Confidence Intervals for Maximin Effects in Inhomogeneous Large-Scale Data

One challenge of large-scale data analysis is that the assumption of an identical distribution for all samples is often not realistic. An optimal linear regression might, for example, be markedly different for distinct groups of the data. Maximin effects have been proposed as a computationally attractive way to estimate effects that are common across all data without fitting a mixture distribution explicitly. So far just point estimators of the common maximin effects have been proposed in Meinshausen and B\"uhlmann (2014). Here we propose asymptotically valid confidence regions for these effects

Multipliers for Continuous Frames in Hilbert Spaces

In this paper we examine the general theory of continuous frame multipliers in Hilbert space. These operators are a generalization of the widely used notion of (discrete) frame multipliers. Well-known examples include Anti-Wick operators, STFT multipliers or Calder\'on- Toeplitz operators. Due to the possible peculiarities of the underlying measure spaces, continuous frames do not behave quite as well as their discrete counterparts. Nonetheless, many results similar to the discrete case are proven for continuous frame multipliers as well, for instance compactness and Schatten class properties. Furthermore, the concepts of controlled and weighted frames are transferred to the continuous setting

Amplification of Cosmological Inhomogeneities by the QCD Transition

The cosmological QCD transition affects primordial density perturbations. If the QCD transition is first order, the sound speed vanishes during the transition and density perturbations fall freely. For scales below the Hubble radius at the transition the primordial Harrison-Zel'dovich spectrum of density fluctuations develops large peaks and dips. These peaks grow with wave number for both the hadron-photon-lepton fluid and for cold dark matter. At the horizon scale the enhancement is small. This by itself does not lead to the formation of black holes at the QCD transition. The peaks in the hadron-photon-lepton fluid are wiped out during neutrino decoupling. For cold dark matter that is kinetically decoupled at the QCD transition (e.g., axions or primordial black holes) these peaks lead to the formation of CDM clumps of masses $10^{-20} M_\odot< M_{\rm clump} < 10^{-10} M_\odot$.Comment: 39 pages, 10 figures, RevTeX; (1) ETH Zuerich, (2) Univ. Frankfurt; improved presentation of 'Introduction' and 'Collisional Damping at Neutrino Decoupling', results unchanged; accepted for publication in Phys. Rev.