this is right

  • Subhalo mass constraints from a single stream improve by factors of 3 to 11 across 10^6 to 10^9 solar masses relative to the one-dimensional power spectrum analysis.
  • GCNN posteriors are simultaneously tighter and better calibrated, so the reported uncertainties can be used for downstream dark-matter parameter inference.
  • Velocity information in the phase space partially breaks the mass-velocity degeneracy that limits power-spectrum analyses.
  • A 300-star sample with full six-dimensional phase space performs about as well as a 3000-star sample with only positions and line-of-sight velocities.
  • Performance plateaus with network size, so future analyses can use thousands rather than hundreds of thousands of training simulations.

Reading between the lines

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

  • If the gains survive realistic multi-subhalo and off-centre encounter simulations, this approach could push single-stream sensitivity toward the 10^5 solar-mass forecast of upcoming surveys and sharpen warm-dark-matter constraints.
  • The coordinate-system sensitivity suggests that choosing or learning a stream-aligned representation could yield further gains without new data.
  • The 300-star result implies an observing strategy for future surveys: complete astrometric and spectroscopic phase space for a few hundred stream members may be worth more than sheer depth on thousands.
  • Applying the trained pipeline to actual GD-1 data would be a direct test, although the paper does not claim this is ready without survey-error modeling.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. This paper presents a proof-of-principle method for inferring the mass and velocity of a dark-matter subhalo that perturbs a GD-1-like stellar stream. The authors generate large training sets with the streamspraydf/galpy particle-spray simulations, then compare two encoders: a 1D angular power spectrum compressed by a dense neural network and a graph convolutional neural network (GCNN) applied directly to the 6D phase space. Both are followed by neural posterior estimation with normalizing flows. The paper reports that the GCNN improves subhalo-mass constraints by factors of roughly 11, 7, and 3 for subhalo masses of 10^8, 10^7, and 10^6 solar masses, respectively, relative to the power-spectrum baseline, and that the GCNN posteriors are better calibrated. It also studies the effect of model size, input coordinate system, reduced phase space, and a smaller number of observed stars, concluding that 300 stars with full 6D data perform about as well as 3000 stars with 3D data.

Significance. If the headline result holds, this is a useful proof-of-principle: it demonstrates that a GCNN can extract more information from the full phase space of a stream than a 1D density power spectrum, and it does so with a careful, controlled comparison on a held-out test set. The paper uses standard simulation-based calibration diagnostics and is explicit about the idealized nature of the simulations. The main value is in showing that field-level compression with graph networks is a viable route for future stream analyses with upcoming surveys. The significance is currently tempered by the fact that the quantitative headline gains are measured against an in-house power-spectrum baseline whose calibration fails, and by the highly idealized single-subhalo, fixed-geometry simulation setup.

major comments (2)
  1. [§2.2.1, §3.2.1, Table 3] The power-spectrum baseline is an in-house pipeline that compresses 179 power-spectrum bins with a dense neural network and then applies neural posterior estimation; it is not the published approximate-Bayesian-computation analysis of Banik et al. (2021b), which the introduction identifies as the current state of the art. The PS posteriors fail the simulation-based calibration test (KS p ≪ 0.05) over exactly the region where the GCNN posteriors pass, and a correctly implemented SBI posterior conditioned on any summary statistic should be well calibrated regardless of the information content of that summary. The calibration failure therefore indicates an implementation or training defect in the PS baseline rather than an intrinsic property of the power spectrum, so the precision ratios in Table 2 and the calibration advantage in Table 3 are likely inflated. Please validate the PS baseline against the published method on the same simulations, or otherwise demonstrate calibrated PS posteriors, and restate the headline gains relative to the validated baseline.
  2. [Abstract, §2.2, Table 2] The headline factors [11,7,3] correspond to the GCNN helio (6-D) variant, which is the best of the five GCNN encoders, rather than to the baseline GCNN architecture described in §2.2.2. Table 2 shows that the baseline GCNN gains relative to the same best model are 1.15–2.02, with the remaining improvement coming from the coordinate-system change (and in the high-mass bin the 300-star model slightly outperforms the nominal best model). The abstract and conclusion should therefore state explicitly that the reported factors use the heliocentric 6-D variant, so that readers do not attribute the full gain to the graph architecture alone.
minor comments (5)
  1. [Table 2, Eq. (8)] The quantity σ_best is defined as the GCNN helio (6-D) standard deviation, yet the GCNN (6-D) 300 stars row reports 0.97 ± 0.11 in the highest-mass bin, i.e., a smaller width than the nominal best model. Please define a per-bin best model or clarify this exception in the table caption.
  2. [§2.1, step 4] The 'target star' selection is not specified in detail; since the interaction time is fixed to -200 Myr and the impact parameter to zero, a sentence explaining how the target star and interaction geometry are chosen would improve reproducibility.
  3. [Throughout] There are several typographical issues: 'T able 1' in the Table 1 caption, 'apparoach' in §4.1, and the broken reference formatting for the Hinton and Williams citation in the bibliography.
  4. [§4.4] The discussion of the coordinate-system improvement is qualitative; a brief statement on whether the improvement is robust across random seeds and training runs would strengthen the claim.
  5. [§3.2.1] The text says posteriors are reported only for the region where the GCNN passes calibration; this restriction should be stated earlier and more prominently in §3, so that readers do not over-interpret comparisons outside that region.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the claimed gains are measured on held-out simulations against an independent encoder baseline, with only a minor non-load-bearing self-citation.

