用距离矩阵动态分析神经网络表征几何演化，区分扩散与相变两类训练过程。
Learning as Observable Matrix Dynamics: Diffusive Relaxations versus Phase Transitions
- 构建固定大小距离矩阵追踪输入表征变化，结合随机矩阵理论诊断几何演化。
- 发现扩散阶段无稳定谱结构，而突变阶段产生可重复的几何指纹。
- 适合研究模型训练机制、几何结构演变的学者，尤其关注表征动力学者。
可观测矩阵动力学（OMD）是一种诊断框架，通过在固定数量的N个输入上构建N×N的距离矩阵M(t)，探测神经网络内部高维表征的动力学特性。该方法结合随机矩阵理论与粒子动力学，揭示了传统标量损失函数忽略的谱重组现象，为训练过程提供深层洞察。通过扩展Bogomolny–Bohigas–Schmit（BBS）理论的环境-隐含分解，对每个快照进行谱顶带结构与环境噪声的诊断，同时引入轨迹级可观测量关联快照，并利用三维MDS嵌入（前三个特征向量）将训练过程可视化为移动粒子云。在七组实验中，扩散区域缺乏稳定的谱顶带结构，而内生或外驱的剧烈重组则形成稳定指纹：对应于平滑或乘积型隐含几何结构（在接近BBS情形下），或有限簇状、傅里叶孤子结构（其他情形）。因此，OMD识别的是表示空间的几何相位，而非单一内在维度。
原文摘要 · Abstract (English)
Observable Matrix Dynamics (OMD) is a diagnostic framework that probes the dynamics of high-dimensional internal representations of inputs by a neural network via a fixed-size $N \times N$ distance matrix $M(t)$ on a held set of $N$ inputs. OMD uses methods of random matrix theory and particle dynamics to explore spectral reorganisations that are missed by scalar loss functions, but are informative of the training process. We read $M(t)$ against a perturbative ambient-versus-latent decomposition extending the Bogomolny--Bohigas--Schmit (BBS) theory of random distance matrices, with per-snapshot diagnostics for the top-of-spectrum band structure and ambient noise, trajectory-level observables linking snapshots, and a 3D MDS embedding (bottom-three eigenvectors) rendering training as a moving particle cloud. Across seven experiments, diffusive regimes lack stable top-of-spectrum band structure, while sharp endogenous or externally driven reorganisations produce stable fingerprints: consistent with smooth or product latent geometries in BBS-adjacent cases, and with finite-cluster or Fourier-soliton structures otherwise. OMD thus reads the geometric regime of a representation rather than reporting a single intrinsic dimension.
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