The Iwasawa decomposition and the Bruhat decomposition of the automorphism group on certain exceptional Jordan algebra

Let $\mathcal{J}^1$ be the real form of a complex simple Jordan algebra such that the automorphism group is $\mathrm{F}_{4(-20)}$. By using some orbit types of $\mathrm{F}_{4(-20)}$ on $\mathcal{J}^1$, for $\mathrm{F}_{4(-20)}$, explicitly, we give the Iwasawa decomposition, the Oshima--Sekiguchi's $K_{\epsilon}-$Iwasawa decomposition, the Matsuki decomposition, and the Bruhat and Gauss decompositions.Comment: v3, 30 pages, 1 figure, major changes from v2 by separating section 9--14, reformatted, and the title changed to appear in Tsukuba J. of Mat

Orbit decomposition of Jordan matrix algebras of order three under the automorphism groups

The orbit decomposition is given under the automorphism group on the real split Jordan algebra of all hermitian matrices of order three corresponding to any real split composition algebra, or the automorphism group on the complexification, explicitly, in terms of the cross product of H. Freudenthal and the characteristic polynomial.Comment: v2, 32 pages, presentation improved, minor errors corrected, and the title changed as appeared in J. Math. Sci. Univ. Toky

Client Selection for Federated Learning with Heterogeneous Resources in Mobile Edge

We envision a mobile edge computing (MEC) framework for machine learning (ML) technologies, which leverages distributed client data and computation resources for training high-performance ML models while preserving client privacy. Toward this future goal, this work aims to extend Federated Learning (FL), a decentralized learning framework that enables privacy-preserving training of models, to work with heterogeneous clients in a practical cellular network. The FL protocol iteratively asks random clients to download a trainable model from a server, update it with own data, and upload the updated model to the server, while asking the server to aggregate multiple client updates to further improve the model. While clients in this protocol are free from disclosing own private data, the overall training process can become inefficient when some clients are with limited computational resources (i.e. requiring longer update time) or under poor wireless channel conditions (longer upload time). Our new FL protocol, which we refer to as FedCS, mitigates this problem and performs FL efficiently while actively managing clients based on their resource conditions. Specifically, FedCS solves a client selection problem with resource constraints, which allows the server to aggregate as many client updates as possible and to accelerate performance improvement in ML models. We conducted an experimental evaluation using publicly-available large-scale image datasets to train deep neural networks on MEC environment simulations. The experimental results show that FedCS is able to complete its training process in a significantly shorter time compared to the original FL protocol

Stabilization of Stochastic Quantum Dynamics via Open and Closed Loop Control

In this paper we investigate parametrization-free solutions of the problem of quantum pure state preparation and subspace stabilization by means of Hamiltonian control, continuous measurement and quantum feedback, in the presence of a Markovian environment. In particular, we show that whenever suitable dissipative effects are induced either by the unmonitored environment or by non Hermitian measurements, there is no need for feedback control to accomplish the task. Constructive necessary and sufficient conditions on the form of the open-loop controller can be provided in this case. When open-loop control is not sufficient, filtering-based feedback control laws steering the evolution towards a target pure state are provided, which generalize those available in the literature