full rationale

The paper's central claim is a benchmark comparison between two data-compression strategies, GCNN and the 1D angular power spectrum, on simulated GD-1-like streams. Both encoders are trained on forward simulations with known input parameters (subhalo mass and velocity), and the reported improvements in posterior precision and calibration are evaluated on a disjoint Latin-hypercube test set drawn from the same simulator. There is no fitted parameter that is renamed as a prediction: the target quantities are the true subhalo parameters of the held-out simulations, and the posteriors are scored against those truths. The power-spectrum baseline is independently trained on the same simulations, so the comparison is not defined in terms of the GCNN output. The paper's own limitation discussion (§4.5) acknowledges that the simulation model fixes several physical ingredients (single subhalo, zero impact parameter, fixed Hernquist scale radius, fixed interaction time); this is a realism/systematic-risk concern, not circularity. The only overlapping-author citation relevant to the paper's motivation is Rogers & Peiris (2021a,b) and Rogers et al. (2022) for Lyman-alpha constraints, which is not load-bearing for the stream-inference derivation. A possible weakness is that the in-house power-spectrum pipeline is not validated against the published Banik et al. (2021b) analysis, and its poor calibration (Table 3) may indicate a suboptimal baseline; that would affect the fairness of the improvement factors, but it does not make the GCNN result equal to its inputs by construction. No circular step was found, so the score is low and reflects only this benchmark-validation caution and a minor non-load-bearing self-citation.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central claim rests on hand-chosen simulation parameters (scale radius, impact parameter, interaction time, progenitor mass, graph connectivity) and on the fidelity of the particle-spray and single-subhalo forward model. No new physical entities are introduced. The improvement factors are conditional on these modeling choices and on the assumption that the training simulations cover the relevant data distribution.

free parameters (5)
  • Subhalo scale radius = 10 pc
    Fixed Hernquist scale radius chosen by hand (§2.1 step 5); perturbation strength and hence the reported gains depend on this choice.
  • Impact parameter = 0
    The closest approach of the subhalo to the stream is set to zero (§2.1 step 5), maximizing the perturbation signal; nonzero values would weaken the constraints.
  • Subhalo interaction time = -200 Myr
    The time of closest approach is fixed in the training simulations (§2.1 step 4), removing a realistic source of uncertainty.
  • Progenitor mass = 10^4 M_sun
    Fixed progenitor mass used in all simulations (§2.1 step 1); stream density and detectability depend on this choice.
  • Number of nearest neighbors (k) = 5
    Graph construction hyperparameter chosen by the authors (§2.2.2); smaller values degrade training, larger values were not tested.
assumptions (4)
  • domain assumption The MWPotential2014 static Milky Way potential is an adequate background for the stream and subhalo orbits.
    Used in every simulation (§2.1 steps 2, 6, 7); real streams experience a time-varying potential and baryonic perturbations.
  • domain assumption The streamspraydf particle-spray method reproduces the observable properties of GD-1 sufficiently for training.
    Assumed in §2.1 and supported by references to Fardal et al. (2015); this is a simplified tidal model, not a full N-body calculation.
  • domain assumption A single subhalo with a Hernquist profile and fixed scale radius adequately represents the perturbation.
    The forward model includes only one subhalo with r_s = 10 pc (§2.1 step 5); realistic halos contain many subhalos with varying profiles.
  • domain assumption The neural network and normalizing flow generalize from the training prior to the test distribution.
    Standard supervised and SBI assumption (§2.2, §2.3); edge-of-prior calibration fails, so the paper restricts to a central well-calibrated area.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Towards characterizing dark matter subhalo perturbations in stellar streams with graph neural networks." pith.science (2026). https://pith.science/paper/46EA6NW7

@misc{pith2026250203522,
  author       = {Pith},
  title        = {Pith review of: Towards characterizing dark matter subhalo perturbations in stellar streams with graph neural networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/46EA6NW7}},
  note         = {Machine review of arXiv:2502.03522}
}
abstract

The phase space of stellar streams is proposed to detect dark substructure in the Milky Way through the perturbations created by passing subhalos - and thus is a powerful test of the cold dark matter paradigm and its alternatives. Using graph convolutional neural network (GCNN) data compression and simulation-based inference (SBI) on a simulated GD-1-like stream, we improve the constraint on the mass of a [$10^8$, $10^7$, $10^6$] $M_\odot$ perturbing subhalo by factors of [11, 7, 3] with respect to the current state-of-the-art density power spectrum analysis. We find that the GCNN produces posteriors that are more accurate (better calibrated) than the power spectrum. We simulate the positions and velocities of stars in a GD-1-like stream and perturb the stream with subhalos of varying mass and velocity. Leveraging the feature encoding of the GCNN to compress the input phase space data, we then use SBI to estimate the joint posterior of the subhalo mass and velocity. We investigate how our results scale with the size of the GCNN, the coordinate system of the input and the effect of incomplete observations. Our results suggest that a survey with $10 \times$ fewer stars (300 stars) with complete 6-D phase space data performs about as well as a deeper survey (3000 stars) with only 3-D data (photometry, spectroscopy). The stronger constraining power and more accurate posterior estimation motivate further development of GCNNs in combining future photometric, spectroscopic and astrometric stream observations.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The Milky Way - Large Magellanic Cloud Interaction with Simulation Based Inference

    astro-ph.GA 2025-10 conditional novelty 5.0 of 10

    Simulation-based inference on outer-halo star velocities gives a Milky Way reflex speed of 26.4 km/s and an LMC enclosed mass of 9.2×10^10 solar masses within 50 kpc.

Reference graph

Works this paper leans on

132 extracted references · 10 canonical work pages · cited by 1 Pith paper

  1. [1]

    , " * write output.state after.block = add.period write newline

    ENTRY address archivePrefix author booktitle chapter doi edition editor eprint howpublished institution journal key month number organization pages publisher school series title misctitle type volume year version url label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts ...

