# Angular Distance

The Experts below are selected from a list of 324 Experts worldwide ranked by ideXlab platform

### Ludwig Schmidt – One of the best experts on this subject based on the ideXlab platform.

• ##### NIPS – Practical and optimal LSH for AngularDistance
, 2015
Co-Authors: Alexandr Andoni, Piotr Indyk, Thijs Laarhoven, Ilya Razenshteyn, Ludwig Schmidt

Abstract:

We show the existence of a Locality-Sensitive Hashing (LSH) family for the Angular Distance that yields an approximate Near Neighbor Search algorithm with the asymptotically optimal running time exponent. Unlike earlier algorithms with this property (e.g., Spherical LSH [1, 2]), our algorithm is also practical, improving upon the well-studied hyperplane LSH [3] in practice. We also introduce a multiprobe version of this algorithm and conduct an experimental evaluation on real and synthetic data sets.

We complement the above positive results with a fine-grained lower bound for the quality of any LSH family for Angular Distance. Our lower bound implies that the above LSH family exhibits a trade-off between evaluation time and quality that is close to optimal for a natural class of LSH functions.

• ##### practical and optimal lsh for AngularDistance
Neural Information Processing Systems, 2015
Co-Authors: Alexandr Andoni, Piotr Indyk, Thijs Laarhoven, Ilya Razenshteyn, Ludwig Schmidt

Abstract:

We show the existence of a Locality-Sensitive Hashing (LSH) family for the Angular Distance that yields an approximate Near Neighbor Search algorithm with the asymptotically optimal running time exponent. Unlike earlier algorithms with this property (e.g., Spherical LSH [1, 2]), our algorithm is also practical, improving upon the well-studied hyperplane LSH [3] in practice. We also introduce a multiprobe version of this algorithm and conduct an experimental evaluation on real and synthetic data sets.

We complement the above positive results with a fine-grained lower bound for the quality of any LSH family for Angular Distance. Our lower bound implies that the above LSH family exhibits a trade-off between evaluation time and quality that is close to optimal for a natural class of LSH functions.

• ##### Practical and Optimal LSH for AngularDistance
arXiv: Data Structures and Algorithms, 2015
Co-Authors: Alexandr Andoni, Piotr Indyk, Thijs Laarhoven, Ilya Razenshteyn, Ludwig Schmidt

Abstract:

We show the existence of a Locality-Sensitive Hashing (LSH) family for the Angular Distance that yields an approximate Near Neighbor Search algorithm with the asymptotically optimal running time exponent. Unlike earlier algorithms with this property (e.g., Spherical LSH [Andoni, Indyk, Nguyen, Razenshteyn 2014], [Andoni, Razenshteyn 2015]), our algorithm is also practical, improving upon the well-studied hyperplane LSH [Charikar, 2002] in practice. We also introduce a multiprobe version of this algorithm, and conduct experimental evaluation on real and synthetic data sets.
We complement the above positive results with a fine-grained lower bound for the quality of any LSH family for Angular Distance. Our lower bound implies that the above LSH family exhibits a trade-off between evaluation time and quality that is close to optimal for a natural class of LSH functions.

### Mohammad Sal Moslehian – One of the best experts on this subject based on the ideXlab platform.

• ##### p-AngularDistance orthogonality
Aequationes Mathematicae, 2019
Co-Authors: Jamal Rooin, Setareh Rajabi, Mohammad Sal Moslehian

Abstract:

We present a new orthogonality which is based on p-Angular Distance in normed linear spaces. This orthogonality generalizes the Singer and isosceles orthogonalities to a vast extent. Some important properties of this orthogonality, such as the $$\alpha$$-existence and the $$\alpha$$-diagonal existence, are established with giving some natural bounds for $$\alpha$$. It is shown that a real normed linear space is an inner product space if and only if the p-Angular Distance orthogonality is either homogeneous or additive. Several examples are presented to illustrate the results.

