Halo Mass Function and the Free Streaming Scale

The nature of structure formation around the particle free streaming scale is still far from understood. Many attempts to simulate hot, warm, and cold dark matter cosmologies with a free streaming cutoff have been performed with cosmological particle-based simulations, but they all suffer from spurious structure formation at scales below their respective free streaming scales -- i.e. where the physics of halo formation is most affected by free streaming. We perform a series of high resolution numerical simulations of different WDM models, and develop an approximate method to subtract artificial structures in the measured halo mass function. The corrected measurements are then used to construct and calibrate an extended Press-Schechter (EPS) model with sharp-$k$ window function and adequate mass assignment. The EPS model gives accurate predictions for the low redshift halo mass function of CDM and WDM models, but it significantly under-predicts the halo abundance at high redshifts. By taking into account the ellipticity of the initial patches and connecting the characteristic filter scale to the smallest ellipsoidal axis, we are able to eliminate this inconsistency and obtain an accurate mass function over all redshifts and all dark matter particle masses covered by the simulations. As an additional application we use our model to predict the microhalo abundance of the standard neutralino-CDM scenario and we give the first quantitative prediction of the mass function over the full range of scales of CDM structure formation.Comment: 16 pages, 10 figures, published in MNRA

The detection and treatment of distance errors in kinematic analyses of stars

We present a new method for detecting and correcting systematic errors in the distances to stars when both proper motions and line-of-sight velocities are available. The method, which is applicable for samples of 200 or more stars that have a significant extension on the sky, exploits correlations between the measured U, V and W velocity components that are introduced by distance errors. We deliver a formalism to describe and interpret the specific imprints of distance errors including spurious velocity correlations and shifts of mean motion in a sample. We take into account correlations introduced by measurement errors, Galactic rotation and changes in the orientation of the velocity ellipsoid with position in the Galaxy. Tests on pseudodata show that the method is more robust and sensitive than traditional approaches to this problem. We investigate approaches to characterising the probability distribution of distance errors, in addition to the mean distance error, which is the main theme of the paper. Stars with the most overestimated distances bias our estimate of the overall distance scale, leading to the corrected distances being slightly too small. We give a formula that can be used to correct for this effect. We apply the method to samples of stars from the SEGUE survey, exploring optimal gravity cuts, sample contamination, and correcting the used distance relations.Comment: published in MNRAS 14 pages, 8 figures, 2 tables, corrected eq.(35), minor editin

High-Dimensional Geometric Streaming in Polynomial Space

Many existing algorithms for streaming geometric data analysis have been plagued by exponential dependencies in the space complexity, which are undesirable for processing high-dimensional data sets. In particular, once $d\geq\log n$, there are no known non-trivial streaming algorithms for problems such as maintaining convex hulls and L\"owner-John ellipsoids of $n$ points, despite a long line of work in streaming computational geometry since [AHV04]. We simultaneously improve these results to $\mathrm{poly}(d,\log n)$ bits of space by trading off with a $\mathrm{poly}(d,\log n)$ factor distortion. We achieve these results in a unified manner, by designing the first streaming algorithm for maintaining a coreset for $\ell_\infty$ subspace embeddings with $\mathrm{poly}(d,\log n)$ space and $\mathrm{poly}(d,\log n)$ distortion. Our algorithm also gives similar guarantees in the \emph{online coreset} model. Along the way, we sharpen results for online numerical linear algebra by replacing a log condition number dependence with a $\log n$ dependence, answering a question of [BDM+20]. Our techniques provide a novel connection between leverage scores, a fundamental object in numerical linear algebra, and computational geometry. For $\ell_p$ subspace embeddings, we give nearly optimal trade-offs between space and distortion for one-pass streaming algorithms. For instance, we give a deterministic coreset using $O(d^2\log n)$ space and $O((d\log n)^{1/2-1/p})$ distortion for $p>2$, whereas previous deterministic algorithms incurred a $\mathrm{poly}(n)$ factor in the space or the distortion [CDW18]. Our techniques have implications in the offline setting, where we give optimal trade-offs between the space complexity and distortion of subspace sketch data structures. To do this, we give an elementary proof of a "change of density" theorem of [LT80] and make it algorithmic.Comment: Abstract shortened to meet arXiv limits; v2 fix statements concerning online condition numbe

On Deterministic Sketching and Streaming for Sparse Recovery and Norm Estimation

We study classic streaming and sparse recovery problems using deterministic linear sketches, including l1/l1 and linf/l1 sparse recovery problems (the latter also being known as l1-heavy hitters), norm estimation, and approximate inner product. We focus on devising a fixed matrix A in R^{m x n} and a deterministic recovery/estimation procedure which work for all possible input vectors simultaneously. Our results improve upon existing work, the following being our main contributions: * A proof that linf/l1 sparse recovery and inner product estimation are equivalent, and that incoherent matrices can be used to solve both problems. Our upper bound for the number of measurements is m=O(eps^{-2}*min{log n, (log n / log(1/eps))^2}). We can also obtain fast sketching and recovery algorithms by making use of the Fast Johnson-Lindenstrauss transform. Both our running times and number of measurements improve upon previous work. We can also obtain better error guarantees than previous work in terms of a smaller tail of the input vector. * A new lower bound for the number of linear measurements required to solve l1/l1 sparse recovery. We show Omega(k/eps^2 + klog(n/k)/eps) measurements are required to recover an x' with |x - x'|_1 <= (1+eps)|x_{tail(k)}|_1, where x_{tail(k)} is x projected onto all but its largest k coordinates in magnitude. * A tight bound of m = Theta(eps^{-2}log(eps^2 n)) on the number of measurements required to solve deterministic norm estimation, i.e., to recover |x|_2 +/- eps|x|_1. For all the problems we study, tight bounds are already known for the randomized complexity from previous work, except in the case of l1/l1 sparse recovery, where a nearly tight bound is known. Our work thus aims to study the deterministic complexities of these problems