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION format.url url empty "" new.block "" url * "" * if FUNCTION format.eprint eprint empty "" archivePrefix empty "" archivePrefix "arXiv" = new.block " " eprint * " " * new.block " " eprint * " " * if if if FUNCTION format.doi doi empty "" " " doi * " " * if FUNCTION format.pid doi empty eprint empty ur...

  3. [4]

    2020, Astronomy & Astrophysics, 641, A6, 10.1051/0004-6361/201833910

    Aghanim, N., Akrami, Y., Ashdown, M., et al. 2020, Astronomy & Astrophysics, 641, A6, 10.1051/0004-6361/201833910

    doi:10.1051/0004-6361/201833910 2020

  4. [5]

    2019, Monthly Notices of the Royal Astronomical Society, 10.1093/mnras/stz1960

    Alsing, J., Charnock, T., Feeney, S., & Wandelt, B. 2019, Monthly Notices of the Royal Astronomical Society, 10.1093/mnras/stz1960

    doi:10.1093/mnras/stz1960 2019

  5. [6]

    2023, Monthly Notices of the Royal Astronomical Society, 525, 3662–3681, 10.1093/mnras/stad2458

    Alvey, J., Gerdes, M., & Weniger, C. 2023, Monthly Notices of the Royal Astronomical Society, 525, 3662–3681, 10.1093/mnras/stad2458

    doi:10.1093/mnras/stad2458 2023

  6. [7]

    2021 a , Journal of Cosmology and Astroparticle Physics, 2021, 043, 10.1088/1475-7516/2021/10/043

    Banik, N., Bovy, J., Bertone, G., Erkal, D., & de Boer, T. 2021 a , Journal of Cosmology and Astroparticle Physics, 2021, 043, 10.1088/1475-7516/2021/10/043

    doi:10.1088/1475-7516/2021/10/043 2021

  7. [8]

    Banik, N., Bovy, J., Bertone, G., Erkal, D., & de Boer, T. J. L. 2021 b , Monthly Notices of the Royal Astronomical Society, 502, 2364–2380, 10.1093/mnras/stab210

    doi:10.1093/mnras/stab210 2021

  8. [9]

    2020, Astronomy & Astrophysics, 642, A191, 10.1051/0004-6361/202038071

    Blanchard, A., Camera, S., Carbone, C., et al. 2020, Astronomy & Astrophysics, 642, A191, 10.1051/0004-6361/202038071

    doi:10.1051/0004-6361/202038071 2020

Show all 132 references
  1. [10]

    D., Bolton, J

    Boera, E., Becker, G. D., Bolton, J. S., & Nasir, F. 2019, The Astrophysical Journal, 872, 101, 10.3847/1538-4357/aafee4

    2019 doi

  2. [11]

    W., Price-Whelan, A

    Bonaca, A., Hogg, D. W., Price-Whelan, A. M., & Conroy, C. 2019, The Astrophysical Journal, 880, 38, 10.3847/1538-4357/ab2873

    2019 doi

  3. [12]

    Bonaca, A., & Price-Whelan, A. M. 2024, Stellar Streams in the Gaia Era, arXiv, 10.48550/ARXIV.2405.19410

    2024 doi arXiv

  4. [13]

    2014, The Astrophysical Journal, 795, 95, 10.1088/0004-637x/795/1/95

    Bovy, J. 2014, The Astrophysical Journal, 795, 95, 10.1088/0004-637x/795/1/95

    2014 doi

  5. [14]

    2015, The Astrophysical Journal Supplement Series, 216, 29, 10.1088/0067-0049/216/2/29

    ---. 2015, The Astrophysical Journal Supplement Series, 216, 29, 10.1088/0067-0049/216/2/29

    2015 doi

  6. [15]

    2016, Physical Review Letters, 116, 10.1103/physrevlett.116.121301

    ---. 2016, Physical Review Letters, 116, 10.1103/physrevlett.116.121301

    2016 doi

  7. [16]

    Bovy , J., Erkal , D., & Sanders , J. L. 2017, , 466, 628, 10.1093/mnras/stw3067

    2017 doi

  8. [17]

    Brooks, R. A. N., Sanders, J. L., Lilleengen, S., Petersen, M. S., & Pontzen, A. 2024, Monthly Notices of the Royal Astronomical Society, 532, 2657–2673, 10.1093/mnras/stae1565

    2024 doi

  9. [18]

    Brown, A. G. A., Vallenari, A., Prusti, T., et al. 2021, Astronomy & Astrophysics, 649, A1, 10.1051/0004-6361/202039657

    2021 doi

  10. [19]

    Buist, H. J. T., & Helmi, A. 2015, Astronomy & Astrophysics, 584, A120, 10.1051/0004-6361/201526203

    2015 doi

  11. [20]

    V., Dutton, A

    Butsky, I., Macciò, A. V., Dutton, A. A., et al. 2016, Monthly Notices of the Royal Astronomical Society, 462, 663–680, 10.1093/mnras/stw1688

    2016 doi

  12. [21]

    2019, Astronomy and Computing, 28, 100307, 10.1016/j.ascom.2019.100307

    Caldeira, J., Wu, W., Nord, B., et al. 2019, Astronomy and Computing, 28, 100307, 10.1016/j.ascom.2019.100307

    2019

  13. [22]

    F., Piedipalumbo, E., & Tortora, C

    Cardone, V. F., Piedipalumbo, E., & Tortora, C. 2005, Monthly Notices of the Royal Astronomical Society, 358, 1325–1336, 10.1111/j.1365-2966.2005.08834.x

    2005

  14. [23]

    Carlberg, R. G. 2018, The Astrophysical Journal, 861, 69, 10.3847/1538-4357/aac88a

