Pith. sign in

REVIEW 2 major objections 5 minor 1 cited by

Denoising Milky Way stellar survey data with normalizing flow models

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read Normalizing flows trained with an importance-sampling loss recover the noise-free six-dimensional phase-space distribution of Gaia-like stars, preserving fine substructures such as the multiple branches of the Hercules stream and the…

desk verdict Solid proof-of-principle for flow-based deconvolution of Gaia-like noise; the Jacobian concern is a misreading, but the validation is in-sample and the method is not yet a general tool. read the letter →

arxiv 2505.16553 v1 pith:5MGNYO7L submitted 2025-05-22 astro-ph.GA astro-ph.IM

Ziyang Yan , Jason L. Sanders This is my paper · ORCID
classification astro-ph.GAastro-ph.IM
keywords normalizingflowsdensityestimationdeconvolutionGaiaDR3heteroscedasticnoisephasespiralHerculesstreamimportancesampling
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 proposes a machine-learning method for estimating the noise-free six-dimensional phase-space density of Milky Way stars from survey data with per-star, heteroscedastic Gaussian uncertainties. The method trains a normalizing flow in two stages: a fast pre-train that ignores noise, then a denoising stage whose loss is the average flow density over importance samples drawn from each star's measurement uncertainty. The authors corrupt a local Gaia DR3 sample with 5 and 10 times the reported uncertainties and show the flow recovers the original distribution rather than the blurred one, including the multiple branches of the Hercules stream and the Gaia phase-space spiral. The comparison with extreme deconvolution indicates the flow retains fine substructure that a Gaussian mixture model smears out. If the approach extends to genuinely noisy samples beyond the solar neighbourhood, it offers a practical tool for studying the Milky Way where Gaia noise is significant.

What carries the argument

The carrying object is the normalizing flow density $P_{\mathrm{NF}}(v)$ built from rational-quadratic-spline layers in an autoregressive architecture. Training proceeds in two stages: a pre-train loop that maximizes the log-density on the observed points alone (Eq. 5), followed by a denoising loop whose loss (Eq. 4) averages the flow density over $K$ Monte-Carlo samples drawn from each star's Gaussian uncertainty in observable space and transformed to Galactocentric coordinates. This importance-sampling average is a Monte-Carlo estimate of the convolution (Eq. 2), so the flow is pulled toward the noise-free density rather than the blurred observed one, with $K$ and the spline bin count $B$ controlling the fidelity of recovery.

What would settle it

Re-run the same training while including the Jacobian determinant of the observable-to-Galactocentric transformation in each importance weight and compare the recovered $v_\phi$ distributions and phase-spiral maps; an observable difference would show the omitted term is not benign. Alternatively, apply the method to a synthetic galaxy with a known distribution function and strong position-dependent Jacobian (distant stars or high transverse velocities) and test the recovered density against the truth.

Watch

Extended reading notes

Core claim

The central claim is that a rational-quadratic-spline autoregressive normalizing flow, trained to maximize the mean of the flow density at importance-sampled points from each star's known uncertainty, approximates the deconvolved 6D phase-space distribution of Gaia-like data. The fitted model, not the corrupted mock sample, reproduces the original solar-neighbourhood Gaia distribution in 1D and 6D projections, sharpening the boundaries of the Hercules stream arches at $v_\phi \approx 200$ km/s and enhancing the phase spiral in $(z, v_z)$. The fidelity degrades gracefully with added noise: at $q=10$ the model still removes noise in the Galactic plane and recovers the strongest kinematic boundaries, although it no longer resolves the substructure of the Hercules stream. The paper also shows that excess spline flexibility with small training sets produces gridded artefacts, and that the method outperforms XDGMM on test loss and visual feature recovery.

Load-bearing premise

The load-bearing premise is that evaluating the flow density at noise-sample points carried into the fitting coordinates, without including the Jacobian determinant of that change of variables, still yields a training objective whose optimum matches the true noise-free density; the paper does not quantify how large this bias is.

Editorial extensions

If this is right

  • The method can be applied to Gaia samples beyond the solar neighbourhood, where reported uncertainties are several times larger, to search for moving groups and phase-spiral structure in the outer disc and halo.
  • Denoised flow densities provide a noise-free surrogate for direct comparison with dynamical models of bar, spiral, and vertical-resonance features.
  • Projecting the denoised flow into action–angle space avoids the artificial features that arise when applying non-linear coordinate transforms directly to noisy data.
  • The two-stage importance-sampling training protocol can be carried over to other stellar surveys that supply per-star Gaussian uncertainties, without changing the core algorithm.
  • A trained flow acts as a compact, resamplable summary of a survey, enabling lightweight sharing and comparison of Milky Way data products.

