发现多种优化问题中临界点普遍存在对称性，揭示深层结构规律。
Ubiquitous Symmetry at Critical Points Across Diverse Optimization Landscapes
- 分析四类新优化场景中的对称性现象
- 所有临界点均表现出非平凡对称性
- 提出新对称性度量，发现隐藏结构
对称性在理解数学结构与优化问题中起关键作用。近期研究发现神经网络损失函数在权重行列置换下不变，局部极小点展现出显著的权重对称性，且未发现无对称性的临界点。本文将此研究扩展至更广泛的实值损失函数空间，引入四个新案例：有限域上的射影情形、八面体图情形、完美匹配情形及粒子吸引情形。结果表明，与神经网络类似，所有观测到的临界点均具有非平凡对称性。最后，我们提出一种新的对称性度量，揭示了先前度量未能捕捉的额外对称结构。
原文摘要 · Abstract (English)
Symmetry plays a crucial role in understanding the properties of mathematical structures and optimization problems. Recent work has explored this phenomenon in the context of neural networks, where the loss function is invariant under column and row permutations of the network weights. It has been observed that local minima exhibit significant symmetry with respect to the network weights (invariance to row and column permutations). And moreover no critical point was found that lacked symmetry. We extend this line of inquiry by investigating symmetry phenomena in real-valued loss functions defined on a broader class of spaces. We will introduce four more cases: the projective case over a finite field, the octahedral graph case, the perfect matching case, and the particle attraction case. We show that as in the neural network case, all the critical points observed have non-trivial symmetry. Finally we introduce a new measure of symmetry in the system and show that it reveals additional symmetry structures not captured by the previous measure.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。