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A graph $mathcal{G}$ is referred to as $mathsf{S}^1$-synchronizing if, roughly speaking, the Kuramoto-like model whose interaction topology is given by $mathcal{G}$ synchronizes almost globally. The Kuramoto model evolves on the unit circle, ie the $1$-sphere $mathsf{S}^1$. This paper concerns generalizations of the Kuramoto-like model and the concept of synchronizing graphs on the Stiefel manifold $mathsf{St}(p,n)$. Previous work on state-space oscillators have largely been influenced by results and techniques that pertain to the $mathsf{S}^1$-case. It has recently been shown that all connected graphs are $mathsf{S}^n$-synchronizing for all $ngeq2$. The previous point of departure may thus have been overly conservative. The $n$-sphere is a special case of the Stiefel manifold, namely $mathsf{St}(1,n+1)$. As such, it is natural to ask for the extent to which the results on $mathsf{S}^{n}$ can be extended to the Stiefel manifold. This paper shows that all connected graphs are $mathsf{St}(p,n)$-synchronizing provided the pair $(p,n)$ satisfies $pleq tfrac{2n}{3}-1$.
The symplectic Stiefel manifold, denoted by $mathrm{Sp}(2p,2n)$, is the set of linear symplectic maps between the standard symplectic spaces $mathbb{R}^{2p}$ and $mathbb{R}^{2n}$. When $p=n$, it reduces to the well-known set of $2ntimes 2n$ symplecti
Riemannian optimization has drawn a lot of attention due to its wide applications in practice. Riemannian stochastic first-order algorithms have been studied in the literature to solve large-scale machine learning problems over Riemannian manifolds.
Strictly enforcing orthonormality constraints on parameter matrices has been shown advantageous in deep learning. This amounts to Riemannian optimization on the Stiefel manifold, which, however, is computationally expensive. To address this challenge
In this paper, we consider the geometric landscape connection of the widely studied manifold and factorization formulations in low-rank positive semidefinite (PSD) and general matrix optimization. We establish an equivalence on the set of first-order
This paper introduces a framework for solving time-autonomous nonlinear infinite horizon optimal control problems, under the assumption that all minimizers satisfy Pontryagins necessary optimality conditions. In detail, we use methods from the field