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Creating accurate and low-noise covariance matrices represents a formidablechallenge in modern-day cosmology. We present a formalism to compress arbitraryobservables into a small number of bins by projection into a model-specificsubspace that minimizes the prior-averaged log-likelihood error. The lowerdimensionality leads to a dramatic reduction in covariance matrix noise,significantly reducing the number of mocks that need to be computed. Given atheory model, a set of priors, and a simple model of the covariance, our methodworks by using singular value decompositions to construct a basis for theobservable that is close to Euclidean; by restricting to the first few basisvectors, we can capture almost all the constraining power in alower-dimensional subspace. Unlike conventional approaches, the method can betailored for specific analyses and captures non-linearities that are notpresent in the Fisher matrix, ensuring that the full likelihood can bereproduced. The procedure is validated with full-shape analyses of powerspectra from BOSS DR12 mock catalogs, showing that the 96-bin power spectra canbe replaced by 12 subspace coefficients without biasing the output cosmology;this allows for accurate parameter inference using only $\sim 100$ mocks. Suchdecompositions facilitate accurate testing of power spectrum covariances; forthe largest BOSS data chunk, we find that: (a) analytic covariances provideaccurate models (with or without trispectrum terms); and (b) using the samplecovariance from the MultiDark-Patchy mocks incurs a $\sim 0.5\sigma$ shift in$\Omega_m$, unless the subspace projection is applied. The method is easilyextended to higher order statistics; the $\sim 2000$-bin bispectrum can becompressed into only $\sim 10$ coefficients, allowing for accurate analysesusing few mocks and without having to increase the bin sizes.

Fewer Mocks and Less Noise: Reducing the Dimensionality of Cosmological Observables with Subspace Projections