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Register a new userA Unified Framework for Photometric Redshifts
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Authors
Tamas Budavari
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We present a rigorous mathematical solution to photometric redshift estimation and the more general inversion problem. The challenge we address is to meaningfully constrain unknown properties of astronomical sources based on given observables, usually multicolor photometry, with the help of a training set that provides an empirical relation between the measurements and the desired quantities. We establish a formalism that blurs the boundary between the traditional empirical and template-fitting algorithms, as both are just special cases that are discussed in detail to put them in context. The new approach enables the development of more sophisticated methods that go beyond the classic techniques to combine their advantages. We look at the directions for further improvement in the methodology, and examine the technical aspects of practical implementations. We show how training sets are to be constructed and used consistently for reliable estimation.
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Upcoming imaging surveys, such as LSST, will provide an unprecedented view of the Universe, but with limited resolution along the line-of-sight. Common ways to increase resolution in the third dimension, and reduce misclassifications, include observing a wider wavelength range and/or combining the broad-band imaging with higher spectral resolution data. The challenge with these approaches is matching the depth of these ancillary data with the original imaging survey. However, while a full 3D map is required for some science, there are many situations where only the statistical distribution of objects (dN/dz) in the line-of-sight direction is needed. In such situations, there is no need to measure the fluxes of individual objects in all of the surveys. Rather a stacking procedure can be used to perform an `ensemble photo-z. We show how a shallow, higher spectral resolution survey can be used to measure dN/dz for stacks of galaxies which coincide in a deeper, lower resolution survey. The galaxies in the deeper survey do not even need to appear individually in the shallow survey. We give a toy model example to illustrate tradeoffs and considerations for applying this method. This approach will allow deep imaging surveys to leverage the high resolution of spectroscopic and narrow/medium band surveys underway, even when the latter do not have the same reach to high redshift.
Machine learning (ML) is a standard approach for estimating the redshifts of galaxies when only photometric information is available. ML photo-z solutions have traditionally ignored the morphological information available in galaxy images or partly included it in the form of hand-crafted features, with mixed results. We train a morphology-aware photometric redshift machine using modern deep learning tools. It uses a custom architecture that jointly trains on galaxy fluxes, colors and images. Galaxy-integrated quantities are fed to a Multi-Layer Perceptron (MLP) branch while images are fed to a convolutional (convnet) branch that can learn relevant morphological features. This split MLP-convnet architecture, which aims to disentangle strong photometric features from comparatively weak morphological ones, proves important for strong performance: a regular convnet-only architecture, while exposed to all available photometric information in images, delivers comparatively poor performance. We present a cross-validated MLP-convnet model trained on 130,000 SDSS-DR12 galaxies that outperforms a hyperoptimized Gradient Boosting solution (hyperopt+XGBoost), as well as the equivalent MLP-only architecture, on the redshift bias metric. The 4-fold cross-validated MLP-convnet model achieves a bias $delta z / (1+z) =-0.70 pm 1 times 10^{-3} $, approaching the performance of a reference ANNZ2 ensemble of 100 distinct models trained on a comparable dataset. The relative performance of the morphology-aware and morphology-blind models indicates that galaxy morphology does improve ML-based photometric redshift estimation.
We study bandits and reinforcement learning (RL) subject to a conservative constraint where the agent is asked to perform at least as well as a given baseline policy. This setting is particular relevant in real-world domains including digital marketing, healthcare, production, finance, etc. For multi-armed bandits, linear bandits and tabular RL, specialized algorithms and theoretical analyses were proposed in previous work. In this paper, we present a unified framework for conservative bandits and RL, in which our core technique is to calculate the necessary and sufficient budget obtained from running the baseline policy. For lower bounds, our framework gives a black-box reduction that turns a certain lower bound in the nonconservative setting into a new lower bound in the conservative setting. We strengthen the existing lower bound for conservative multi-armed bandits and obtain new lower bounds for conservative linear bandits, tabular RL and low-rank MDP. For upper bounds, our framework turns a certain nonconservative upper-confidence-bound (UCB) algorithm into a conservative algorithm with a simple analysis. For multi-armed bandits, linear bandits and tabular RL, our new upper bounds tighten or match existing ones with significantly simpler analyses. We also obtain a new upper bound for conservative low-rank MDP.
We propose a novel type filter for multicolor imaging to improve on the photometric redshift estimation of galaxies. An extra filter - specific to a certain photometric system - may be utilized with high efficiency. We present a case study of the Hubble Space Telescopes Advanced Camera for Surveys and show that one extra exposure could cut down the mean square error on photometric redshifts by 34% over the z<1.3 redshift range.
Photometric redshift estimation is an indispensable tool of precision cosmology. One problem that plagues the use of this tool in the era of large-scale sky surveys is that the bright galaxies that are selected for spectroscopic observation do not have properties that match those of (far more numerous) dimmer galaxies; thus, ill-designed empirical methods that produce accurate and precise redshift estimates for the former generally will not produce good estimates for the latter. In this paper, we provide a principled framework for generating conditional density estimates (i.e. photometric redshift PDFs) that takes into account selection bias and the covariate shift that this bias induces. We base our approach on the assumption that the probability that astronomers label a galaxy (i.e. determine its spectroscopic redshift) depends only on its measured (photometric and perhaps other) properties x and not on its true redshift. With this assumption, we can explicitly write down risk functions that allow us to both tune and compare methods for estimating importance weights (i.e. the ratio of densities of unlabeled and labeled galaxies for different values of x) and conditional densities. We also provide a method for combining multiple conditional density estimates for the same galaxy into a single estimate with better properties. We apply our risk functions to an analysis of approximately one million galaxies, mostly observed by SDSS, and demonstrate through multiple diagnostic tests that our method achieves good conditional density estimates for the unlabeled galaxies.
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