pith. sign in

arxiv: 2606.11295 · v1 · pith:WVGP3S4Bnew · submitted 2026-06-09 · 🌌 astro-ph.CO · cs.LG

Interpretable Neural Marked Statistics for Cosmological Inference

Federico Semenzato , Benjamin D. Wandelt , Michele Liguori , Alvise Raccanelli This is my paper

Pith reviewed 2026-06-27 12:02 UTC · model grok-4.3

classification 🌌 astro-ph.CO cs.LG
keywords marked statisticsneural networkscontrastive learningcosmological inferencenon-Gaussian informationmatter density fieldσ8 constraintΩm-σ8 degeneracy
0
0 comments X

The pith

A neural marking scheme trained with contrastive learning tightens σ8 constraints by 2.9 times and Ωm by 1.8 times over classical marks at k_max=0.2 h Mpc^{-1}.

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

The paper develops a neural approach to marked statistics that applies learnable transformations to the matter density field instead of fixed functions. A contrastive objective trains these marks to align with cosmological parameters, allowing direct interpretation of information gains at the morphological level. This yields tighter marginalized constraints on σ8 and Ωm while breaking their degeneracy in the Fisher matrix, and lowers overall parameter estimation error across the prior. The work targets the recovery of late-time non-Gaussian signals that power spectra alone cannot access for future surveys.

Core claim

The authors introduce a neural marking scheme of interpretable, physically motivated transformations applied to the density field. They optimize the marks with a contrastive learning objective that aligns the resulting summaries with the underlying cosmological parameters. At k_max=0.2 h Mpc^{-1} the neural marks produce 2.9 times tighter constraints on σ8 and 1.8 times tighter constraints on Ωm relative to classical marks, break the Ωm-σ8 degeneracy at the Fisher level, and reduce parameter MSE by a factor of 1.45 over the best classical mark. The learned latent space geometry aligns with the Ωm and σ8 directions.

What carries the argument

Neural marking scheme consisting of learnable transformations on the density field, optimized by a contrastive learning objective that aligns marked summaries with cosmological parameters.

If this is right

  • Marked statistics can be extended from fixed functional forms to learnable, interpretable transformations.
  • The Ωm-σ8 degeneracy is broken at the Fisher information level by the neural marks.
  • Parameter mean-squared error across the cosmological prior is reduced by 1.45 times relative to the best classical mark.
  • The latent geometry of the marks recovers the dominant axes of cosmological information.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same contrastive alignment technique could be applied to design other summary statistics beyond marked two-point functions.
  • Morphological interpretation of the learned marks may identify which density features carry the extra non-Gaussian information.
  • Real-data application to galaxy surveys would test whether the simulated gains persist after observational systematics.
  • The approach suggests a general route for constructing parameter-aligned summaries that could be combined with higher-order statistics.

Load-bearing premise

The contrastive objective produces marks whose information gain generalizes beyond the specific simulation suite and training cosmologies without arising from overfitting to the parameter directions in the loss.

What would settle it

Applying the trained neural marks to an independent simulation suite with cosmologies drawn from a different prior range and checking whether the reported factors of improvement in σ8 and Ωm constraints are recovered.

Figures

Figures reproduced from arXiv: 2606.11295 by Alvise Raccanelli, Benjamin D. Wandelt, Federico Semenzato, Michele Liguori.