Reading between the lines

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

  • Since the loss in Eq. (4) omits the Jacobian of the observable-to-Galactocentric transformation, its bias should grow with distance or with stars on fast orbits; multiplying each importance sample by the local Jacobian would make the loss an unbiased estimator of the marginal likelihood and is the most direct correction to test.
  • The paper assumes the reported Gaia uncertainties are exact; injecting deliberately mismatched noise (for example, 20 percent over- or under-estimated errors) and measuring the contrast of the recovered phase spiral would quantify how sensitive the method is to noise-model misspecification.
  • A natural extension is to let the number of importance samples $K$ scale with per-star uncertainty, so tiny-error stars contribute few samples and high-error stars contribute many; the paper notes this is possible but does not test it.
  • For halo applications the selection function cannot be separated from the noise model; coupling the denoising flow with a selection-function model would be needed before the method is used far from the solar neighbourhood.
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. The paper proposes a normalizing-flow method for denoising density estimation of stellar survey data with heteroscedastic uncertainties, and demonstrates it on Gaia DR3 RVS data in a local solar-neighborhood volume. The method trains an autoregressive rational-quadratic-spline flow in Galactocentric coordinates using a Monte Carlo importance-sampling loss: for each star, noise samples are drawn in observable space, transformed to Galactocentric coordinates, and the flow density is averaged over them. The authors validate on mock datasets created by inflating the reported Gaia uncertainties by factors q=5 and q=10, comparing against a no-deconvolution baseline, a ground-truth flow fitted to the original Gaia sample, and extreme deconvolution (XDGMM). They report qualitative recovery of the Hercules stream and the phase spiral, plus quantitative test-loss comparisons and a study of hyperparameters K, B, and training-set size.

Significance. If the training objective is correct, the method is a useful advance: it offers a flexible, non-Gaussian alternative to XDGMM for heteroscedastic noise, and the demonstration on real Gaia data with publicly available code is a valuable proof of concept. The paper is appropriately framed as a proof of principle, includes a detailed hyperparameter analysis, and credits the limitations of treating the already-noisy Gaia sample as ground truth. However, the central training objective as implemented is not the convolution likelihood, because the Jacobian of the observable-to-Galactocentric transformation is omitted from the importance-sampling estimator. Until that is corrected and the main figures are reproduced with the corrected loss, the quantitative and qualitative claims must be treated with caution.

