Taxicab geometry

Summary

A taxicab geometry or a Manhattan geometry is a geometry whose usual distance function or metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian coordinates. The taxicab metric is also known as rectilinear distance, L1 distance, L1 distance or \ell_1 norm (see Lp space), snake distance, city block distance, Manhattan distance or Manhattan length. The latter names refer to the rectilinear street layout on the island of Manhattan, where the shortest path a taxi travels between two points is the sum of the absolute values of distances that it travels on avenues and on streets. The geometry has been used in regression analysis since the 18th century, and is often referred to as LASSO. The geometric interpretation dates to non-Euclidean geometry of the 19th century and is due to Hermann Minkowski. In \mathbb{R}^2 , the taxicab distance between two points (x

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The growing demands on compact and high-definition single-photon avalanche diode (SPAD) arrays have motivated researchers to explore pixel miniaturization techniques to achieve sub-10 μm pixels. The scaling of the SPAD pixel size has an impact on key performance metrics, and it is, thereby, critical to conduct a systematic analysis of the underlying tradeoffs in miniaturized SPADs. On the basis of the general assumptions and constraints for layout geometry, we performed an analytical formulation of the scaling laws for the key metrics, such as the fill factor (FF), photon detection probability (PDP), dark count rate (DCR), correlated noise, and power consumption. Numerical calculations for various parameter sets indicated that some of the metrics, such as the DCR and power consumption, were improved by pixel miniaturization, whereas other metrics, such as the FF and PDP, were degraded. Comparison of the theoretically estimated scaling trends with previously published experimental results suggests that the scaling law analysis is in good agreement with practical SPAD devices. Our scaling law analysis could provide a useful tool to conduct a detailed performance comparison between various process, device, and layout configurations, which is essential for pushing the limit of SPAD pixel miniaturization toward sub-2 μm-pitch SPADs.

2021

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Supervised and unsupervised kernel-based algorithms widely used in the physical sciences depend upon the notion of similarity. Their reliance on pre-defined distance metrics-e.g. the Euclidean or Manhattan distance-are problematic especially when used in combination with high-dimensional feature vectors for which the similarity measure does not well-reflect the differences in the target property. Metric learning is an elegant approach to surmount this shortcoming and find a property-informed transformation of the feature space. We propose a new algorithm for metric learning specifically adapted for kernel ridge regression (KRR): metric learning for kernel ridge regression (MLKRR). It is based on the Metric Learning for Kernel Regression framework using the Nadaraya-Watson estimator, which we show to be inferior to the KRR estimator for typical physics-based machine learning tasks. The MLKRR algorithm allows for superior predictive performance on the benchmark regression task of atomisation energies of QM9 molecules, as well as generating more meaningful low-dimensional projections of the modified feature space.

IOP Publishing Ltd2022

Electron quantization in arbitrarily shaped gold islands on MgO thin films

Wolf-Dieter Schneider

Low-temperature scanning tunneling microscopy has been employed to analyze the formation of quantum well states (QWS) in two-dimensional gold islands, containing between 50 and 200 atoms, on MgO thin films. The energy position and symmetry of the eigenstates are revealed from conductance spectroscopy and imaging. The majority of the QWS originates from overlapping Au 6p orbitals in the individual atoms and is unoccupied. Their characteristic is already reproduced with simple particle-in-a-box models that account for the symmetry of the islands (rectangular, triangular, or linear). However, better agreement is achieved when considering the true atomic structure of the aggregates via a density functional tight-binding approach. Based on a statistically relevant number of single-island data, we have established a correlation between the island geometry and the gap between the highest-occupied and the lowest-unoccupied molecular orbital in the finite-sized islands. The linear eccentricity is identified as a suitable descriptor for this relationship, as it combines information on both island size and island shape. Finally, the depth of the confinement potential is determined from the spatial extension of QWS beyond the physical boundaries of the Au islands. Our paper demonstrates how electron quantization effects can be analyzed in detail in metal nanostructures. The results may help elucidating the interplay between electronic and chemical properties of oxide-supported clusters as used in heterogeneous catalysis.

Amer Physical Soc2013