自适应调整图构建的带宽，提升小样本分类与标签传播性能。
Geometry-Aware Graph Construction via Adaptive Spectral Bandwidth Control
- 基于最小生成树估计局部内在维度，动态设定每节点带宽。
- 在CIFAR-100上，自适应带宽使留一法分类和标签传播准确率提升。
- 方法在不同编码器下均优于固定带宽与现有自适应方法。
基于核函数的图方法（如谱聚类、扩散映射、稀疏核回归图）依赖高斯带宽σ，其取值直接影响局部核算子的谱特性。当σ过小，核函数过度估计局部复杂度，将每个样本视为独立方向；当σ过大，多个方向被合并，条件数发散，几何区分能力丧失。本文提出一种自适应带宽选择策略，使核函数的谱复杂度与底层流形的内在复杂度一致。通过联合匹配核的有效秩与最小生成树估计的局部内在维度，在流形一致的对数-对数尺度区间内实现带宽优化。在六个编码器的CIFAR-100半监督嵌入上评估，自适应带宽在留一法分类与标签传播任务中持续优于固定带宽方法及现有自适应方法。
原文摘要 · Abstract (English)
Kernelized graph methods - spectral clustering, diffusion maps, and sparse kernel -regression graphs - that use Gaussian kernels depend on the choice of Gaussian bandwidth sigma, which governs the spectral character of the local kernel operator. When sigma is too small, the kernel overestimates local complexity and treats each sample as an independent direction; when sigma is too large, the kernel collapses multiple directions together, the condition number diverges, and all geometric discrimination is lost. We propose a choice of scale to make the spectral complexity of the kernel consistent with the intrinsic complexity of the underlying manifold. We propose a per-node bandwidth criterion that operationalizes this principle by jointly matching the kernel's effective rank to the local intrinsic dimension estimated via minimum spanning tree, anchoring the search in the manifold-consistent log-log scaling regime. We evaluate SSL embeddings from six encoders on CIFAR-100, showing that adaptive bandwidth consistently improves leave-one-out (LOO) classification and label propagation (LP) accuracy over fixed-bandwidth methods and competing adaptive methods.
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