major comments (2)
  1. [Section 3.1 and Algorithm 1; Eq. (4)] The Monte Carlo estimator in Eq. (4) does not compute the convolution likelihood in Galactocentric coordinates. For a flow density P_NF(v_gc), the likelihood of an observation w_i in observable space is p(w_i) = ∫ P_NF(v_gc) p_n(w_i - T^{-1}(v_gc)) dv_gc, where T is the observable-to-Galactocentric map. Sampling v_obs ~ N(w_i, Σ_i) and averaging P_NF(T(v_obs)) gives an estimator of ∫ P_NF(v_gc) p_n(w_i - T^{-1}(v_gc)) |det J_{T^{-1}}(v_gc)| dv_gc, which contains an extra Jacobian factor. The unbiased estimator instead requires weighting each sample by |det J_T(v_obs)| (equivalently dividing by |det J_{T^{-1}}(v_gc)|). Because T includes 1/parallax and proper-motion distance scaling, the Jacobian varies across the sample and across noise realizations for a single star, so the fitted flow is not the true GC-space density. The reported recovery of the Hercules stream (Fig. 2) and phase spiral (Fig. 3) could therefore be shaped by this misspecified objective rather than by genuine deconvolution. I request that the authors correct the loss by including the Jacobian determinant, retrain the models, and verify that the qualitative conclusions persist.
  2. [Section 3.4; Tables 1 and 2; Fig. 8] The quantitative evaluation selects hyperparameters on the same test set used to report final performance. The number of denoising samples K (Table 1) and the spline bin number B (Fig. 8) are chosen by comparing test losses on the held-out 30% of the Gaia data, and the same test set is then used for the benchmarks and the XDGMM comparison. This makes the reported test losses optimistic and means that the comparisons between configurations are not independent. I recommend introducing a separate validation split for hyperparameter selection and reporting test losses on a final untouched split, or explicitly stating that the quoted values are validation-selected and should be interpreted as upper bounds on performance.
minor comments (5)
  1. [Section 2.1 and Algorithm 1] The notation in Eq. (3) is ambiguous: it writes the expectation as E_{v∼p_n_i(w_i−v)}[P_NF(v)], but the samples are drawn in observable space and then transformed to Galactocentric coordinates before evaluating P_NF. Please make this explicit in the equations, since the correct treatment of the coordinate change is essential to the method.
  2. [Section 3.1, Eq. (6)] The quality cut is written as error_over_parallax > 0.5, which selects stars with relative parallax errors greater than 50% and contradicts the description of a high-quality local sample. This is almost certainly a typo for '< 0.5'. Please correct it, as the cut defines the dataset used in all experiments.
  3. [Table 1] Some entries lack standard deviations: the q=10 column is reported without error bars, and the Ground truth row has no uncertainty. Since these values are used to support the denoising claims, please report run-to-run uncertainties for all configurations.
  4. [Section 3.4.3] The sentence 'These artefacts are irrelative to the sample number K' contains a typo ('irrelative' should be 'irrelevant'), and the figure captions in Figs. 6 and 7 should be reworded for clarity.
  5. [Section 2.1] The statement that Gaussian deconvolution is 'well-posed' conflicts with the immediately following discussion of Carroll & Hall (1988), which describes logarithmic convergence rates and sensitivity to sharp features. Please reword to avoid the contradiction, e.g., by saying the problem is well-posed in the strict sense of unique identifiability but severely ill-conditioned in practice.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the denoising loss is a Monte Carlo convolution likelihood and the validation uses held-out data with external baselines.

full rationale

The paper's derivation chain is self-contained. Eq. (4) is a Monte Carlo estimate of the convolution in Eq. (2): sampling v from the noise kernel p_ni(w_i - v) and averaging P_NF(v) estimates (P_NF * p_ni)(w_i), so maximizing this loss is a legitimate deconvolution objective rather than a quantity defined in terms of the claimed noise-free density. The claimed denoising results are then tested on a held-out 30% of the original Gaia sample, with the 'ground truth' flow fitted only on the training split, and are compared against an external XDGMM baseline. Hyperparameters (K, B) are chosen via cross-validated test loss, not by fitting the reported substructure recovery. There are no load-bearing self-citations and no known result is merely renamed. The omission of a Jacobian determinant when moving from observable to Galactocentric coordinates in Algorithm 1/Eq. (4) is a potential bias in what density the flow actually fits, but it is a correctness/approximation concern, not circularity: the optimized density is not equivalent to its input by construction. The Appendix A2 caveat that multiple deconvolved distributions are consistent with finite noisy data is an identifiability/uncertainty statement, not circular reasoning.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The method relies on known noise models, an expressive flow architecture, and an unstated Jacobian approximation. Hyperparameters K, B, and q are chosen by hand and tuned on the test set. No new physical entities are introduced.

free parameters (3)
  • denoising samples per star K = 128
    Chosen from test-loss optimization in Table 1/Fig. 4; the central qualitative results use K=128.
  • spline bin number B = 32
    Chosen from the pre-train test-loss curve in Fig. 8; higher B gives finer resolution but causes artifacts on small samples.
  • noise amplification factor q = 5 and 10
    Chosen to mimic noise levels outside the solar neighbourhood; q=5 recovers features, q=10 does not.
assumptions (4)
  • domain assumption The noise distributions p_{n_i} are known exactly for every star.
    Section 2.1 states the problem is well-posed only if the noise distributions are known perfectly; the paper uses reported Gaia uncertainties without validation.
  • ad hoc to paper The observable-to-Galactocentric transformation can be applied to noise samples without a Jacobian correction in the importance-sampling loss.
    Algorithm 1 and Eq. (4) evaluate the flow density at transformed noise samples; no Jacobian term appears.
  • domain assumption The normalizing flow is sufficiently expressive to represent the true underlying distribution.
    Section 2.2 selects an autoregressive rational-quadratic-spline flow; expressivity is assumed, with bin number as the main lever.
  • domain assumption Selection effects and completeness can be ignored for the mock-data test.
    Section 3.1 acknowledges quality cuts introduce biases but treats the selected Gaia sample as an example ground truth; real applications would require modeling the selection function.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Denoising Milky Way stellar survey data with normalizing flow models." pith.science (2026). https://pith.science/paper/5MGNYO7L