• ##### Geometric aspects of $p$-Angular and skew $p$-AngularDistances
arXiv: Functional Analysis, 2017
Co-Authors: Jamal Rooin, S. Habibzadeh, Mohammad Sal Moslehian

Abstract:

Corresponding to the concept of $p$-Angular Distance $\alpha_p[x,y]:=\left\lVert\lVert x\rVert^{p-1}x-\lVert y\rVert^{p-1}y\right\rVert$, we first introduce the notion of skew $p$-Angular Distance $\beta_p[x,y]:=\left\lVert \lVert y\rVert^{p-1}x-\lVert x\rVert^{p-1}y\right\rVert$ for non-zero elements of $x, y$ in a real normed linear space and study some of significant geometric properties of the $p$-Angular and the skew $p$-Angular Distances. We then give some results comparing two different $p$-Angular Distances with each other. Finally, we present some characterizations of inner product spaces related to the $p$-Angular and the skew $p$-Angular Distances. In particular, we show that if $p>1$ is a real number, then a real normed space $\mathcal{X}$ is an inner product space, if and only if for any $x,y\in \mathcal{X}\smallsetminus{\lbrace 0\rbrace}$, it holds that $\alpha_p[x,y]\geq\beta_p[x,y]$.

• ##### Dunkl–Williams inequality for operators \\ associated with $p$-AngularDistance
arXiv: Operator Algebras, 2010

Abstract:

We present several operator versions of the Dunkl–Williams inequality with respect to the $p$-Angular Distance for operators. More precisely, we show that if $A, B \in \mathbb{B}(\mathscr{H})$ such that $|A|$ and $|B|$ are invertible, $\frac{1}{r}+\frac{1}{s}=1\,\,(r>1)$ and $p\in\mathbb{R}$, then \begin{equation*} |A|A|^{p-1}-B|B|^{p-1}|^{2} \leq |A|^{p-1}(r|A-B|^{2}+s||A|^{1-p}|B|^{p}-|B||^2)|A|^{p-1}.%\nonumber \end{equation*} In the case that $0 0$, then $$|(U|A|^{p}-V|B|^{p})|A|^{1-p}|^{2}\leq (1+t)|A-B|^{2}+(1+\frac{1}{t})||B|^{p}|A|^{1-p}-|B||^2 \,.$$ We obtain several equivalent conditions, when the case of equalities hold.

### Nynke Hofstra – One of the best experts on this subject based on the ideXlab platform.

• ##### spatial variability in correlation decay Distance and influence on AngularDistance weighting interpolation of daily precipitation over europe
International Journal of Climatology, 2009
Co-Authors: Nynke Hofstra, Mark New

Abstract:

AngularDistance weighting (ADW) is a common approach for interpolation of an irregular network of meteorological observations to a regular grid. A widely used version of ADW employs the correlation decay Distance (CDD) to (1) select stations that should contribute to each grid-point estimate and (2) define the Distance component of the station weights. We show, for Europe, that the CDD of daily precipitation varies spatially, as well as by season and synoptic state, and is also anisotropic. However, ADW interpolation using CDDs that varies spatially by season or synoptic state yield only small improvements in interpolation skill, relative to the use of a fixed CDD across the entire domain. If CDDs are optimized through cross validation, a larger improvement in interpolation skill is achieved. Improvements are larger for the determination of the state of precipitation (wet/dry) than for the magnitude. These or other attempts to improve interpolation skill appear to be fundamentally limited by the available station network

• ##### Spatial variability in correlation decay Distance and influence on Angular‐Distance weighting interpolation of daily precipitation over Europe
International Journal of Climatology, 2008
Co-Authors: Nynke Hofstra, Mark New

Abstract:

AngularDistance weighting (ADW) is a common approach for interpolation of an irregular network of meteorological observations to a regular grid. A widely used version of ADW employs the correlation decay Distance (CDD) to (1) select stations that should contribute to each grid-point estimate and (2) define the Distance component of the station weights. We show, for Europe, that the CDD of daily precipitation varies spatially, as well as by season and synoptic state, and is also anisotropic. However, ADW interpolation using CDDs that varies spatially by season or synoptic state yield only small improvements in interpolation skill, relative to the use of a fixed CDD across the entire domain. If CDDs are optimized through cross validation, a larger improvement in interpolation skill is achieved. Improvements are larger for the determination of the state of precipitation (wet/dry) than for the magnitude. These or other attempts to improve interpolation skill appear to be fundamentally limited by the available station network