    2018 doi

  15. [24]

    G., Grillmair, C

    Carlberg, R. G., Grillmair, C. J., & Hetherington, N. 2012, The Astrophysical Journal, 760, 75, 10.1088/0004-637x/760/1/75

    2012 doi

  16. [25]

    P., & Wells, M

    Casella, G., Robert, C. P., & Wells, M. T. 2004, Lecture notes-monograph series, 342

    2004

  17. [26]

    Chen, H., Speagle, J., & Rogers, K. K. 2023, in 37th Conference on Neural Information Processing Systems . 2311.16238

    2023 arXiv

  18. [27]

    Y., & Ash , N

    Chen , Y., Valluri , M., Gnedin , O. Y., & Ash , N. 2024, arXiv e-prints, arXiv:2408.01496, 10.48550/arXiv.2408.01496

    2024 doi arXiv

  19. [28]

    K., Witte, S

    Cole, A., Miller, B. K., Witte, S. J., et al. 2022, Journal of Cosmology and Astroparticle Physics, 2022, 004, 10.1088/1475-7516/2022/09/004

    2022 doi

  20. [29]

    R., Gelman, A., & Rubin, D

    Cook, S. R., Gelman, A., & Rubin, D. B. 2006, Journal of Computational and Graphical Statistics, 15, 675–692, 10.1198/106186006x136976

    2006 doi

  21. [30]

    Cooley, J., et al. 2022. 2209.07426

    2022 arXiv

  22. [31]

    P., Koposov, S

    Cooper, A. P., Koposov, S. E., Allende Prieto, C., et al. 2023, The Astrophysical Journal, 947, 37, 10.3847/1538-4357/acb3c0

    2023 doi

  23. [32]

    2020, Discovering Symbolic Models from Deep Learning with Inductive Biases, arXiv, 10.48550/ARXIV.2006.11287

    Cranmer, M., Sanchez-Gonzalez, A., Battaglia, P., et al. 2020, Discovering Symbolic Models from Deep Learning with Inductive Biases, arXiv, 10.48550/ARXIV.2006.11287

    2020 doi arXiv

  24. [33]

    Croft, R. A. C., Weinberg, D. H., Bolte, M., et al. 2002, The Astrophysical Journal, 581, 20–52, 10.1086/344099

    2002 doi

  25. [34]

    Croft, R. A. C., Weinberg, D. H., Pettini, M., Hernquist, L., & Katz, N. 1999, The Astrophysical Journal, 520, 1–23, 10.1086/307438

    1999 doi

  26. [35]

    J., & Macke, J

    Deistler, M., Goncalves, P. J., & Macke, J. H. 2022, Truncated proposals for scalable and hassle-free simulation-based inference, arXiv, 10.48550/ARXIV.2210.04815

    2022 doi arXiv

  27. [36]

    2007, The Astrophysical Journal, 667, 859–877, 10.1086/520573

    Diemand, J., Kuhlen, M., & Madau, P. 2007, The Astrophysical Journal, 667, 859–877, 10.1086/520573

    2007 doi

  28. [37]

    2008, Nature, 454, 735–738, 10.1038/nature07153

    Diemand, J., Kuhlen, M., Madau, P., et al. 2008, Nature, 454, 735–738, 10.1038/nature07153

    2008 doi

  29. [38]

    J., & Gratton, R

    Diggle, P. J., & Gratton, R. J. 1984, Journal of the Royal Statistical Society Series B: Statistical Methodology, 46, 193–212, 10.1111/j.2517-6161.1984.tb01290.x

    1984

  30. [39]

    2019, arXiv e-prints, arXiv:1902.01055, 10.48550/arXiv.1902.01055

    Drlica-Wagner , A., Mao , Y.-Y., Adhikari , S., et al. 2019, arXiv e-prints, arXiv:1902.01055, 10.48550/arXiv.1902.01055

    2019 doi arXiv

  31. [40]

    O., & Hart, P

    Duda, R. O., & Hart, P. E. 1974, in A Wiley-Interscience publication. https://api.semanticscholar.org/CorpusID:12946615

    1974

  32. [41]

    2021, Monthly Notices of the Royal Astronomical Society, 507, 1746–1761, 10.1093/mnras/stab1762

    Eifler, T., Miyatake, H., Krause, E., et al. 2021, Monthly Notices of the Royal Astronomical Society, 507, 1746–1761, 10.1093/mnras/stab1762

    2021 doi

  33. [42]

    2015, Monthly Notices of the Royal Astronomical Society, 450, 1136–1149, 10.1093/mnras/stv655

    Erkal, D., & Belokurov, V. 2015, Monthly Notices of the Royal Astronomical Society, 450, 1136–1149, 10.1093/mnras/stv655

    2015 doi

  34. [43]

    Erkal, D., Belokurov, V., Bovy, J., & Sanders, J. L. 2016, Monthly Notices of the Royal Astronomical Society, 463, 102–119, 10.1093/mnras/stw1957

    2016 doi

  35. [44]

    E., & Belokurov , V

    Erkal , D., Koposov , S. E., & Belokurov , V. 2017, , 470, 60, 10.1093/mnras/stx1208

    2017 doi

  36. [45]

    Erkal, D., Belokurov, V., Laporte, C. F. P., et al. 2019, Monthly Notices of the Royal Astronomical Society, 487, 2685–2700, 10.1093/mnras/stz1371

    2019 doi

  37. [46]

    A., Huang, S., & Weinberg, M

    Fardal, M. A., Huang, S., & Weinberg, M. D. 2015, Monthly Notices of the Royal Astronomical Society, 452, 301–319, 10.1093/mnras/stv1198

    2015 doi

  38. [47]

