Replicating weak-lensing summary-statistic covariances with normalizing flows

TL;DR

This work evaluates normalizing-flow generative models for reproducing weak-lensing summary statistics derived from simulations. By training neural spline flows and multi-scale NF architectures on SLICS κ-maps, the authors assess the fidelity of the mean, variance, and covariance of $C_\ell^{\kappa\kappa}$, the $\kappa$-PDF, and Minkowski functionals. They find that means and variances are recovered at percent-level accuracy, but off-diagonal covariances are systematically under-estimated, with mitigation strategies such as data augmentation and noisy training reducing the gap to roughly 5–15%. The results demonstrate the potential of NF as fast emulators for weak-lensing statistics while highlighting the need for robust covariance modeling to avoid overconfident cosmological inferences; future work may leverage larger models and diffusion-based approaches to further improve covariance replication.

Abstract

We explore the ability of normalizing flow (NF) generative models to reproduce weak-lensing summary statistics when trained on a set of cosmological simulations. Our analysis focuses on how accurately NF models recover the mean, standard deviation, and covariance of key statistics derived from convergence ($κ$) maps: The angular power spectrum $C_{\ell}$, probability density function, and Minkowski functionals of weak lensing convergence $κ$-maps. We test two scenarios for training: (1) on the data vectors and (2) on the full $κ$-maps. In both cases, the NF models reproduce the mean and variance of the target statistics within percent-level accuracy. However, the accuracy of the off-diagonal elements of the covariance matrix is underestimated by up to $\sim25\%$. We study several mitigation strategies and find that data augmentation and training with noisy fields help improve covariance recovery to $\mathcal{O}(10\%)$. Our study demonstrates that while the means and variances of weak lensing statistics can be well modeled by NF, covariances can be significantly underestimated if mitigation strategies are not applied.

Figures (9)

• Figure 1: Diagram of multiscale NF network used for learning $\kappa$-map features. Images are passed through the flow with an initial size of $N_{\rm pix} = 256\times 256$ with a clear non-Gaussian distribution. After all the transformation, the latent space represents a Gaussian random field with the same image dimensions (number of pixels). The image representation is schemed in the center, showing how the original image, a convergence map, loses resolution by passing coupling, squeezing, and splitting transforms until it becomes indistinguishable from Gaussian random noise. The nature of the NF network requires the flow to be invertible, allowing for the generation of $\kappa$-maps from an initial Gaussian random field.
• Figure 2: Replication of summary statistics using NF network and SLICS simulations (ground truth) dataset. We calculate the mean of the angular power spectrum $C_{\ell}^{\kappa\kappa}$, $\kappa$-PDF (top panels) and three Minkowski functionals $V_0$, $V_1$ and $V_2$ (bottom panels) for the SLICS simulation (black lines) and the synthetic data vectors generated using the Neural Spline Flow network. The bottom sub-panels include the relative difference compared to the ground truth data vector. We add the standard deviation of SLICS (black dashed lines) and the synthetic data (red error bars).
• Figure 3: Ratio of the covariance matrix of the summary statistic showed in Fig. \ref{['fig:DV_stats']}, $C_{\ell}^{\kappa\kappa}$ (left), $\kappa$-PDF (middle), and Minkowski functionals (right) for NF generated data vectors and SLIC simulations.
• Figure 4: Angular Power spectrum $C_{\ell}^{\kappa\kappa}$ of NF generated convergence maps compared to the $\kappa$-maps from SLICS simulations. These are calculated from maps with different pixel sizes: $64\times 64$ (red), $128\times 128$ (blue), and $256\times 256$ green. We add the relative difference w.r.t. the ground truth in the bottom panel.
• Figure 5: Ratio of covariance for $C_{\ell}^{\kappa\kappa}$ in Fig. \ref{['fig:C_ell_maps_resol']}. We select 12 bins for each resolution map to help the comparison between the cases: $64\times 64$ (left), $128\times 128$ (middle), and $256\times 256$ right.