Rotation Averaging is a non-convex optimization problem that determines
orientations of a collection of cameras from their images of a 3D scene. The
problem has been studied using a variety of distances and robustifiers. The
intrinsic (or geodesic) distance on SO(3) is geometrically meaningful; but
while some extrinsic distance-based solvers admit (conditional) guarantees of
correctness, no comparable results have been found under the intrinsic metric.
In this paper, we study the spatial distribution of local minima. First, we
do a novel empirical study to demonstrate sharp transitions in qualitative
behavior: as problems become noisier, they transition from a single
(easy-to-find) dominant minimum to a cost surface filled with minima. In the
second part of this paper we derive a theoretical bound for when this
transition occurs. This is an extension of the results of [24], which used
local convexity as a proxy to study the difficulty of problem. By recognizing
the underlying quotient manifold geometry of the problem we achieve an n-fold
improvement over prior work. Incidentally, our analysis also extends the prior
$l_2$ work to general $l_p$ costs. Our results suggest using algebraic
connectivity as an indicator of problem difficulty.

本文研究的问题是通过相机拍摄的三维场景图像确定相机方向的非凸优化问题，主要关注旋转平均的空间分布和局部极小值以及使用代数连通性作为问题难度的指标。

围绕内在度量旋转平均值的最小值分布

On the Distribution of Minima in Intrinsic-Metric Rotation Averaging

This paper shows how Graph Neural Networks can be used for learning
distributed coordination mechanisms in connected teams of robots. We capture
the relational aspect of robot coordination by modeling the robot team as a
graph, where each robot is a node, and edges represent communication links.
During training, robots learn how to pass messages and update internal states,
so that a target behavior is reached. As a proxy for more complex problems,
this short paper considers the problem where each robot must locally estimate
the algebraic connectivity of the team's network topology.

本文展示了如何利用图神经网络来学习连接的机器人团队的分布式协调机制，并通过将机器人队伍建模为图形来捕捉机器人协调的关系。训练过程中，机器人学习如何传递信息和更新内部状态，以达到目标行为，然后通过局部估计团队网络拓扑的代理方法来考虑更复杂的问题。

用于学习机器人团队协调的图神经网络

Graph Neural Networks for Learning Robot Team Coordination

This article deals with the spectra of Laplacians of weighted graphs. In this
context, two objects are of fundamental importance for the dynamics of complex
networks: the second eigenvalue of such a spectrum (called algebraic
connectivity) and its associated eigenvector, the so-called Fiedler vector.
Here we prove that, given a Laplacian matrix, it is possible to perturb the
weights of the existing edges in the underlying graph in order to obtain simple
eigenvalues and a Fiedler vector composed of only non-zero entries. These
structural genericity properties with the constraint of not adding edges in the
underlying graph are stronger than the classical ones, for which arbitrary
structural perturbations are allowed. These results open the opportunity to
understand the impact of structural changes on the dynamics of complex systems.

研究 Laplacian 加权图的频谱，证明可以变化权重而得到简单特征值和 Fiedler 向量，这与经典的结构变动研究不同，打开了理解复杂系统动力学影响的机会。