用扩散算子分析神经网络层间特征几何，揭示类别间关系演化规律。
Diffusion Operator Geometry of Feedforward Representations
- 构建高斯核马尔可夫算子描述类别间一步转移过程。
- 深层网络中类别传输更持久且结构关系保持稳定。
- 相比k近邻图，该方法在扰动下更鲁棒，适合研究模型内部表征。
前馈神经网络通过学习到的表示对数据进行变换，其几何结构决定了不同类别在各层间的分离与关联方式。本文通过扩散算子研究这一几何特性：为每个特征云快照分配一个高斯核马尔可夫算子，从而获得类别间单步迁移的平滑描述，从中可提取谱、边界及局部几何信息。我们定义了经验类链，并给出了其为精确马尔可夫商的条件，推导出对应总体转移概率和基于期望类别亲和度的简化重叠链。对于具有共享协方差的平衡高斯类条件快照，这些亲和度具有闭式表达，由正则化马氏距离控制，可显式给出泄漏率与粗粒度谱行为。进一步证明，算子可观测量在特征扰动下变化平滑，而硬邻域图受邻近顺序边际约束。在CIFAR-10和CIFAR-100的ResNet-18表示上实验发现，随着深度增加，类别传输逐渐变得持久，同时保持类间结构关系；且扩散类链在相同扰动下比k近邻版本更稳定。
原文摘要 · Abstract (English)
Feedforward neural networks transform data through learned representations whose geometry shapes how classes separate and relate across successive layers. We study that geometry through diffusion operators. Each feature-cloud snapshot is assigned a Gaussian-kernel Markov operator, giving a smooth description of one-step transport between classes from which spectral, boundary, and local geometric information can be read. We define the empirical class chain, state the condition under which it is an exact Markov quotient, and derive both the corresponding population transition and a simpler overlap chain based on expected class affinities. For balanced shared-covariance Gaussian class-conditional snapshots these affinities have closed forms controlled by a regularized Mahalanobis separation, which yields explicit expressions for leakage and coarse spectral behaviour. We further show that operator observables vary smoothly under feature perturbations, whereas hard neighborhood graphs are controlled by neighbor-order margins. Experiments on CIFAR-10 and CIFAR-100 ResNet-18 representations find that class transport becomes increasingly persistent with depth while retaining structured relations between classes, and that the diffusion class chain is more stable than its $k$-nearest-neighbor counterpart under matched perturbations.
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