Testing and Learning on Distributions with Symmetric Noise Invariance

Kernel embeddings of distributions and the Maximum Mean Discrepancy (MMD), the resulting distance between distributions, are useful tools for fully nonparametric two-sample testing and learning on distributions. However, it is rarely that all possible differences between samples are of interest -- discovered differences can be due to different types of measurement noise, data collection artefacts or other irrelevant sources of variability. We propose distances between distributions which encode invariance to additive symmetric noise, aimed at testing whether the assumed true underlying processes differ. Moreover, we construct invariant features of distributions, leading to learning algorithms robust to the impairment of the input distributions with symmetric additive noise.Comment: 22 page

The Excursion Set Theory of Halo Mass Functions, Halo Clustering, and Halo Growth

I review the excursion set theory (EST) of dark matter halo formation and clustering. I recount the Press-Schechter argument for the mass function of bound objects and review the derivation of the Press-Schechter mass function in EST. The EST formalism is powerful and can be applied to numerous problems. I review the EST of halo bias and the properties of void regions. I spend considerable time reviewing halo growth in the EST. This section culminates with descriptions of two Monte Carlo methods for generating halo mass accretion histories. In the final section, I emphasize that the standard EST approach is the result of several simplifying assumptions. Dropping these assumptions can lead to more faithful predictions and a more versatile formalism. One such assumption is the constant height of the barrier for nonlinear collapse. I review implementations of the excursion set approach with arbitrary barrier shapes. An application of this is the now well-known improvement to standard EST that follows from the ellipsoidal-collapse barrier. Additionally, I emphasize that the statement that halo accretion histories are independent of halo environments is a simplifying assumption, rather than a prediction of the theory. I review the method for constructing correlated random walks of the density field in more general cases. I construct a simple toy model with correlated walks and I show that excursion set theory makes a qualitatively simple and general prediction for the relation between halo accretion histories and halo environments: regions of high density preferentially contain late-forming halos and conversely for regions of low density. I conclude with a brief discussion of this prediction in the context of recent numerical studies of the environmental dependence of halo properties. (Abridged)Comment: 62 pages, 19 figures. Review article based on lectures given at the Sixth Summer School of the Helmholtz Institute for Supercomputational Physics. Accepted for Publication in IJMPD. Comments Welcom

A Fully Self-Consistent Treatment of Collective Fluctuations in Quantum Liquids

The problem of calculating collective density fluctuations in quantum liquids is revisited. A fully quantum mechanical self-consistent treatment based on a quantum mode-coupling theory [E. Rabani and D.R. Reichman, J. Chem. Phys.116, 6271 (2002)] is presented. The theory is compared with the maximum entropy analytic continuation approach and with available experimental results. The quantum mode-coupling theory provides semi-quantitative results for both short and long time dynamics. The proper description of long time phenomena is important in future study of problems related to the physics of glassy quantum systems, and to the study of collective fluctuations in Bose fluids.Comment: 9 pages, 4 figure

Tailoring Anderson localization by disorder correlations in 1D speckle potentials

We study Anderson localization of single particles in continuous, correlated, one-dimensional disordered potentials. We show that tailored correlations can completely change the energy-dependence of the localization length. By considering two suitable models of disorder, we explicitly show that disorder correlations can lead to a nonmonotonic behavior of the localization length versus energy. Numerical calculations performed within the transfer-matrix approach and analytical calculations performed within the phase formalism up to order three show excellent agreement and demonstrate the effect. We finally show how the nonmonotonic behavior of the localization length with energy can be observed using expanding ultracold-atom gases