Non-negative Matrix Factorization for Self-calibration of Photometric Redshift Scatter in Weak-lensing Surveys

Abstract

Photo-z error is one of the major sources of systematics degrading the accuracy of weak lensing cosmological inferences. Zhang et al. (2010) proposed a self-calibration method combining galaxy-galaxy correlations and galaxy-shear correlations between different photo-z bins. Fisher matrix analysis shows that it can determine the rate of photo-z outliers at a level of 0.01-1% merely using photometric data and do not rely on any prior knowledge. In this paper, we develop a new algorithm to implement this method by solving a constrained nonlinear optimization problem arising in the self-calibration process. Based on the techniques of fixed-point iteration and non-negative matrix factorization, the proposed algorithm can efficiently and robustly reconstruct the scattering probabilities between the true-z and photo-z bins. The algorithm has been tested extensively by applying it to mock data from simulated stage IV weak lensing projects. We find that the algorithm provides a successful recovery of the scatter rates at the level of 0.01-1%, and the true mean redshifts of photo-z bins at the level of 0.001, which may satisfy the requirements in future lensing surveys.

Self-calibration of photometric redshift scatter from DECaLS DR8 power spectrum and validation with simulated catalogues

“…Algorithm 2 is based on NMF, which can further iterate over the matrix reconstructed by Algorithm 1 to further reduce the J and find the global optimal solution. In Zhang et al (2017), the NMF based algorithm successfully obtained the correct and accurate solution from a mock data with theoretical signal and Gaussian measurement noise. The configuration considered there is quiet optimal, with the sky coverage 𝑓 sky = 0.5, the galaxy number density n𝑔 = 40 arcmin −2 and the galaxy shape noise 𝛾 rms = 0.2.…”

“…On the contrary, the self-calibration techniques do not rely on external spectroscopic samples (Schneider et al 2006;Benjamin et al 2010;Zhang et al 2010Zhang et al , 2017Schaan et al 2020;Stölzner et al 2021). In the absence of photo-𝑧 errors, the galaxy-galaxy correlation between two distinguish redshift bins should vanish under the limber approximation, and there is no correlation between the low-redshift cosmic shear and high-redshift galaxy distribution.…”

“…However, finding a stable and efficient algorithm to solve the non-convex nonlinear optimization problem with numerous unknown parameters and multiple constraints is a non-trivial task. Zhang et al (2017) developed a non-negative matrix factorization (NMF) based algorithm and the validation succeeded on the theoretical data. Further validation and optimization are required before applying to surveys.…”

“…In this work, we construct lightcone simulations to systematically investigate the performance of the self-calibration algorithm based on Zhang et al (2017). To deal with the complexity in the simulated galaxy catalogues and observations, the algorithm is improved by modifying the randomness of the initial guess, adjusting the convergence criterion, and assigning weightings on different scales.…”

Calibrating systematic errors in the distance determination with the luminosity-distance space large scale structure of dark sirens and its potential applications

“…One early application is the SZ tomography (Zhang & Pen 2001;Shao et al 2011), where cross-correlating wit galaxies help recover the redshift information of the thermal Sunyaev Zel'dovich (SZ) effect. It is also widely used in the cross calibration (Newman 2008) and the self-calibration of galaxy photometric redshift errors (Schneider et al 2006;Zhang et al 2010Zhang et al , 2017. Similar techniques have been designed to identification of dark siren redshifts (Namikawa et al 2016;Oguri 2016;Nair et al 2018;Mukherjee & Wandelt 2018;Zhang 2018b).…”