    T., Naidu, R

    Gialluca, M. T., Naidu, R. P., & Bonaca, A. 2021, The Astrophysical Journal Letters, 911, L32, 10.3847/2041-8213/abf491

    2021 doi

  39. [48]

    2016, Deep Learning (MIT Press)

    Goodfellow, I., Bengio, Y., & Courville, A. 2016, Deep Learning (MIT Press)

    2016

  40. [49]

    S., Nonnenmacher, M., & Macke, J

    Greenberg, D. S., Nonnenmacher, M., & Macke, J. H. 2019, Automatic Posterior Transformation for Likelihood-Free Inference, arXiv, 10.48550/ARXIV.1905.07488

    2019 doi arXiv

  41. [50]

    J., & Dionatos, O

    Grillmair, C. J., & Dionatos, O. 2006, The Astrophysical Journal, 643, L17–L20, 10.1086/505111

    2006 doi

  42. [51]

    U., Cor, J., & er

    Gutmann, M. U., Cor, J., & er. 2016, Journal of Machine Learning Research, 17, 1. http://jmlr.org/papers/v17/15-017.html

    2016

  43. [52]

    2021, Monthly Notices of the Royal Astronomical Society, 507, 1999–2011, 10.1093/mnras/stab2181

    Hermans, J., Banik, N., Weniger, C., Bertone, G., & Louppe, G. 2021, Monthly Notices of the Royal Astronomical Society, 507, 1999–2011, 10.1093/mnras/stab2181

    2021 doi

  44. [53]

    2019, Likelihood-free MCMC with Amortized Approximate Ratio Estimators, arXiv, 10.48550/ARXIV.1903.04057

    Hermans, J., Begy, V., & Louppe, G. 2019, Likelihood-free MCMC with Amortized Approximate Ratio Estimators, arXiv, 10.48550/ARXIV.1903.04057

    2019 doi arXiv

  45. [54]

    1990, The Astrophysical Journal, 356, 359, 10.1086/168845

    Hernquist, L. 1990, The Astrophysical Journal, 356, 359, 10.1086/168845

    1990 doi

  46. [55]

    E., et al

    Hilmi , T., Erkal , D., Koposov , S. E., et al. 2024, arXiv e-prints, arXiv:2404.02953, 10.48550/arXiv.2404.02953

    2024 doi arXiv

  47. [56]

    Hinton, D. R. G., & Williams, R. 1986, Nature. rumelhart1986learning

    1986

  48. [57]

    F., Wetzel, A., Kereš, D., et al

    Hopkins, P. F., Wetzel, A., Kereš, D., et al. 2018, Monthly Notices of the Royal Astronomical Society, 480, 800–863, 10.1093/mnras/sty1690

    2018 doi

  49. [58]

    1989, Neural networks, 2, 359

    Hornik, K., Stinchcombe, M., & White, H. 1989, Neural networks, 2, 359

    1989

  50. [59]

    2021, in First International Meeting for Applied Geoscience & Energy Expanded Abstracts (Society of Exploration Geophysicists), 10.1190/segam2021-3584127.1

    Huang, X., Alkhalifah, T., & Song, C. 2021, in First International Meeting for Applied Geoscience & Energy Expanded Abstracts (Society of Exploration Geophysicists), 10.1190/segam2021-3584127.1

    2021 doi

  51. [60]

    2021, , 914, 123, 10.3847/1538-4357/abfcc2

    Ibata , R., Malhan , K., Martin , N., et al. 2021, , 914, 123, 10.3847/1538-4357/abfcc2

    2021 doi

  52. [61]

    A., Lewis , G

    Ibata , R. A., Lewis , G. F., Irwin , M. J., & Quinn , T. 2002, , 332, 915, 10.1046/j.1365-8711.2002.05358.x

    2002

  53. [62]

    A., Malhan , K., & Martin , N

    Ibata , R. A., Malhan , K., & Martin , N. F. 2019, , 872, 152, 10.3847/1538-4357/ab0080

    2019 doi

  54. [63]

    2018, Monthly Notices of the Royal Astronomical Society, 480, 5342–5351, 10.1093/mnras/sty2226

    Jethwa, P., Torrealba, G., Navarrete, C., et al. 2018, Monthly Notices of the Royal Astronomical Society, 480, 5342–5351, 10.1093/mnras/sty2226

    2018 doi

  55. [64]

    V., Spergel, D

    Johnston, K. V., Spergel, D. N., & Haydn, C. 2002, The Astrophysical Journal, 570, 656–664, 10.1086/339791

    2002 doi

  56. [65]

    P., & Ba, J

    Kingma, D. P., & Ba, J. 2014, Adam: A Method for Stochastic Optimization, arXiv, 10.48550/ARXIV.1412.6980

    2014 doi arXiv

  57. [66]

    N., & Welling, M

    Kipf, T. N., & Welling, M. 2016, Semi-Supervised Classification with Graph Convolutional Networks, arXiv, 10.48550/ARXIV.1609.02907

    2016 doi arXiv

  58. [67]

    Kolmogorov, A. 1933, G. Ist. Ital. Attuari

    1933

  59. [68]

    E., Irwin , M., Belokurov , V., et al

    Koposov , S. E., Irwin , M., Belokurov , V., et al. 2014, , 442, L85, 10.1093/mnrasl/slu060

    2014 doi

  60. [69]

    E., Rix , H.-W., & Hogg , D

    Koposov , S. E., Rix , H.-W., & Hogg , D. W. 2010, , 712, 260, 10.1088/0004-637X/712/1/260

    2010 doi

  61. [70]

    H., & Helmi, A

    Koppelman, H. H., & Helmi, A. 2021, Astronomy & Astrophysics, 649, A55, 10.1051/0004-6361/202039968