Figure 1
Figure 1. Figure 1: The learned mark aims to extract interpretable features from the density field. (a) Spherical-harmonic filters (ℓ = 0, 1, 2) produce density-like, gradient-like, and quadrupole-like responses. Scalar contractions form rotation invariant fields which are independently processed by MLPs and linearly combined with a small cross interaction. The resulting mark M produces a marked field ∆. (b) The Pδδ embedder … view at source ↗
Figure 2
Figure 2. Figure 2: End-to-end introspection of the learned mark on a representative held-out simulation. Top row: real-space projection of the per-ℓ kernels reconstructed from the trained Gℓ(k), for ℓ = 0, 1, 2. Middle row: the input density field δ and the four rotation-invariant maps E0, E1, E2, I3 extracted from the spherical-harmonic-filtered field. Bottom row: the learned scalar response contributions ha(E0), ha(E1), ha… view at source ↗
Figure 3
Figure 3. Figure 3: Marginalized posterior contours from the Fisher infor￾mation matrix at the fiducial cosmology, for Pδδ alone (blue) and the learned mark {Pδδ, Pδ∆, P∆∆} (green). The learned mark tightens the one-dimensional posteriors on every parameter and rotates the Ωm–σ8 contour, breaking the canonical degeneracy and enabling simultaneous improvement on both parameters. and in elongated regions, and I3 adds only a wea… view at source ↗
Figure 5
Figure 5. Figure 5: Two-dimensional PCA projection of the complementary query embedding z⊥ for held-out simulations, colored by σ8 (left) and Ωm (right). σ8 varies smoothly along the first principal direc￾tion and Ωm along an approximately orthogonal one. The first two PCs therefore align with the most well-constrained parameters of the problem. gains on σ8 and Ωm. On the other hand, with a further held-out test set span￾ning… view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of learned and classical marks on both Fisher and held-out generalization at kmax = 0.20 h Mpc−1 . Top: Improvement in the marginalized confidence intervals relative to a set of classical marks from (Massara et al., 2021) with R = 10 h −1Mpc. Bottom: Mean-squared error from a MLP regressor on Latin-hypercube simulations unseen during training; the learned mark beats the unmarked baseline and the… view at source ↗
Figure 6
Figure 6. Figure 6: Marginalized Ωm (left) and σ8 (right) constraints as a function of kmax, for Pδδ (black), a reference classical mark (gray), the isotropic ablation ℓmax = 0 (blue), and the full learned mark ℓmax = 2 (red). The Fisher matrix is evaluated for the full set of summaries {Pδδ, Pδ∆, P∆∆}. The learned marks converge toward the classical baseline as kmax grows. A. Dependence on kmax and Channel Ablation In this a… view at source ↗
read the original abstract

Recovering cosmological information beyond the power spectrum is a central goal for upcoming cosmological surveys, since late-time non-Gaussian signal in the matter density cannot be accessed through two-point statistics alone. Marked statistics fold part of this information back into the two-point level by reweighting the field with non-linear functions. We propose a neural marking scheme to generalize this process through a set of interpretable, physically motivated transformations that directly allow to interpret the gain in cosmological information at the morphological level. We employ a contrastive learning objective to align learnable marked summaries with the underlying cosmological parameters. At $k_{\max}=0.2\,h\mathrm{Mpc}^{-1}$, our neural mark tightens the marginalized constraint on $\sigma_8$ by $2.9\times$ and on $\Omega_m$ by $1.8\times$ compared to classical marks, breaking the $\Omega_m-\sigma_8$ degeneracy at the Fisher information level. It further reduces the parameter MSE across our cosmological parameter prior by $1.45\times$ over the best classical mark. The learned latent geometry aligns with the $\Omega_m$ and $\sigma_8$ directions in parameter space, indicating that the contrastive objective recovers the dominant axes of cosmological information. Our approach opens the door to more powerful, interpretable summary statistics for cosmological inference.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 0 minor

Summary. The manuscript proposes a neural marking scheme that uses contrastive learning to produce interpretable marked statistics from the matter density field. It claims that at k_max=0.2 h Mpc^{-1} the neural mark tightens marginalized constraints on σ8 by 2.9× and on Ωm by 1.8× relative to classical marks, reduces parameter MSE by 1.45× across the prior, breaks the Ωm-σ8 degeneracy at the Fisher level, and yields a latent geometry aligned with the Ωm and σ8 axes.

Significance. If the reported gains are shown to arise from generalizable morphological features rather than the training objective, the method would offer a route to more powerful yet interpretable two-point summaries that capture non-Gaussian information. The emphasis on interpretability is a potential strength, but only if the alignment with cosmological parameters is independently verified.

major comments (3)
  1. [Abstract] Abstract: the statement that 'the contrastive objective recovers the dominant axes of cosmological information' follows directly from the construction of the loss, which supervises alignment with Ωm and σ8; this is not an independent test of whether the marks discover transferable morphological features.
  2. [Abstract] Abstract: the headline factors (2.9× on σ8, 1.8× on Ωm, 1.45× MSE reduction) and degeneracy-breaking claim are presented without any description of held-out cosmologies, cross-validation splits, or explicit checks that the information gain persists for parameter values outside the training prior.
  3. [Abstract] Abstract: the assertion that the neural mark 'breaks the Ωm-σ8 degeneracy at the Fisher information level' is unsupported by any reported Fisher-matrix comparison, covariance structure, or table of information content; the quantitative improvement cannot be evaluated without these elements.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive comments on the abstract. We agree that several claims require additional context or clarification to be fully supported. We will revise the abstract accordingly and address each point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the statement that 'the contrastive objective recovers the dominant axes of cosmological information' follows directly from the construction of the loss, which supervises alignment with Ωm and σ8; this is not an independent test of whether the marks discover transferable morphological features.

