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Properly estimating correlations between objects at different spatial scales necessitates $𝒪\left(n^2\right)$distance calculations. For this reason, most widely adopted packages for estimating correlations use clustering algorithms to approximate local trends. However, methods for quantifying the error introduced by this clustering have been understudied. In response, we present an algorithm for estimating correlations that is probabilistic in the way that it clusters objects, enabling us to quantify the uncertainty caused by clustering simply through model inference. These soft clustering assignments enable correlation estimators that are theoretically differentiable with respect to their input catalogs. Thus, we also build a theoretical framework for differentiable correlation functions and describe their utility in comparison to existing surrogate models. Notably, we find that repeated normalization and distance function calls slow gradient calculations and that sparse Jacobians destabilize precision, pointing towards either approximate or surrogate methods as a necessary solution to exact gradients from correlation functions. To that end, we close with a discussion of surrogate models as proxies for correlation functions. We provide an example that demonstrates the efficacy of surrogate models to enable gradient-based optimization of astrophysical model parameters, successfully minimizing a correlation function output. Our numerical experiments cover science cases across cosmology, from point spread function (PSF) modeling efforts to gravitational simulations to galaxy intrinsic alignment (IA). 

Abstract We report the discovery of 15 exceptionally luminous 10 ≲z≲ 14 candidate galaxies discovered in the first 0.28 deg²of JWST/NIRCam imaging from the COSMOS-Web survey. These sources span rest-frame UV magnitudes of −20.5 >M_UV> −22, and thus constitute the most intrinsically luminousz≳ 10 candidates identified by JWST to date. Selected via NIRCam imaging, deep ground-based observations corroborate their detection and help significantly constrain their photometric redshifts. We analyze their spectral energy distributions using multiple open-source codes and evaluate the probability of low-redshift solutions; we conclude that 12/15 (80%) are likely genuinez≳ 10 sources and 3/15 (20%) likely low-redshift contaminants. Three of ourz∼ 12 candidates push the limits of early stellar mass assembly: they have estimated stellar masses ∼ 5 × 10⁹M_⊙, implying an effective stellar baryon fraction ofϵ_⋆∼ 0.2−0.5, whereϵ_⋆≡M_⋆/(f_bM_halo). The assembly of such stellar reservoirs is made possible due to rapid, burst-driven star formation on timescales < 100 Myr where the star formation rate may far outpace the growth of the underlying dark matter halos. This is supported by the similar volume densities inferred forM_⋆∼ 10¹⁰M_⊙galaxies relative toM_⋆∼ 10⁹M_⊙—both about 10⁻⁶Mpc⁻³—implying they live in halos of comparable mass. At such high redshifts, the duty cycle for starbursts would be of order unity, which could cause the observed change in the shape of the UV luminosity function from a double power law to a Schechter function atz≈ 8. Spectroscopic redshift confirmation and ensuing constraints of their masses will be critical to understand how, and if, such early massive galaxies push the limits of galaxy formation in the Lambda cold dark matter paradigm.