@misc{pith2026250516553,
  author       = {Pith},
  title        = {Pith review of: Denoising Milky Way stellar survey data with normalizing flow models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5MGNYO7L}},
  note         = {Machine review of arXiv:2505.16553}
}
read the original abstract

The Gaia dataset has revealed many intricate Milky Way substructures in exquisite detail, including moving groups and the phase spiral. Precise characterisation of these features and detailed comparisons to theoretical models require engaging with Gaia's heteroscedastic noise model, particularly in more distant parts of the Galactic disc and halo. We propose a general, novel machine-learning approach using normalizing flows for denoising density estimation, with particular focus on density estimation from stellar survey data such as that from Gaia. Normalizing flows transform a simple base distribution into a complex target distribution through bijective transformations resulting in a highly expressive and flexible model. The denoising is performed using importance sampling. We demonstrate that this general procedure works excellently on Gaia data by reconstructing detailed local velocity distributions artificially corrupted with noise. For example, we show the multiple branches of the Hercules stream and the phase-space spiral can both be well captured by our model. We discuss hyperparameter choice to optimally recover substructure and compare our approach to extreme deconvolution. The model therefore promises to be a robust tool for studying the Milky Way's kinematics in Galactic locations where the noise from Gaia is significant.

Figures

Figures reproduced from arXiv: 2505.16553 by the authors.

Figure 1
Figure 1. 1d 𝑣𝜙 distribution for models trained using the 𝑞 = 5 (left) and 𝑞 = 10 (right) training sets. For the 𝑞 = 5 setup, the flow model successfully recovers the Hercules Stream feature at 200 km/s, whilst for 𝑞 = 10 the Hercules feature is not as cleanly resolved. data successfully recovers the 1d 𝑣𝜙 distribution from the mock data distribution. There is minor overfitting at the peak, but the model also recovers the Her… view at source ↗
Figure 2
Figure 2. The velocity distribution within 𝑧 = ±0.05 kpc for the considered ‘ground-truth’ Gaia training dataset of 536109 stars (𝜂 = 0.7, top row), a mock training dataset generated by scattering the training dataset by 𝑞 = 5 times their uncertainties (middle row) and samples from the fitted normalizing flow model using 𝐾 = 128 samples per star from the uncertainty distributions. Note the similarity between the top and botto… view at source ↗
Figure 3
Figure 3. Phase spiral recovered from the 𝑞 = 5 mock dataset of 536109 stars (𝜂 = 0.7). We show two projections: (𝑧, 𝑣𝑧 , 𝑣𝑟 ) top row and (𝑧, 𝑣𝑧 , 𝑣𝜙 ) bottom row. The left column shows the ‘ground-truth’ Gaia dataset, the middle column the mock dataset and the right column samples from the normalizing flow model of the denoised distribution. The normalizing flow model enhances the spiral from the mock data and corrects the … view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Test loss per star for the models trained on the 𝑞 = 5 (red dots) and 𝑞 = 10 (green crosses) datasets of 536109 stars (𝜂 = 0.7). The test loss is computed on the held-out test set and is reported relative to the best-case scenario loss computed using the ‘ground truth’…
Figure 6
Figure 6. Figure 6: Phase-space spiral plot when using only 1% of the training data (7659 stars). For better comparison between 3, the first and second column shows the spiral plot Gaia data and the mock data of 𝜂 = 0.7, 536109 stars in total. And the third column shows the 536109 samples…
Figure 7
Figure 7. Figure 7: Velocity distribution of the flow model using only 1% of the training data (7659 stars). Similarly, the first and second column shows the spiral plot Gaia data and the mock data of 𝜂 = 0.7, 536109 stars in total. And the third column shows the 536109 samples that gener…
Figure 9
Figure 9. Figure 9: Comparison of the test loss per star for the (XD)GMM models (left, plotted against number of Gaussian components) and normalizing flow models (right, plotted against number of denoising samples, 𝐾). In the left plot, the yellow triangles show results for a GMM fit to t…
Figure 8
Figure 8. Figure 8: Test loss against model bin number 𝐵 after 100 epochs of the pre-train stage. 𝜂 is the relative train data size. The test loss of models with 𝜂 ≲ 0.1 show a clear ‘V’ shape that indicative of over-smoothing at small 𝐵 and an overly flexible model at large 𝐵. For 𝜂 ≳ 0.…
Figure 10
Figure 10. Figure 10: Comparison of the model XDGMM and normalizing flow velocity distributions within the Galactic height range |𝑧 | < 2kpc. The top row is the ‘ground-truth’ Gaia test data set, the second row is the samples from the flow model using 𝑞 = 5 and 𝜂 = 0.7 and the last row is …