    Authors: We acknowledge that the contrastive loss is explicitly constructed to align latent representations with Ωm and σ8. The contribution lies in achieving this alignment through a constrained set of interpretable, physically motivated marking transformations rather than arbitrary functions. The resulting latent geometry then provides a morphological interpretation of which features drive the alignment. We will revise the abstract wording to clarify that the contrastive objective enables recovery of the dominant axes via these interpretable marks, avoiding any implication of unsupervised discovery. revision: yes

  2. Referee: [Abstract] Abstract: the headline factors (2.9× on σ8, 1.8× on Ωm, 1.45× MSE reduction) and degeneracy-breaking claim are presented without any description of held-out cosmologies, cross-validation splits, or explicit checks that the information gain persists for parameter values outside the training prior.

    Authors: The quantitative results are obtained from Fisher forecasts and MSE evaluations performed on held-out cosmologies within the simulation suite, using cross-validation splits to assess generalization. We will add a concise description of this validation procedure to the abstract so that the reported gains are presented with the necessary methodological context. revision: yes

  3. Referee: [Abstract] Abstract: the assertion that the neural mark 'breaks the Ωm-σ8 degeneracy at the Fisher information level' is unsupported by any reported Fisher-matrix comparison, covariance structure, or table of information content; the quantitative improvement cannot be evaluated without these elements.

    Authors: The degeneracy breaking is demonstrated by direct comparison of the Fisher information matrices (and resulting parameter covariances) between the neural marks and classical marks; the relevant matrices, off-diagonal elements, and information-content tables appear in the main text. We will revise the abstract to explicitly reference this Fisher-matrix analysis and the supporting covariance comparisons. revision: yes

Circularity Check

1 steps flagged

Contrastive objective directly enforces Ωm-σ8 alignment, rendering reported latent geometry recovery tautological

specific steps
  1. self definitional [Abstract]
    "We employ a contrastive learning objective to align learnable marked summaries with the underlying cosmological parameters. [...] The learned latent geometry aligns with the Ωm and σ8 directions in parameter space, indicating that the contrastive objective recovers the dominant axes of cosmological information."

    The contrastive objective is defined to produce alignment with Ωm and σ8. The subsequent claim that observed alignment indicates the objective recovers the dominant axes therefore follows by construction from the loss definition rather than from any separate derivation or external validation.

full rationale

The paper trains marked summaries via contrastive loss explicitly constructed to align with target cosmological parameters Ωm and σ8. It then presents the resulting alignment of latent geometry with those same axes as evidence that the objective 'recovers the dominant axes of cosmological information.' This step reduces directly to the training construction rather than an independent test. The reported Fisher gains (2.9× on σ8, 1.8× on Ωm) and MSE reduction are measured against classical marks and are not themselves by construction, so the circularity is localized to the interpretation of the learned geometry and does not collapse the entire quantitative claim. No self-citation chains or ansatz smuggling appear in the provided text.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

Based solely on the abstract, the central claim rests on the effectiveness of a learned neural transformation whose weights are optimized by a contrastive loss; no explicit free parameters, axioms, or invented entities beyond the neural mark itself are stated.

free parameters (1)
  • neural network weights
    Weights of the marking network are fitted via the contrastive objective on simulation data.
axioms (2)
  • domain assumption Marked statistics can capture non-Gaussian information when the field is reweighted by suitable non-linear functions
    Foundation for using marked power spectra in cosmology.
  • domain assumption A contrastive objective can align learned summaries with underlying cosmological parameters
    Core training assumption stated in the abstract.
invented entities (1)
  • neural mark no independent evidence
    purpose: Generalize classical marked statistics with learnable, interpretable transformations
    New scheme introduced by the paper.

pith-pipeline@v0.9.1-grok · 5778 in / 1487 out tokens · 38452 ms · 2026-06-27T12:02:04.345142+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

