Formal Proofs of Orthogonality for Class-Incremental Learning for Wireless Device Identification in IoT

<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>

Formal Proofs of Orthogonality for Class-Incremental Learning for Wireless Device Identification in IoT

<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>

Some Results on the Norm Attainment Set for Bounded Linear Operators on Smooth Banach Spaces

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.

Mappings preserving approximate orthogonality in Hilbert $C^*$-modules

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.