For Kernel Range Spaces a Constant Number of Queries Are Sufficient

We introduce the notion of an $\varepsilon$-cover for a kernel range space. A kernel range space concerns a set of points $X \subset \mathbb{R}^d$ and the space of all queries by a fixed kernel (e.g., a Gaussian kernel $K(p,\cdot) = \exp(-\|p-\cdot\|^2)$). For a point set $X$ of size $n$, a query returns a vector of values $R_p \in \mathbb{R}^n$, where the $i$th coordinate $(R_p)_i = K(p,x_i)$ for $x_i \in X$. An $\varepsilon$-cover is a subset of points $Q \subset \mathbb{R}^d$ so for any $p \in \mathbb{R}^d$ that $\frac{1}{n} \|R_p - R_q\|_1\leq \varepsilon$ for some $q \in Q$. This is a smooth analog of Haussler's notion of $\varepsilon$-covers for combinatorial range spaces (e.g., defined by subsets of points within a ball query) where the resulting vectors $R_p$ are in $\{0,1\}^n$ instead of $[0,1]^n$. The kernel versions of these range spaces show up in data analysis tasks where the coordinates may be uncertain or imprecise, and hence one wishes to add some flexibility in the notion of inside and outside of a query range. Our main result is that, unlike combinatorial range spaces, the size of kernel $\varepsilon$-covers is independent of the input size $n$ and dimension $d$. We obtain a bound of $(1/\varepsilon)^{\tilde O(1/\varepsilon^2)}$, where $\tilde{O}(f(1/\varepsilon))$ hides log factors in $(1/\varepsilon)$ that can depend on the kernel. This implies that by relaxing the notion of boundaries in range queries, eventually the curse of dimensionality disappears, and may help explain the success of machine learning in very high-dimensions. We also complement this result with a lower bound of almost $(1/\varepsilon)^{\Omega(1/\varepsilon)}$, showing the exponential dependence on $1/\varepsilon$ is necessary.Comment: 27 page

Microplastics in agricultural soils from a semi-arid region and their transport by wind erosion.

Despite the importance of agricultural soils, little is known about the fate of microplastics (MPs) in this environment. In the present study, MPs have been determined in soils and wind-eroded sediments from two vegetable-growing fields in the Fars province of Iran, one using plastic mulch for water retention (Field 1) and the other using wastewater for irrigation (Field 2). MPs were heterogeneously distributed in the surface (0-5 cm) and subsurface (5-15 cm) soils of both fields, with a maximum concentration overall of about 1.1 MP g-1 and no significant differences in concentrations between either fields or depths. Fibres represented the principal shape of MPs, but spherules, presumably from wastewater, also made a significant (∼25%) contribution to MPs in Field 2. Analysis of selected samples by Raman spectroscopy and scanning electron microscopy revealed that polyethylene terephthalate (PET) and nylon were the most abundant polymers and that MPs exhibited varying degrees of weathering. Concentrations of MPs in this study are within the range reported previously for agricultural soils, although the absence of PET observed in earlier studies is attributed to the use of insufficiently dense solutions to isolate plastics. Deployment of a portable wind tunnel revealed threshold wind velocities for soil erosion of up to 7 and 12 m s-1 and MP erosion rates up to about 0.4 and 1.1 MP m-2 s-1 for Fields 1 and 2, respectively. Erosion rates are considerably greater than published depositional rates for MPs and suggest that agricultural soils act as both a temporary sink and dynamic secondary source of MPs that should be considered in risk assessments and global transport budgets

Microplastics captured by snowfall: A study in Northern Iran.

Samples of fresh snow (n = 34) have been collected from 29 locations in various urban and remote regions of northern Iran following a period of sustained snowfall and the thawed contents examined for microplastics (MPs) according to established techniques. MP concentrations ranged from undetected to 86 MP L-1 (mean and median concentrations ~20 MP and 12 MP L-1, respectively) and there was no significant difference in MP concentration between sample location type or between different depths of snow (or time of deposition) sampled at selected sites. Fibres were the dominant shape of MP and μ-Raman spectroscopy of selected samples revealed a variety of polymer types, with nylon most abundant. Scanning electron microscopy coupled with energy-dispersive X-ray analysis showed that some MPs were smooth and unweathered while others were more irregular and exhibited significant photo-oxidative and mechanical weathering as well as contamination by extraneous geogenic particles. These characteristics reflect the importance of both local and distal sources to the heterogeneous pool of MPs in precipitated snow. The mean and median concentrations of MPs in the snow samples were not dissimilar to the published mean and median concentrations for MPs in rainfall collected from an elevated location in southwest Iran. However, compared with rainfall, MPs in snow appear to be larger and more diverse in their shape and composition (and include rubber particulates), possibly because of the greater size but lower terminal velocities of snowflakes relative to raindrops. Snowfall represents a significant means by which MPs are scavenged from the atmosphere and transferred to soil and surface waters that warrants further attention