    2021 doi

  62. [72]

    Küpper, A. H. W., Macleod, A., & Heggie, D. C. 2008, Monthly Notices of the Royal Astronomical Society, 387, 1248, 10.1111/j.1365-2966.2008.13323.x

    2008

  63. [73]

    2018, Physical Review D, 98, 10.1103/physrevd.98.063511

    Leclercq, F. 2018, Physical Review D, 98, 10.1103/physrevd.98.063511

    2018 doi

  64. [74]

    2023, Machine Learning: Science and Technology, 4, 01LT01, 10.1088/2632-2153/acbb53

    Lemos, P., Cranmer, M., Abidi, M., et al. 2023, Machine Learning: Science and Technology, 4, 01LT01, 10.1088/2632-2153/acbb53

    2023 doi

  65. [75]

    2024, Physical Review D, 109, 10.1103/physrevd.109.083536

    Lemos, P., Parker, L., Hahn, C., et al. 2024, Physical Review D, 109, 10.1103/physrevd.109.083536

    2024 doi

  66. [76]

    S., Koposov, S

    Li, T. S., Koposov, S. E., Zucker, D. B., et al. 2019, Monthly Notices of the Royal Astronomical Society, 490, 3508–3531, 10.1093/mnras/stz2731

    2019 doi

  67. [77]

    S., Koposov, S

    Li, T. S., Koposov, S. E., Erkal, D., et al. 2021, The Astrophysical Journal, 911, 149, 10.3847/1538-4357/abeb18

    2021 doi

  68. [78]

    2023, Monthly Notices of the Royal Astronomical Society, 524, 6167–6180, 10.1093/mnras/stad2262

    Lin, K., von wietersheim Kramsta, M., Joachimi, B., & Feeney, S. 2023, Monthly Notices of the Royal Astronomical Society, 524, 6167–6180, 10.1093/mnras/stad2262

    2023 doi

  69. [79]

    2024, arXiv preprint arXiv:2402.02054

    Liu, J., Mao, H., Chen, Z., et al. 2024, arXiv preprint arXiv:2402.02054

    2024 arXiv

  70. [80]

    2021, in Proceedings of Machine Learning Research, Vol

    Lueckmann, J.-M., Boelts, J., Greenberg, D., Goncalves, P., & Macke, J. 2021, in Proceedings of Machine Learning Research, Vol. 130, Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, ed. A. Banerjee & K. Fukumizu (PMLR), 343--351. http...

    2021

  71. [81]

    Lynden-Bell, D., & Lynden-Bell, R. M. 1995, Monthly Notices of the Royal Astronomical Society, 275, 429–442, 10.1093/mnras/275.2.429

    1995 doi

  72. [82]

    V., Kang, X., Fontanot, F., et al

    Macciò, A. V., Kang, X., Fontanot, F., et al. 2010, Monthly Notices of the Royal Astronomical Society, 402, 1995–2008, 10.1111/j.1365-2966.2009.16031.x

    2010

  73. [83]

    A., Carlberg, R

    Malhan, K., Ibata, R. A., Carlberg, R. G., Valluri, M., & Freese, K. 2019, The Astrophysical Journal, 881, 106, 10.3847/1538-4357/ab2e07

    2019 doi

  74. [84]

    2024, The Astrophysical Journal, 964, 104, 10.3847/1538-4357/ad1885

    Malhan, K., & Rix, H.-W. 2024, The Astrophysical Journal, 964, 104, 10.3847/1538-4357/ad1885

    2024 doi

  75. [85]

    M., & Brinkmann, J

    Mandelbaum, R., Seljak, U., Kauffmann, G., Hirata, C. M., & Brinkmann, J. 2006, Monthly Notices of the Royal Astronomical Society, 368, 715–731, 10.1111/j.1365-2966.2006.10156.x

    2006

  76. [86]

    2010, Reports on Progress in Physics, 73, 086901, 10.1088/0034-4885/73/8/086901

    Massey, R., Kitching, T., & Richard, J. 2010, Reports on Progress in Physics, 73, 086901, 10.1088/0034-4885/73/8/086901

    2010 doi

  77. [87]

    2000, The Astrophysical Journal, 543, 1–23, 10.1086/317079

    McDonald, P., Miralda‐Escude, J., Rauch, M., et al. 2000, The Astrophysical Journal, 543, 1–23, 10.1086/317079

    2000 doi

  78. [88]

    D., Beckman, R

    McKay, M. D., Beckman, R. J., & Conover, W. J. 1979, Technometrics, 21, 239, 10.2307/1268522

    1979 doi

  79. [89]

    2013, Efficient Estimation of Word Representations in Vector Space, arXiv, 10.48550/ARXIV.1301.3781

    Mikolov, T., Chen, K., Corrado, G., & Dean, J. 2013, Efficient Estimation of Word Representations in Vector Space, arXiv, 10.48550/ARXIV.1301.3781

    2013 doi arXiv

  80. [90]

    K., Weniger, C., & Forré, P

    Miller, B. K., Weniger, C., & Forré, P. 2022, Contrastive Neural Ratio Estimation for Simulation-based Inference, arXiv, 10.48550/ARXIV.2210.06170

    2022 doi arXiv

  81. [92]

    2022, Machine Learning: Science and Technology, 3, 01LT03, 10.1088/2632-2153/ac494a

    Mishra-Sharma, S. 2022, Machine Learning: Science and Technology, 3, 01LT03, 10.1088/2632-2153/ac494a

    2022 doi

  82. [93]

    Nagai, R

    Miyamoto, M. ; Nagai, R. 1975, Astronomical Society of Japan, 27, 533

    1975

  83. [94]

