关于内度量旋转平均中极小值的分布
On the Distribution of Minima in Intrinsic-Metric Rotation Averaging
摘要：
旋转平均是一个非凸优化问题，它从3D场景的图像中确定相机集合的方向。这个问题已经研究了使用各种距离和鲁棒。SO（3）上的内在（或测地线）距离在几何上是有意义的;但是，虽然一些基于外在距离的求解器承认（有条件）保证正确性，但在内在度量下没有找到可比的结果。本文研究了局部极小点的空间分布。首先，我们做了一个新的实证研究来证明定性行为的急剧转变：随着问题变得越来越嘈杂，它们从一个单一的（容易找到的）占主导地位的最小值过渡到一个充满最小值的成本表面。在本文的第二部分，我们推导出一个理论界时，这种转变发生。本文推广了文[24]的结果，文[24]以局部凸性为代理来研究问题的难度。通过认识到潜在的商流形几何的问题，我们实现了n倍的改善，比以前的工作。顺便说一句，我们的分析还将先前的$l_2$工作扩展到一般的$l_p$成本。我们的研究结果表明，使用代数连接作为问题难度的指标。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
