Standard Adjacency Spectral Embedding (ASE) relies on a global low-rank assumption often incompatible with the sparse, transitive structure of real-world networks, causing local geometric features to be 'smeared'. To address this, we introduce Local Adjacency Spectral Embedding (LASE), which uncovers locally low-dimensional structure via weighted spectral decomposition. Under a latent position model with a kernel feature map, we treat the image of latent positions as a locally low-dimensional set in infinite-dimensional feature space. We establish finite-sample bounds quantifying the trade-off between the statistical cost of localisation and the reduced truncation error achieved by targeting a locally low-dimensional region of the embedding. Furthermore, we prove that sufficient localisation induces rapid spectral decay and the emergence of a distinct spectral gap, theoretically justifying low-dimensional local embeddings. Experiments on synthetic and real networks show that LASE improves local reconstruction and visualisation over global and subgraph baselines, and we introduce UMAP-LASE for assembling overlapping local embeddings into high-fidelity global visualisations.

翻译：标准的邻接谱嵌入（ASE）依赖于全局低秩假设，该假设常与真实世界网络的稀疏性、传递性结构不相容，导致局部几何特征被“模糊化”。为解决此问题，我们提出了局部邻接谱嵌入（LASE），该方法通过加权谱分解来揭示局部低维结构。在具有核特征映射的潜在位置模型下，我们将潜在位置的像视为无限维特征空间中的一个局部低维集合。我们建立了有限样本界，量化了局部化带来的统计成本与通过针对嵌入的局部低维区域所实现的截断误差减少之间的权衡。此外，我们证明了充分的局部化会诱导快速的谱衰减和显著谱间隙的出现，从而从理论上为低维局部嵌入提供了依据。在合成网络和真实网络上的实验表明，LASE在局部重建和可视化方面优于全局及子图基线方法，并且我们提出了UMAP-LASE，用于将重叠的局部嵌入组装成高保真度的全局可视化。