Translation and Rotation Equivariant Normalizing Flow (TRENF) for OptimalCosmological Analysis

Our universe is homogeneous and isotropic, and its perturbations aretranslation and rotation equivariant. We develop a generative Normalizing Flow(NF) architecture which explicitly incorporates these symmetries, defining thedata likelihood via a sequence of Fourier space based convolutions andnonlinear transforms, evaluating their Jacobian at each layer. This allows totrain the NF by maximizing the data likelihood p(x|y) as a function of thelabels y, such as cosmological parameters. In contrast to other generativemodels the NF approach has no loss of information since it preserves the fulldimensionality of the data, and gives direct access to the data likelihoodp(x|y). We apply this to outputs of cosmological N-body simulations and showthat the summary statistics of the generated samples agree well with thesimulations. We show that the reverse mapping is visually indistinguishablefrom a Gaussian white noise: when this is perfectly achieved the resultingp(x|y) likelihood analysis becomes optimal. On simple Gaussian examples we showthat this approach maximizes the information in the data and saturates theFisher information content in the labels y. On N-body simulation outputs weshow that this leads to significant improvements in constraining power over thestandard summary statistics such as the power spectrum. Finally, we develop ageneralization of this NF that can handle effects that break the symmetry ofthe data, such as the survey mask.
 

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