Near-Neighbor Preserving Dimension Reduction for Doubling Subsets of l_1

Randomized dimensionality reduction has been recognized as one of the fundamental techniques in handling high-dimensional data. Starting with the celebrated Johnson-Lindenstrauss Lemma, such reductions have been studied in depth for the Euclidean (l_2) metric, but much less for the Manhattan (l_1) metric. Our primary motivation is the approximate nearest neighbor problem in l_1. We exploit its reduction to the decision-with-witness version, called approximate near neighbor, which incurs a roughly logarithmic overhead. In 2007, Indyk and Naor, in the context of approximate nearest neighbors, introduced the notion of nearest neighbor-preserving embeddings. These are randomized embeddings between two metric spaces with guaranteed bounded distortion only for the distances between a query point and a point set. Such embeddings are known to exist for both l_2 and l_1 metrics, as well as for doubling subsets of l_2. The case that remained open were doubling subsets of l_1. In this paper, we propose a dimension reduction by means of a near neighbor-preserving embedding for doubling subsets of l_1. Our approach is to represent the pointset with a carefully chosen covering set, then randomly project the latter. We study two types of covering sets: c-approximate r-nets and randomly shifted grids, and we discuss the tradeoff between them in terms of preprocessing time and target dimension. We employ Cauchy variables: certain concentration bounds derived should be of independent interest

Characterisation of the Perseid meteoroid stream through SPOSH observations between 2010–2016

We have characterised the Perseid meteoroid stream from data acquired in a series of observing campaigns between 2010 and 2016. The data presented in this work were obtained by the Smart Panoramic Optical Sensor Head (SPOSH), an all-sky camera system designed to image faint transient noctilucent phenomena on dark planetary hemispheres. For the data reduction, a sophisticated software package was developed that utilises the high geometric and photometric quality of images obtained by the camera system. We identify 934 meteors as Perseids, observed over a long period between late July (~124°) and mid-to-late August (~147°). The maximum meteor activity contributing to the annual shower was found at λ⊙ = 140°.08 ± 0°.07. The radiant of the shower was estimated at RA = 47°.2 and Dec = 57°.5 with a median error of 0°.6 and 0°.2, respectively. The mean population index of the shower between solar longitudes of 120°.68 and 145°.19 was r = 2.36 ± 0.05, showing strong temporal variation. A predicted outburst in shower activity for the night of August 11–12, 2016 was confirmed, with a peak observed 12.75 hr before the annual maximum at 23:30 ± 15′ UT. We measure a peak flux of 6.1 × 10−4 km−2 hr−1 for meteoroids of mass 1.6 × 10−2 g or more, appearing in the time period between 23:00 and 00:00 UT. We estimate the measured flux of the outburst meteoroids to be approximately twice as high as the annual meteoroid flux of the same mass. The population index of r = 2.19 ± 0.08, computed from the outburst Perseids in 2016, is higher than the value of r = 1.92 ± 0.06 derived from meteors observed in 2015 belonging to the annual Perseid shower which was active near the time of the outburst. A dust trail with an unusually high population index of r = 3.58 ± 0.24 was encountered in 2013 between solar longitudes 136°.261 and 137°.442. The relatively high r-value implies an encounter with a dust trail rich in low-mass particles