38 extracted references · 33 canonical work pages · 3 internal anchors

  1. [1]

    Adame, A. G. et al. DESI 2024 VII: cosmological constraints from the full-shape modeling of clustering measurements . JCAP, 07: 0 028, 2025 a . doi:10.1088/1475-7516/2025/07/028

    work page doi:10.1088/1475-7516/2025/07/028 2024

  2. [2]

    Adame, A. G. et al. DESI 2024 VI: cosmological constraints from the measurements of baryon acoustic oscillations . JCAP, 02: 0 021, 2025 b . doi:10.1088/1475-7516/2025/02/021

    work page doi:10.1088/1475-7516/2025/02/021 2024

  3. [3]

    Aghamousa, A. et al. The DESI Experiment Part I: Science,Targeting, and Survey Design . 10 2016

    2016

  4. [4]

    and Wandelt, B

    Bairagi, A. and Wandelt, B. PatchNet: A hierarchical approach for neural field-level inference from Quijote simulations . JCAP, 03: 0 028, 2026. doi:10.1088/1475-7516/2026/03/028

    work page doi:10.1088/1475-7516/2026/03/028 2026

  5. [5]

    2025, title The BIG SOBOL SEQUENCE: How many simulations do we need for simulation-based inference in cosmology? , , 703, A301, 10.1051/0004-6361/202554602

    Bairagi, A., Wandelt, B., and Villaescusa-Navarro, F. The BIG SOBOL SEQUENCE: How many simulations do we need for simulation-based inference in cosmology? Astron. Astrophys., 703: 0 A301, 2025. doi:10.1051/0004-6361/202554602

    work page doi:10.1051/0004-6361/202554602 2025

  6. [6]

    Large scale structure of the universe and cosmological perturbation theory

    Bernardeau, F., Colombi, S., Gaztanaga, E., and Scoccimarro, R. Large scale structure of the universe and cosmological perturbation theory . Phys. Rept., 367: 0 1--248, 2002. doi:10.1016/S0370-1573(02)00135-7

    work page doi:10.1016/s0370-1573(02)00135-7 2002

  7. [7]

    Observed galaxy number counts on the lightcone up to second order: II

    Bertacca, D., Maartens, R., and Clarkson, C. Observed galaxy number counts on the lightcone up to second order: II. Derivation . JCAP, 11: 0 013, 2014. doi:10.1088/1475-7516/2014/11/013

    work page doi:10.1088/1475-7516/2014/11/013 2014

  8. [8]

    Charnock , T., Lavaux , G., and Wandelt , B. D. Automatic physical inference with information maximizing neural networks . , 97 0 (8): 0 083004, April 2018. doi:10.1103/PhysRevD.97.083004

    work page doi:10.1103/physrevd.97.083004 2018

  9. [9]

    A Simple Framework for Contrastive Learning of Visual Representations

    Chen , T., Kornblith , S., Norouzi , M., and Hinton , G. A Simple Framework for Contrastive Learning of Visual Representations . arXiv e-prints, art. arXiv:2002.05709, feb 2020. doi:10.48550/arXiv.2002.05709

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2002.05709 2002

  10. [10]

    A., Alonso, D., and Liu, J

    Cowell, J. A., Alonso, D., and Liu, J. Optimizing marked power spectra for cosmology . Mon. Not. Roy. Astron. Soc., 535 0 (4): 0 3129--3140, 2024. doi:10.1093/mnras/stae2492

    work page doi:10.1093/mnras/stae2492 2024

  11. [11]

    The imprints of primordial non-gaussianities on large-scale structure: scale dependent bias and abundance of virialized objects

    Dalal, N., Dor \'e , O., Huterer, D., and Shirokov, A. The imprints of primordial non-gaussianities on large-scale structure: scale dependent bias and abundance of virialized objects . Phys. Rev. D, 77: 0 123514, 2008. doi:10.1103/PhysRevD.77.123514

    work page doi:10.1103/physrevd.77.123514 2008

  12. [12]

    Dor \'e , O. et al. Cosmology with the SPHEREX All-Sky Spectral Survey . 12 2014

    2014

  13. [13]

    and White, M

    Ebina, H. and White, M. An analytically tractable marked power spectrum . JCAP, 01: 0 150, 2025. doi:10.1088/1475-7516/2025/01/150

    work page doi:10.1088/1475-7516/2025/01/150 2025

  14. [14]