    2021, Physical Review Letters, 126, 10.1103/physrevlett.126.091101

    Nadler, E., Drlica-Wagner, A., Bechtol, K., et al. 2021, Physical Review Letters, 126, 10.1103/physrevlett.126.091101

    2021 doi

  84. [95]

    O., Birrer , S., Gilman , D., et al

    Nadler , E. O., Birrer , S., Gilman , D., et al. 2021, , 917, 7, 10.3847/1538-4357/abf9a3

    2021 doi

  85. [96]

    F., Frenk, C

    Navarro, J. F., Frenk, C. S., & White, S. D. M. 1996, The Astrophysical Journal, 462, 563, 10.1086/177173

    1996 doi

  86. [97]

    2024, Astronomy & Astrophysics, 689, A153, 10.1051/0004-6361/202348485

    Nayak, P., Walther, M., Gruen, D., & Adiraju, S. 2024, Astronomy & Astrophysics, 689, A153, 10.1051/0004-6361/202348485

    2024 doi

  87. [98]

    J., & Carlin, J

    Newberg, H. J., & Carlin, J. L. 2016, Tidal Streams in the Local Group and Beyond: Observations and Implications (Springer International Publishing), 10.1007/978-3-319-19336-6

    2016 doi

  88. [99]

    G., et al

    Ngan, W., Bozek, B., Carlberg, R. G., et al. 2015, The Astrophysical Journal, 803, 75, 10.1088/0004-637x/803/2/75

    2015 doi

  89. [100]

    2024, How much information can be extracted from galaxy clustering at the field level?, arXiv, 10.48550/ARXIV.2403.03220

    Nguyen, N.-M., Schmidt, F., Tucci, B., Reinecke, M., & Kostić, A. 2024, How much information can be extracted from galaxy clustering at the field level?, arXiv, 10.48550/ARXIV.2403.03220

    2024 doi arXiv

  90. [101]

    2023, Physical Review D, 107, 10.1103/physrevd.107.043015

    Nguyen, T., Mishra-Sharma, S., Williams, R., & Necib, L. 2023, Physical Review D, 107, 10.1103/physrevd.107.043015

    2023 doi

  91. [102]

    2022, The Astrophysical Journal, 940, 22, 10.3847/1538-4357/ac93ee

    Nibauer, J., Belokurov, V., Cranmer, M., Goodman, J., & Ho, S. 2022, The Astrophysical Journal, 940, 22, 10.3847/1538-4357/ac93ee

    2022 doi

  92. [103]

    K., Rockosi , C

    Odenkirchen , M., Grebel , E. K., Rockosi , C. M., et al. 2001, , 548, L165, 10.1086/319095

    2001 doi

  93. [104]

    2016, in Advances in Neural Information Processing Systems, ed

    Papamakarios, G., & Murray, I. 2016, in Advances in Neural Information Processing Systems, ed. D. Lee, M. Sugiyama, U. Luxburg, I. Guyon, & R. Garnett, Vol. 29 (Curran Associates, Inc.). https://proceedings.neurips.cc/paper_files/paper/2016/file/6aca97005c68f1206823815f6610286...

    2016

  94. [105]

    J., Mohamed, S., & Lakshminarayanan, B

    Papamakarios, G., Nalisnick, E., Rezende, D. J., Mohamed, S., & Lakshminarayanan, B. 2019, 10.48550/ARXIV.1912.02762

    2019 doi arXiv

  95. [106]

    2017, Masked Autoregressive Flow for Density Estimation, arXiv, 10.48550/ARXIV.1705.07057

    Papamakarios, G., Pavlakou, T., & Murray, I. 2017, Masked Autoregressive Flow for Density Estimation, arXiv, 10.48550/ARXIV.1705.07057

    2017 doi arXiv

  96. [107]

    C., & Murray, I

    Papamakarios, G., Sterratt, D. C., & Murray, I. 2018, Sequential Neural Likelihood: Fast Likelihood-free Inference with Autoregressive Flows, arXiv, 10.48550/ARXIV.1805.07226

    2018 doi arXiv

  97. [108]

    M., Koposov , S

    Patrick , J. M., Koposov , S. E., & Walker , M. G. 2022, , 514, 1757, 10.1093/mnras/stac1478

    2022 doi

  98. [109]

    M., & Bonaca, A

    Price-Whelan, A. M., & Bonaca, A. 2018, The Astrophysical Journal Letters, 863, L20, 10.3847/2041-8213/aad7b5

    2018 doi

  99. [110]

    2022, Monthly Notices of the Royal Astronomical Society, 511, 2339–2348, 10.1093/mnras/stac238

    Qian, Y., Arshad, Y., & Bovy, J. 2022, Monthly Notices of the Royal Astronomical Society, 511, 2339–2348, 10.1093/mnras/stac238

    2022 doi

  100. [111]

    K., Dvorkin, C., & Peiris, H

    Rogers, K. K., Dvorkin, C., & Peiris, H. V. 2022, Phys. Rev. Lett., 128, 171301, 10.1103/PhysRevLett.128.171301

    2022 doi

  101. [112]

    K., & Peiris, H

    Rogers, K. K., & Peiris, H. V. 2021 a , Physical Review Letters, 126, 10.1103/physrevlett.126.071302

    2021 doi

  102. [113]

    2021 b , Phys

    ---. 2021 b , Phys. Rev. D, 103, 043526, 10.1103/PhysRevD.103.043526

    2021 doi

  103. [114]

    2018, , 862, 114, 10.3847/1538-4357/aacdab

    Shipp , N., Drlica-Wagner , A., Balbinot , E., et al. 2018, , 862, 114, 10.3847/1538-4357/aacdab

    2018 doi

  104. [115]