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. Deep Potential: Recovering the gravitational potential and local pattern speed in the solar neighborhood with GDR3 using normalizing flows

    astro-ph.GA 2025-07 conditional novelty 6.0 of 10

    Using normalizing flows and a neural network on Gaia DR3 data, the authors recover a local pattern speed of 28.2 km/s/kpc and a total matter density of 0.086 solar masses per cubic parsec within 1 kpc of the Sun.

Reference graph

Works this paper leans on

49 extracted references · 9 canonical work pages · cited by 1 Pith paper

  1. [1]

    W., Leistedt B., Price-Whelan A

    Anderson L., Hogg D. W., Leistedt B., Price-Whelan A. M., Bovy J., 2018, @doi [ ] 10.3847/1538-3881/aad7bf , https://ui.adsabs.harvard.edu/abs/2018AJ....156..145A 156, 145

    work page doi:10.3847/1538-3881/aad7bf 2018

  2. [2]

    Antoja T., et al., 2018, @doi [ ] 10.1038/s41586-018-0510-7 , https://ui.adsabs.harvard.edu/abs/2018Natur.561..360A 561, 360

    doi:10.1038/s41586-018-0510-7 2018

  3. [3]

    Antoja T., Ramos P., Garc \' a-Conde B., Bernet M., Laporte C. F. P., Katz D., 2023, @doi [ ] 10.1051/0004-6361/202245518 , https://ui.adsabs.harvard.edu/abs/2023A&A...673A.115A 673, A115

    doi:10.1051/0004-6361/202245518 2023

  4. [4]

    Ascasibar Y., Binney J., 2005, @doi [ ] 10.1111/j.1365-2966.2004.08480.x , https://ui.adsabs.harvard.edu/abs/2005MNRAS.356..872A 356, 872

    arXiv 2005

  5. [5]

    Astropy Collaboration et al., 2022, @doi [ ] 10.3847/1538-4357/ac7c74 , https://ui.adsabs.harvard.edu/abs/2022ApJ...935..167A 935, 167

    doi:10.3847/1538-4357/ac7c74 2022

  6. [6]

    W., Roweis S

    Bovy J., Hogg D. W., Roweis S. T., 2011, @doi [Annals of Applied Statistics] 10.1214/10-AOAS439 , https://ui.adsabs.harvard.edu/abs/2011AnApS...5.1657B 5, 1657

    doi:10.1214/10-aoas439 2011

  7. [7]

    J., Hall P., 1988, Journal of the American Statistical Association, 83, 1184

    Carroll R. J., Hall P., 1988, Journal of the American Statistical Association, 83, 1184

    work page 1988

  8. [8]

    Ciuca I., Ting Y.-S., 2022, in Machine Learning for Astrophysics. p. 17 ( @eprint arXiv 2207.02785 ), @doi 10.48550/arXiv.2207.02785

    work page Pith review arXiv doi:10.48550/arxiv.2207.02785 2022

Show all 49 references
  1. [9]

    Cropper M., et al., 2018, @doi [ ] 10.1051/0004-6361/201832763 , https://ui.adsabs.harvard.edu/abs/2018A&A...616A...5C 616, A5

    2018 doi

  2. [10]

    Darvish B., Mobasher B., Sobral D., Scoville N., Aragon-Calvo M., 2015, @doi [ ] 10.1088/0004-637X/805/2/121 , https://ui.adsabs.harvard.edu/abs/2015ApJ...805..121D 805, 121

    2015 doi

  3. [11]

    arXiv:0805.2216

    Delaigle A., Meister A., 2008, @doi [arXiv e-prints] 10.48550/arXiv.0805.2216 , https://ui.adsabs.harvard.edu/abs/2008arXiv0805.2216D p. arXiv:0805.2216

    2008 doi arXiv

  4. [12]