    The Marked Power Spectrum as a Practical Bispectrum Measure for Galaxy Redshift Surveys

    Ebina, H., White, M., and Chaussidon, E. The Marked Power Spectrum as a Practical Bispectrum Measure for Galaxy Redshift Surveys . 3 2026

    2026

  15. [15]

    Elbers, W. et al. Constraints on neutrino physics from DESI DR2 BAO and DR1 full shape . Phys. Rev. D, 112 0 (8): 0 083513, 2025. doi:10.1103/w9pk-xsk7

    work page doi:10.1103/w9pk-xsk7 2025

  16. [16]

    nbodykit: an open-source, massively parallel toolkit for large-scale structure

    Hand, N., Feng, Y., Beutler, F., Li, Y., Modi, C., Seljak, U., and Slepian, Z. nbodykit: an open-source, massively parallel toolkit for large-scale structure . Astron. J., 156 0 (4): 0 160, 2018. doi:10.3847/1538-3881/aadae0

    work page doi:10.3847/1538-3881/aadae0 2018

  17. [17]

    Heavens, A. F. and Taylor, A. N. A Spherical Harmonic Analysis of Redshift Space . Mon. Not. Roy. Astron. Soc., 275: 0 483--497, 1995. doi:10.1093/mnras/275.2.483

    work page doi:10.1093/mnras/275.2.483 1995

  18. [18]

    Lemos, P. et al. Field-level simulation-based inference of galaxy clustering with convolutional neural networks . Phys. Rev. D, 109 0 (8): 0 083536, 2024. doi:10.1103/PhysRevD.109.083536

    work page doi:10.1103/physrevd.109.083536 2024

  19. [19]

    and Pastor, S

    Lesgourgues, J. and Pastor, S. Massive neutrinos and cosmology. Phys. Rept., 429: 0 307--379, 2006. doi:10.1016/j.physrep.2006.04.001

    work page doi:10.1016/j.physrep.2006.04.001 2006

  20. [20]

    and Matarrese, S

    Lucchin, F. and Matarrese, S. The Effect of nonGaussian statistics on the mass multiplicity of cosmic structures . Astrophys. J., 330: 0 535--544, 1988. doi:10.1086/166492

    work page doi:10.1086/166492 1988

  21. [21]

    L., Charnock, T., Alsing, J., and Wandelt, B

    Makinen, T. L., Charnock, T., Alsing, J., and Wandelt, B. D. Lossless, scalable implicit likelihood inference for cosmological fields . JCAP, 11 0 (11): 0 049, 2021. doi:10.1088/1475-7516/2021/11/049. [Erratum: JCAP 04, E02 (2023)]

    work page doi:10.1088/1475-7516/2021/11/049 2021

  22. [22]

    L., Sui, C., Wandelt, B

    Makinen, T. L., Sui, C., Wandelt, B. D., Porqueres, N., and Heavens, A. Hybrid summary statistics, 2025. URL https://arxiv.org/abs/2410.07548

    arXiv 2025

  23. [23]

    Non-gaussian features of primordial fluctuations in single field inflationary models

    Maldacena and Martin, J. Non-gaussian features of primordial fluctuations in single field inflationary models. JHEP, 05: 0 013, 2003. doi:10.1088/1126-6708/2003/05/013

    work page doi:10.1088/1126-6708/2003/05/013 2003

  24. [24]

    Marinucci, M. et al. The constraining power of the marked power spectrum: an analytical study . JCAP, 09: 0 036, 2025. doi:10.1088/1475-7516/2025/09/036

    work page doi:10.1088/1475-7516/2025/09/036 2025

  25. [25]

    Massara, E., Villaescusa-Navarro, F., Ho, S., Dalal, N., and Spergel, D. N. Using the Marked Power Spectrum to Detect the Signature of Neutrinos in Large-Scale Structure . Phys. Rev. Lett., 126 0 (1): 0 011301, 2021. doi:10.1103/PhysRevLett.126.011301

    work page doi:10.1103/physrevlett.126.011301 2021

  26. [26]

    and Verde, L

    Matarrese, S. and Verde, L. The effect of primordial non-Gaussianity on halo bias . Astrophys. J. Lett., 677: 0 L77--L80, 2008. doi:10.1086/587840

    work page doi:10.1086/587840 2008

  27. [27]

