tokens: absorption or emission-line features can be present alongside smoothly varying blackbody or other continuum radiation (see e.g. Tennyson 2019 ). In each of these general settings, we observe a signal plus noise and would like to denoise the signal as accurately as possible. Indeed the set of statistical tools available for addressing this general problem is quite vast. Commonly used non-parametric regression methods include kernel smoothers (e.g. Hall et al. 2002 ; Croft et al. 2002 ), local polynomial regression (LOESS; e.g. Maron Howes 2003 ; Persson et al. 2004 ), splines (e.g. Peiris Verde 2010 ; Contreras et al. 2010 ; Dhawan et al. 2015 ), Gaussian process regression (e.g. Gibson et al. 2012 ; Aigrain, Parviainen Pope 2016 ; Gómez-Valent Amendola 2018 ), and wavelet decompositions (e.g. Fligge Solanki 1997 ; Theuns Zaroubi 2000 ; Golkhou Butler 2014 ). A rich and elegant statistical literature exists on the theoretical and practical achievements of these methods (see e.g. Györfi et al. 2002 ; Wasserman 2006 ; Hastie, Tibshirani Friedman 2009 for general references). However, when the underlying signal is spatially heterogeneous , i.e. exhibits varying degrees of smoothness, the power of classical statistical literature is quite limited. Kernels, LOESS, smoothing splines, and Gaussian process regression belong to a broad family of non-parametric methods called linear smoothers , which has been shown to be uniformly suboptimal for estimating spatially heterogeneous signals (Nemirovskii, Polyak Tsybakov 1985 ; Nemirovskii 1985 ; Donoho Johnstone 1998 ). The common limitation of these methods is that they are not locally adaptive; i.e. by construction, they do not adapt to local degrees of smoothness in a signal. In particular, continuing with the example of a smoothly varying signal with occasional sharp features, a linear smoother will tend to oversmooth the sharp features and/or overfit the smooth regions in its effort to optimally balance statistical bias and variance. Considerable effort has been made to address this problem by locally varying the hyperparameter(s) of a linear smoother. For example, locally varying the kernel bandwidth (e.g. Muller Stadtmuller 1987 ; Fan Gijbels 1992 , 1995 ; Lepski, Mammen Spokoiny 1997 ; Gijbels Mammen 1998 ) irregularly varying spline knot locations (e.g. De Boor 1974 ; Jupp 1978 ; Dimatteo, Genovese Kass 2001 ), and constructing non-stationary covariance functions for Gaussian process regression (e.g. Schmidt O’Hagan 2003 ; Paciorek Schervish 2004 , 2006 ). However, since hyperparameters typically need to be estimated from the data, such exponential increases in the hyperparameter complexity severely limit the practicality of choosing the hyperparameters in a fully data-driven, generalizable, and computationally efficient fashion. Wavelet decompositions offer an elegant solution to the problem of estimating spatially heterogeneous signals, providing both statistical optimality (e.g. Donoho
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