approximate orthogonality
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2021 ◽  
Author(s):  
Yongxin Liu ◽  
Jian Wang ◽  
Jianqiang Li ◽  
Shuteng Niu ◽  
Houbing Song

<div> <div> <div> <p>This document provides a formal proof and supple- mentary information of the paper: Class-Incremental Learning for Wireless Device Identification in IoT. The original paper focuses on providing a novel and efficient incremental learning algorithm. In this document, we explicitly explain why the mem- ory representations (latent device fingerprints in our application) in Artificial Neural Networks approximate orthogonality with insights for the invention of our Channel Separation Incremental Learning algorithm. </p> </div> </div> </div>


2021 ◽  
Author(s):  
Yongxin Liu ◽  
Jian Wang ◽  
Jianqiang Li ◽  
Shuteng Niu ◽  
Houbing Song

<div> <div> <div> <p>This document provides a formal proof and supple- mentary information of the paper: Class-Incremental Learning for Wireless Device Identification in IoT. The original paper focuses on providing a novel and efficient incremental learning algorithm. In this document, we explicitly explain why the mem- ory representations (latent device fingerprints in our application) in Artificial Neural Networks approximate orthogonality with insights for the invention of our Channel Separation Incremental Learning algorithm. </p> </div> </div> </div>


2021 ◽  
pp. 35-44
Author(s):  
Saied A. Johnny ◽  
Buthainah A. A. Ahmed

In this paper, we give new results and proofs that include the notion of norm attainment set of bounded linear operators on a smooth Banach spaces and using these results to characterize a bounded linear operators on smooth Banach spaces that preserve of approximate - -orthogonality. Noting that this work takes brief sidetrack in terms of approximate - -orthogonality relations characterizations of a smooth Banach spaces. 


2019 ◽  
Vol 244 (1) ◽  
pp. 43-97
Author(s):  
Livio Flaminio ◽  
Krzysztof Frączek ◽  
Joanna Kułaga-Przymus ◽  
Mariusz Lemańczyk

2018 ◽  
Vol 122 (2) ◽  
pp. 257 ◽  
Author(s):  
Mohammad Sal Moslehian ◽  
Ali Zamani

We introduce a notion of approximate orthogonality preserving mappings between Hilbert $C^*$-modules. We define the concept of $(\delta , \varepsilon )$-orthogonality preserving mapping and give some sufficient conditions for a linear mapping to be $(\delta , \varepsilon )$-orthogonality preserving. In particular, if $\mathscr {E}$ is a full Hilbert $\mathscr {A}$-module with $\mathbb {K}(\mathscr {H})\subseteq \mathscr {A} \subseteq \mathbb {B}(\mathscr {H})$ and $T, S\colon \mathscr {E}\to \mathscr {E}$ are two linear mappings satisfying $|\langle Sx, Sy\rangle | = \|S\|^2|\langle x, y\rangle |$ for all $x, y\in \mathscr {E}$ and $\|T - S\| \leq \theta \|S\|$, then we show that $T$ is a $(\delta , \varepsilon )$-orthogonality preserving mapping. We also prove whenever $\mathbb {K}(\mathscr {H})\subseteq \mathscr {A} \subseteq \mathbb {B}(\mathscr {H})$ and $T\colon \mathscr {E} \to \mathscr {F}$ is a nonzero $\mathscr {A}$-linear $(\delta , \varepsilon )$-orthogonality preserving mapping between $\mathscr {A}$-modules, then \[ \bigl \|\langle Tx, Ty\rangle - \|T\|^2\langle x, y\rangle \bigr \|\leq \frac {4(\varepsilon - \delta )}{(1 - \delta )(1 + \varepsilon )} \|Tx\|\|Ty\|\qquad (x, y\in \mathscr {E}). \] As a result, we present some characterizations of the orthogonality preserving mappings.


2017 ◽  
Vol 531 ◽  
pp. 305-317 ◽  
Author(s):  
Jacek Chmieliński ◽  
Tomasz Stypuła ◽  
Paweł Wójcik

2011 ◽  
Vol 434 (9) ◽  
pp. 2089-2108 ◽  
Author(s):  
Christos Chorianopoulos ◽  
Panayiotis Psarrakos

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