    A., Bovy J., Myers A

    DiPompeo M. A., Bovy J., Myers A. D., Lang D., 2015, @doi [ ] 10.1093/mnras/stv1562 , https://ui.adsabs.harvard.edu/abs/2015MNRAS.452.3124D 452, 3124

    2015 doi

  5. [13]

    arXiv:1410.8516

    Dinh L., Krueger D., Bengio Y., 2014, @doi [arXiv e-prints] 10.48550/arXiv.1410.8516 , https://ui.adsabs.harvard.edu/abs/2014arXiv1410.8516D p. arXiv:1410.8516

    2014 doi arXiv

  6. [14]

    A., Yu Y., Murray I., 2020, @doi [arXiv e-prints] 10.48550/arXiv.2006.09396 , https://ui.adsabs.harvard.edu/abs/2020arXiv200609396D p

    Dockhorn T., Ritchie J. A., Yu Y., Murray I., 2020, @doi [arXiv e-prints] 10.48550/arXiv.2006.09396 , https://ui.adsabs.harvard.edu/abs/2020arXiv200609396D p. arXiv:2006.09396

    2020 doi arXiv

  7. [15]

    arXiv:1906.04032

    Durkan C., Bekasov A., Murray I., Papamakarios G., 2019, @doi [arXiv e-prints] 10.48550/arXiv.1906.04032 , https://ui.adsabs.harvard.edu/abs/2019arXiv190604032D p. arXiv:1906.04032

    2019 doi arXiv

  8. [16]

    Fan J., 1991, @doi [The Annals of Statistics] 10.1214/aos/1176348248 , 19, 1257

    1991

  9. [17]

    J., Buddelmeijer H., Trager S

    Ferdosi B. J., Buddelmeijer H., Trager S. C., Wilkinson M. H. F., Roerdink J. B. T. M., 2011, @doi [ ] 10.1051/0004-6361/201116878 , https://ui.adsabs.harvard.edu/abs/2011A&A...531A.114F 531, A114

    2011 doi

  10. [18]

    Gaia Collaboration et al., 2016, @doi [ ] 10.1051/0004-6361/201629272 , https://ui.adsabs.harvard.edu/abs/2016A&A...595A...1G 595, A1

    2016 doi

  11. [19]

    Gaia Collaboration et al., 2018, @doi [ ] 10.1051/0004-6361/201832865 , https://ui.adsabs.harvard.edu/abs/2018A&A...616A..11G 616, A11

    2018 doi

  12. [20]

    Gaia Collaboration et al., 2023a, @doi [ ] 10.1051/0004-6361/202243940 , https://ui.adsabs.harvard.edu/abs/2023A&A...674A...1G 674, A1

    doi

  13. [21]

    Gaia Collaboration et al., 2023b, @doi [ ] 10.1051/0004-6361/202243797 , https://ui.adsabs.harvard.edu/abs/2023A&A...674A..37G 674, A37

    doi

  14. [22]

    Grathwohl W., Chen R. T. Q., Bettencourt J., Sutskever I., Duvenaud D., 2018, @doi [arXiv e-prints] 10.48550/arXiv.1810.01367 , https://ui.adsabs.harvard.edu/abs/2018arXiv181001367G p. arXiv:1810.01367

    2018 doi arXiv

  15. [23]

    M., Ting Y.-S., 2020, @doi [arXiv e-prints] 10.48550/arXiv.2011.04673 , https://ui.adsabs.harvard.edu/abs/2020arXiv201104673G p

    Green G. M., Ting Y.-S., 2020, @doi [arXiv e-prints] 10.48550/arXiv.2011.04673 , https://ui.adsabs.harvard.edu/abs/2020arXiv201104673G p. arXiv:2011.04673

    2020 doi arXiv

  16. [24]

    M., Ting Y.-S., Kamdar H., 2023, @doi [ ] 10.3847/1538-4357/aca3a7 , https://ui.adsabs.harvard.edu/abs/2023ApJ...942...26G 942, 26

    Green G. M., Ting Y.-S., Kamdar H., 2023, @doi [ ] 10.3847/1538-4357/aca3a7 , https://ui.adsabs.harvard.edu/abs/2023ApJ...942...26G 942, 26

    2023 doi

  17. [25]

    R., et al., 2020, @doi [ ] 10.1038/s41586-020-2649-2 , https://ui.adsabs.harvard.edu/abs/2020Natur.585..357H 585, 357

