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

Minimax rates for latent position estimation in the generalized random dot product graph