Fewer Mocks and Less Noise: Reducing the Dimensionality of Cosmological Observables with Subspace Projections. (arXiv:2009.03311v2 [astro-ph.CO] UPDATED)
<a href="http://arxiv.org/find/astro-ph/1/au:+Philcox_O/0/1/0/all/0/1">Oliver H. E. Philcox</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Ivanov_M/0/1/0/all/0/1">Mikhail M. Ivanov</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Zaldarriaga_M/0/1/0/all/0/1">Matias Zaldarriaga</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Simonovic_M/0/1/0/all/0/1">Marko Simonovic</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Schmittfull_M/0/1/0/all/0/1">Marcel Schmittfull</a>
Creating accurate and low-noise covariance matrices represents a formidable
challenge in modern-day cosmology. We present a formalism to compress arbitrary
observables into a small number of bins by projecting into a subspace that
minimizes the log-likelihood error. The lower dimensionality leads to a
dramatic reduction in covariance matrix noise, significantly reducing the
number of mocks that need to be computed. Given a theory model, a set of
priors, and a simple model of the covariance, our method works by using
singular value decompositions to construct a basis for the observable that is
close to Euclidean; by restricting to the first few basis vectors, we can
capture almost all the signal-to-noise in a lower-dimensional subspace. Unlike
conventional approaches, the method can be tailored for specific analyses and
captures non-linearities that are not present in the Fisher matrix, ensuring
that the full likelihood can be reproduced. The procedure is validated with
full-shape analyses of power spectra from BOSS DR12 mock catalogs, showing that
the 96-bin power spectra can be replaced by 12 subspace coefficients without
biasing the output cosmology; this allows for accurate parameter inference
using only $sim 100$ mocks. Such decompositions facilitate accurate testing of
power spectrum covariances; for the largest BOSS data chunk, we find that: (a)
analytic covariances provide accurate models (with or without trispectrum
terms); and (b) using the sample covariance from the MultiDark-Patchy mocks
incurs a $sim 0.5sigma$ bias in $Omega_m$, unless the subspace projection is
applied. The method is easily extended to higher order statistics; the $sim
2000$ bin bispectrum can be compressed into only $sim 10$ coefficients,
allowing for accurate analyses using few mocks and without having to increase
the bin sizes.
Creating accurate and low-noise covariance matrices represents a formidable
challenge in modern-day cosmology. We present a formalism to compress arbitrary
observables into a small number of bins by projecting into a subspace that
minimizes the log-likelihood error. The lower dimensionality leads to a
dramatic reduction in covariance matrix noise, significantly reducing the
number of mocks that need to be computed. Given a theory model, a set of
priors, and a simple model of the covariance, our method works by using
singular value decompositions to construct a basis for the observable that is
close to Euclidean; by restricting to the first few basis vectors, we can
capture almost all the signal-to-noise in a lower-dimensional subspace. Unlike
conventional approaches, the method can be tailored for specific analyses and
captures non-linearities that are not present in the Fisher matrix, ensuring
that the full likelihood can be reproduced. The procedure is validated with
full-shape analyses of power spectra from BOSS DR12 mock catalogs, showing that
the 96-bin power spectra can be replaced by 12 subspace coefficients without
biasing the output cosmology; this allows for accurate parameter inference
using only $sim 100$ mocks. Such decompositions facilitate accurate testing of
power spectrum covariances; for the largest BOSS data chunk, we find that: (a)
analytic covariances provide accurate models (with or without trispectrum
terms); and (b) using the sample covariance from the MultiDark-Patchy mocks
incurs a $sim 0.5sigma$ bias in $Omega_m$, unless the subspace projection is
applied. The method is easily extended to higher order statistics; the $sim
2000$ bin bispectrum can be compressed into only $sim 10$ coefficients,
allowing for accurate analyses using few mocks and without having to increase
the bin sizes.
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