    Harris C. R., et al., 2020, @doi [ ] 10.1038/s41586-020-2649-2 , https://ui.adsabs.harvard.edu/abs/2020Natur.585..357H 585, 357

    2020 doi

  18. [26]

    D., 2007, @doi [Computing in Science and Engineering] 10.1109/MCSE.2007.55 , https://ui.adsabs.harvard.edu/abs/2007CSE.....9...90H 9, 90

    Hunter J. D., 2007, @doi [Computing in Science and Engineering] 10.1109/MCSE.2007.55 , https://ui.adsabs.harvard.edu/abs/2007CSE.....9...90H 9, 90

    2007 doi

  19. [27]

    M., Ghosh S., 2024, @doi [ ] 10.1093/mnras/stae011 , https://ui.adsabs.harvard.edu/abs/2024MNRAS.52712284K 527, 12284

    Kalda T., Green G. M., Ghosh S., 2024, @doi [ ] 10.1093/mnras/stae011 , https://ui.adsabs.harvard.edu/abs/2024MNRAS.52712284K 527, 12284

    2024 doi

  20. [28]

    F., Schindler J.-T., Tamanas J., Nanni R., 2024, @doi [arXiv e-prints] 10.48550/arXiv.2412.03029 , https://ui.adsabs.harvard.edu/abs/2024arXiv241203029K p

    Kang Y., Hennawi J. F., Schindler J.-T., Tamanas J., Nanni R., 2024, @doi [arXiv e-prints] 10.48550/arXiv.2412.03029 , https://ui.adsabs.harvard.edu/abs/2024arXiv241203029K p. arXiv:2412.03029

    2024 doi arXiv

  21. [29]

    P., Ba J., 2014, @doi [arXiv e-prints] 10.48550/arXiv.1412.6980 , https://ui.adsabs.harvard.edu/abs/2014arXiv1412.6980K p

    Kingma D. P., Ba J., 2014, @doi [arXiv e-prints] 10.48550/arXiv.1412.6980 , https://ui.adsabs.harvard.edu/abs/2014arXiv1412.6980K p. arXiv:1412.6980

    2014 doi arXiv

  22. [30]

    P., Dhariwal P., 2018, @doi [arXiv e-prints] 10.48550/arXiv.1807.03039 , https://ui.adsabs.harvard.edu/abs/2018arXiv180703039K p

    Kingma D. P., Dhariwal P., 2018, @doi [arXiv e-prints] 10.48550/arXiv.1807.03039 , https://ui.adsabs.harvard.edu/abs/2018arXiv180703039K p. arXiv:1807.03039

    2018 doi arXiv

  23. [31]

    P., Welling M., 2013, @doi [arXiv e-prints] 10.48550/arXiv.1312.6114 , https://ui.adsabs.harvard.edu/abs/2013arXiv1312.6114K p

    Kingma D. P., Welling M., 2013, @doi [arXiv e-prints] 10.48550/arXiv.1312.6114 , https://ui.adsabs.harvard.edu/abs/2013arXiv1312.6114K p. arXiv:1312.6114

    2013 doi arXiv

  24. [32]

    M., Kraus A

    Krolikowski D. M., Kraus A. L., Rizzuto A. C., 2021, @doi [ ] 10.3847/1538-3881/ac0632 , https://ui.adsabs.harvard.edu/abs/2021AJ....162..110K 162, 110

    2021 doi

  25. [33]

    S., 2024, @doi [arXiv e-prints] 10.48550/arXiv.2408.11414 , https://ui.adsabs.harvard.edu/abs/2024arXiv240811414L p

    Li Z., Han J., Wang W., Qian Y.-Z., Li Q., Jing Y., Li T. S., 2024, @doi [arXiv e-prints] 10.48550/arXiv.2408.11414 , https://ui.adsabs.harvard.edu/abs/2024arXiv240811414L p. arXiv:2408.11414

    2024 doi arXiv

  26. [34]

    H., Putney E., Buckley M

    Lim S. H., Putney E., Buckley M. R., Shih D., 2025, @doi [ ] 10.1088/1475-7516/2025/01/021 , https://ui.adsabs.harvard.edu/abs/2025JCAP...01..021L 2025, 021

    2025 doi

  27. [35]

    arXiv:2003.01941

    Meng C., Song Y., Song J., Ermon S., 2020, @doi [arXiv e-prints] 10.48550/arXiv.2003.01941 , https://ui.adsabs.harvard.edu/abs/2020arXiv200301941M p. arXiv:2003.01941

