Alexander Van Werde's contribution to the Discussion of ‘Vintage Factor Analysis with Varimax Performs Statistical Inference’ by Rohe & Zeng

原文摘要

I would like to point out that the author’s results have potential applications in the theory of random graphs beyond those for block models which are explicitly remarked upon in the paper. In particular, the results could be used to resolve the nonidentifiability associated with random dot product graphs (Athreya et al., 2017, Remark 1). The authors are most likely already aware of this connection: they have previously worked on generalized random dot product graphs in Rohe et al. (2018). Thus, I would like to invite the authors to share further insights concerning the connection between this work and random dot product graphs in their reply. Some concrete questions are as follows: Q1: Does the assumption of leptokurtic entries for the latent positions appear to be satisfied in real-world graphs? Q2: Are there other considerations which one should take into account when applying the theory of the current paper to random dot product graphs? For instance, are there situations where one should be careful? For the sake of the reader unfamiliar with the topic, I will now sketch the connection between random dot product graphs and the current paper. A random dot product graph with latent positions X1,…,Xn∈Rk is a random graph whose adjacency matrix A∈Rn×n has independent entries with Aij∼Bernoulli(⟨Xi,Xj⟩)⁠. Here, it should be assumed that the latent positions are such that ⟨Xi,Xj⟩∈[0,1] for all i,j∈{1,…,n}⁠. The key problem in the theory is to estimate the latent positions X1,…,Xn given an observation of A. There is however an unidentifiability in the model: for any orthogonal transformation R∈O(k) it holds that ⟨RXi,RXj⟩=⟨Xi,Xj⟩⁠. Thus, if no additional assumptions are made concerning the nature of the latent positions, then estimation is only possible up to an orthogonal transformation. If, however, it is further assumed that the latent positions are generated as Xi=BZi where Z1,…,Zn∈Rk are i.i.d. random vectors with independent and leptokurtic components of variance one and B∈Rk×k is a fixed matrix. Then, with Z:=[Z1,…,Zn]T it holds that which falls in the scope of the current paper. Hence, estimation of the Zi and the kernel BTB is possible up to permutations and sign flips. Viewing the Zi as the true latent positions may lead to more interpretable representations of the graph since there is less ambiguity and more sparsity. The corresponding random graph model, allowing for a kernel BTB⁠, is called a generalized random dot product graph.

摘要来源：OpenAlex