    2023, The Astrophysical Journal, 949, 44, 10.3847/1538-4357/acc582

    Shipp, N., Panithanpaisal, N., Necib, L., et al. 2023, The Astrophysical Journal, 949, 44, 10.3847/1538-4357/acc582

    2023 doi

  105. [116]

    1948, Annals of Mathematical Statistics

    Smirnov, N. 1948, Annals of Mathematical Statistics

    1948

  106. [117]

    Snoek, J., Larochelle, H., & Adams, R. P. 2012, Practical Bayesian Optimization of Machine Learning Algorithms, arXiv, 10.48550/ARXIV.1206.2944

    2012 doi arXiv

  107. [118]

    G., & Turner, C

    Tabak, E. G., & Turner, C. V. 2012, Communications on Pure and Applied Mathematics, 66, 145–164, 10.1002/cpa.21423

    2012 doi

  108. [119]

    G., & Vanden-Eijnden, E

    Tabak, E. G., & Vanden-Eijnden, E. 2010, Communications in Mathematical Sciences, 8, 217–233, 10.4310/cms.2010.v8.n1.a11

    2010 doi

  109. [120]

    2018, Validating Bayesian Inference Algorithms with Simulation-Based Calibration, arXiv, 10.48550/ARXIV.1804.06788

    Talts, S., Betancourt, M., Simpson, D., Vehtari, A., & Gelman, A. 2018, Validating Bayesian Inference Algorithms with Simulation-Based Calibration, arXiv, 10.48550/ARXIV.1804.06788

    2018 doi arXiv

  110. [121]

    2022, , 925, 118, 10.3847/1538-4357/ac399b

    Tavangar , K., Ferguson , P., Shipp , N., et al. 2022, , 925, 118, 10.3847/1538-4357/ac399b

    2022 doi

  111. [122]

    2020, Journal of Open Source Software, 5, 2505, 10.21105/joss.02505

    Tejero-Cantero, A., Boelts, J., Deistler, M., et al. 2020, Journal of Open Source Software, 5, 2505, 10.21105/joss.02505

    2020 doi

  112. [123]

    Vale, A., & Ostriker, J. P. 2004, Monthly Notices of the Royal Astronomical Society, 353, 189–200, 10.1111/j.1365-2966.2004.08059.x

    2004

  113. [124]

    2022, Snowmass2021 Cosmic Frontier White Paper: Prospects for obtaining Dark Matter Constraints with DESI, arXiv, 10.48550/ARXIV.2203.07491

    Valluri, M., Chabanier, S., Irsic, V., et al. 2022, Snowmass2021 Cosmic Frontier White Paper: Prospects for obtaining Dark Matter Constraints with DESI, arXiv, 10.48550/ARXIV.2203.07491

    2022 doi arXiv

  114. [125]

    2017, Attention Is All You Need, arXiv, 10.48550/ARXIV.1706.03762

    Vaswani, A., Shazeer, N., Parmar, N., et al. 2017, Attention Is All You Need, arXiv, 10.48550/ARXIV.1706.03762

    2017 doi arXiv

  115. [126]

    Vegetti, S., Koopmans, L. V. E., Bolton, A., Treu, T., & Gavazzi, R. 2010, Monthly Notices of the Royal Astronomical Society, 408, 1969–1981, 10.1111/j.1365-2966.2010.16865.x

    2010

  116. [127]

    2023, Strong gravitational lensing as a probe of dark matter, arXiv, 10.48550/ARXIV.2306.11781

    Vegetti, S., Birrer, S., Despali, G., et al. 2023, Strong gravitational lensing as a probe of dark matter, arXiv, 10.48550/ARXIV.2306.11781

    2023 doi arXiv

  117. [128]

    2023, Physical Review D, 108, 10.1103/physrevd.108.023502

    Villasenor, B., Robertson, B., Madau, P., & Schneider, E. 2023, Physical Review D, 108, 10.1103/physrevd.108.023502

    2023 doi

  118. [129]

    A., Stinson, G

    Wang, L., Dutton, A. A., Stinson, G. S., et al. 2015, Monthly Notices of the Royal Astronomical Society, 454, 83–94, 10.1093/mnras/stv1937

    2015 doi

  119. [130]

    J., & Bovy, J

    Webb, J. J., & Bovy, J. 2019, Monthly Notices of the Royal Astronomical Society, 485, 5929–5938, 10.1093/mnras/stz867

    2019 doi

  120. [131]

    H., & Tinker, J

    Wechsler, R. H., & Tinker, J. L. 2018, Annual Review of Astronomy and Astrophysics, 56, 435–487, 10.1146/annurev-astro-081817-051756

    2018 doi

  121. [132]

    Weinberg, D. H. 2003, in AIP Conference Proceedings (AIP), 10.1063/1.1581786

    2003 doi

  122. [133]

    F., & Jespersen, C

    Wu, J. F., & Jespersen, C. K. 2023, Learning the galaxy-environment connection with graph neural networks, arXiv, 10.48550/ARXIV.2306.12327

    2023 doi arXiv

  123. [134]

    2024 a , A Comprehensive Review of the Oversmoothing in Graph Neural Networks (Springer Nature Singapore), 451–465, 10.1007/978-981-99-9637-7_33

    Zhang, X., Xu, Y., He, W., Guo, W., & Cui, L. 2024 a , A Comprehensive Review of the Oversmoothing in Graph Neural Networks (Springer Nature Singapore), 451–465, 10.1007/978-981-99-9637-7_33

    2024 doi

  124. [135]

    2024 b , Scientific Reports, 14, 10.1038/s41598-024-57137-4

    Zhang, Z., Lin, C., & Wang, B. 2024 b , Scientific Reports, 14, 10.1038/s41598-024-57137-4

    2024 doi

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.