    2020 doi arXiv

  28. [36]

    C., Figueras F., Roca-F \`a brega S., Luri X., 2019, @doi [ ] 10.1051/0004-6361/201935105 , https://ui.adsabs.harvard.edu/abs/2019A&A...624L...1M 624, L1

    Mor R., Robin A. C., Figueras F., Roca-F \`a brega S., Luri X., 2019, @doi [ ] 10.1051/0004-6361/201935105 , https://ui.adsabs.harvard.edu/abs/2019A&A...624L...1M 624, L1

    2019 doi

  29. [37]

    arXiv:1808.03856

    M \"u ller T., McWilliams B., Rousselle F., Gross M., Nov \'a k J., 2018, @doi [arXiv e-prints] 10.48550/arXiv.1808.03856 , https://ui.adsabs.harvard.edu/abs/2018arXiv180803856M p. arXiv:1808.03856

    2018 doi arXiv

  30. [38]

    arXiv:1705.07057

    Papamakarios G., Pavlakou T., Murray I., 2017, @doi [arXiv e-prints] 10.48550/arXiv.1705.07057 , https://ui.adsabs.harvard.edu/abs/2017arXiv170507057P p. arXiv:1705.07057

    2017 doi arXiv

  31. [39]

    arXiv:1912.02762

    Papamakarios G., Nalisnick E., Jimenez Rezende D., Mohamed S., Lakshminarayanan B., 2019, @doi [arXiv e-prints] 10.48550/arXiv.1912.02762 , https://ui.adsabs.harvard.edu/abs/2019arXiv191202762P p. arXiv:1912.02762

    2019 doi arXiv

  32. [40]

    arXiv:1912.01703

    Paszke A., et al., 2019, @doi [arXiv e-prints] 10.48550/arXiv.1912.01703 , https://ui.adsabs.harvard.edu/abs/2019arXiv191201703P p. arXiv:1912.01703

    2019 doi arXiv

  33. [41]

    Poggio E., et al., 2021, @doi [ ] 10.1051/0004-6361/202140687 , https://ui.adsabs.harvard.edu/abs/2021A&A...651A.104P 651, A104

    2021 doi

  34. [42]

    Ramos P., et al., 2022, @doi [ ] 10.1051/0004-6361/202142830 , https://ui.adsabs.harvard.edu/abs/2022A&A...666A..64R 666, A64

    2022 doi

  35. [43]

    A., Murray I., 2019, @doi [arXiv e-prints] 10.48550/arXiv.1911.11663 , https://ui.adsabs.harvard.edu/abs/2019arXiv191111663R p

    Ritchie J. A., Murray I., 2019, @doi [arXiv e-prints] 10.48550/arXiv.1911.11663 , https://ui.adsabs.harvard.edu/abs/2019arXiv191111663R p. arXiv:1911.11663

    2019 doi arXiv

  36. [44]

    Rozet F., Divo F., Schnake S., 2024, probabilists/zuko: Zuko 1.1.0 , @doi 10.5281/zenodo.7625672

    2024 doi

  37. [45]

    Sch \"o nrich R., Dehnen W., 2018, @doi [ ] 10.1093/mnras/sty1256 , https://ui.adsabs.harvard.edu/abs/2018MNRAS.478.3809S 478, 3809

    2018 doi

  38. [46]

    Sharma S., Steinmetz M., 2006, @doi [ ] 10.1111/j.1365-2966.2006.11043.x , https://ui.adsabs.harvard.edu/abs/2006MNRAS.373.1293S 373, 1293

    2006

  39. [47]

    arXiv:2004.00464

    Sick B., Hothorn T., D \"u rr O., 2020, @doi [arXiv e-prints] 10.48550/arXiv.2004.00464 , https://ui.adsabs.harvard.edu/abs/2020arXiv200400464S p. arXiv:2004.00464

    2020 doi arXiv

  40. [48]

    arXiv:2312.06017

    Wolf L., Buck T., 2023, @doi [arXiv e-prints] 10.48550/arXiv.2312.06017 , https://ui.adsabs.harvard.edu/abs/2023arXiv231206017W p. arXiv:2312.06017

    2023 doi arXiv

  41. [49]

    write newline

    " write newline "" before.all 'output.state := FUNCTION fin.entry write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence 'output.stat...

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.