    Mellier, Y. et al. Euclid. I. Overview of the Euclid mission . 5 2024

    2024

  28. [28]

    Srinivasan, Matthew Tancik, Jonathan T

    Mildenhall , B., Srinivasan , P. P., Tancik , M., Barron , J. T., Ramamoorthi , R., and Ng , R. NeRF: Representing Scenes as Neural Radiance Fields for View Synthesis . arXiv e-prints, art. arXiv:2003.08934, March 2020. doi:10.48550/arXiv.2003.08934

    work page doi:10.48550/arxiv.2003.08934 2003

  29. [29]

    J., & Tout, C

    Percival, W. J., Verde, L., and Peacock, J. A. Fourier analysis of luminosity-dependent galaxy clustering . Mon. Not. Roy. Astron. Soc., 347: 0 645, 2004. doi:10.1111/j.1365-2966.2004.07245.x

    work page doi:10.1111/j.1365-2966.2004.07245.x 2004

  30. [30]

    Philcox , O. H. E., Massara , E., and Spergel , D. N. What does the marked power spectrum measure? Insights from perturbation theory . , 102 0 (4): 0 043516, August 2020. doi:10.1103/PhysRevD.102.043516

    work page doi:10.1103/physrevd.102.043516 2020

  31. [31]

    Learning Transferable Visual Models From Natural Language Supervision

    Radford , A., Kim , J. W., Hallacy , C., Ramesh , A., Goh , G., Agarwal , S., Sastry , G., Askell , A., Mishkin , P., Clark , J., Krueger , G., and Sutskever , I. Learning Transferable Visual Models From Natural Language Supervision . arXiv e-prints, art. arXiv:2103.00020, February 2021. doi:10.48550/arXiv.2103.00020

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2103.00020 2021

  32. [32]

    Scoccimarro, R., Couchman, H. M. P., and Frieman, J. A. The Bispectrum as a Signature of Gravitational Instability in Redshift-Space . Astrophys. J., 517: 0 531--540, 1999. doi:10.1086/307220

    work page doi:10.1086/307220 1999

  33. [33]

    and Komatsu, E

    Sefusatti, E. and Komatsu, E. The bispectrum of galaxies from high-redshift galaxy surveys: primordial non-Gaussianity and non-linear galaxy bias . Phys. Rev. D, 76: 0 083004, 2007. doi:10.1103/PhysRevD.76.083004

    work page doi:10.1103/physrevd.76.083004 2007

  34. [34]

    1997, title Measuring Cosmological Parameters with Galaxy Surveys , , 79, 3806, 10.1103/PhysRevLett.79.3806

    Tegmark, M. Measuring cosmological parameters with galaxy surveys . Phys. Rev. Lett., 79: 0 3806--3809, 1997. doi:10.1103/PhysRevLett.79.3806

    work page doi:10.1103/physrevlett.79.3806 1997

  35. [35]

    Representation Learning with Contrastive Predictive Coding

    van den Oord , A., Li , Y., and Vinyals , O. Representation Learning with Contrastive Predictive Coding . arXiv e-prints, art. arXiv:1807.03748, July 2018. doi:10.48550/arXiv.1807.03748

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1807.03748 2018

  36. [36]

    , keywords =

    Verde, L., Wang, L.-M., Heavens, A., and Kamionkowski, M. Large scale structure, the cosmic microwave background, and primordial non-gaussianity. Mon.Not.Roy.Astron.Soc., 313: 0 L141, 2000. doi:10.1046/j.1365-8711.2000.03191.x

    work page doi:10.1046/j.1365-8711.2000.03191.x 2000

  37. [37]

    Ingredients for 21 cm intensity mapping

    Villaescusa-Navarro et al. Ingredients for 21 cm intensity mapping. The Astrophysical Journal, 866 0 (2): 0 135, oct 2018. ISSN 1538-4357. doi:10.3847/1538-4357/aadba0

    work page doi:10.3847/1538-4357/aadba0 2018

  38. [38]

    A marked correlation function for constraining modified gravity models

    White, M. A marked correlation function for constraining modified gravity models . JCAP, 11: 0 057, 2016. doi:10.1088/1475-7516/2016/11/057

    work page doi:10.1088/1475-7516/2016/11/057 2016