Detecting approximate replicate components of a high-dimensional random vector with latent structure

Abstract

High-dimensional feature vectors are likely to contain sets of measurements that are approximate replicates of one another. In complex applications, or automated data collection, these feature sets are not known a priori, and need to be determined.

This work proposes a class of latent factor models on the observed, high-dimensional, random vector $X\in {\mathbb{R}}^p$, for defining, identifying and estimating the index set of its approximately replicate components. The model class is parametrized by a $p\times K$ loading matrix A that contains a hidden sub-matrix whose rows can be partitioned into groups of parallel vectors. Under this model class, a set of approximate replicate components of X corresponds to a set of parallel rows in A: these entries of X are, up to scale and additive error, the same linear combination of the K latent factors; the value of K is itself unknown.

The problem of finding approximate replicates in X reduces to identifying, and estimating, the location of the hidden sub-matrix within A, and of the partition $\mathcal{H}$ of its row index set H. Both H and $\mathcal{H}$ can be fully characterized in terms of a new family of criteria based on the correlation matrix of X, and their identifiability, as well as that of the unknown latent dimension K, are obtained as consequences. The constructive nature of the identifiability arguments enables computationally efficient procedures, with consistency guarantees.

Furthermore, when the loading matrix A has a particular sparse structure, provided by the errors-in-variable parametrization, the difficulty of the problem is elevated. The task becomes that of separating out groups of parallel rows that are proportional to canonical basis vectors from other, possibly dense, parallel rows in A. This is met under a scale assumption, via a principled way of selecting the target row indices, guided by the successive maximization of Schur complements of appropriate covariance matrices. The resulting procedure is an enhanced version of that developed for recovering general parallel rows in A. It is also computationally efficient, consistent. It has immediate applications to latent space overlapping clustering and the estimation of loading matrices that satisfy a canonical parametrization.