Latent space models play an important role in the modeling and analysis of network data. Under these models, each node has an associated latent point in some (typically low-dimensional) geometric space, and network formation is driven by this unobserved geometric structure. The random dot product graph (RDPG) and its generalization (GRDPG) are latent space models under which this latent geometry is taken to be Euclidean. These latent vectors can be efficiently and accurately estimated using well-studied spectral embeddings. In this paper, we develop a minimax lower bound for estimating the latent positions in the RDPG and the GRDPG models under the two-to-infinity norm, and show that a particular spectral embedding method achieves this lower bound. We also derive a minimax lower bound for the related task of subspace estimation under the two-to-infinity norm that holds in general for low-rank plus noise network models, of which the RDPG and GRDPG are special cases. The lower bounds are achieved by a novel construction based on Hadamard matrices.

翻译：潜变量模型在网络数据的建模与分析中发挥着重要作用。在该类模型下，每个节点在某个（通常为低维）几何空间中关联一个潜变量点，网络的形成由这种未观测的几何结构驱动。随机点积图（RDPG）及其推广形式（GRDPG）是潜变量模型的特例，其中潜变量几何结构被设定为欧几里得空间。这些潜变量向量可通过成熟的谱嵌入方法实现高效且精确的估计。本文针对RDPG和GRDPG模型在二至无穷范数下的潜变量位置估计问题，推导了极小极大下界，并证明特定谱嵌入方法能够达到该下界。此外，本文还针对子空间估计这一相关任务，在二至无穷范数下推导了适用于低秩加噪声网络模型（RDPG与GRDPG为其特例）的极小极大下界。上述下界的构建采用了基于阿达马矩阵的新颖方法。