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This work introduces a novel estimation method, called LOVE, of the entriesand structure of a loading matrix A in a sparse latent factor model X = AZ + E,for an observable random vector X in Rp, with correlated unobservable factors Z\in RK, with K unknown, and independent noise E. Each row of A is scaled andsparse. In order to identify the loading matrix A, we require the existence ofpure variables, which are components of X that are associated, via A, with oneand only one latent factor. Despite the fact that the number of factors K, thenumber of the pure variables, and their location are all unknown, we onlyrequire a mild condition on the covariance matrix of Z, and a minimum of onlytwo pure variables per latent factor to show that A is uniquely defined, up tosigned permutations. Our proofs for model identifiability are constructive, andlead to our novel estimation method of the number of factors and of the set ofpure variables, from a sample of size n of observations on X. This is the firststep of our LOVE algorithm, which is optimization-free, and has lowcomputational complexity of order p2. The second step of LOVE is an easilyimplementable linear program that estimates A. We prove that the resultingestimator is minimax rate optimal up to logarithmic factors in p. The modelstructure is motivated by the problem of overlapping variable clustering,ubiquitous in data science. We define the population level clusters as groupsof those components of X that are associated, via the sparse matrix A, with thesame unobservable latent factor, and multi-factor association is allowed.Clusters are respectively anchored by the pure variables, and form overlappingsub-groups of the p-dimensional random vector X. The Latent model approach toOVErlapping clustering is reflected in the name of our algorithm, LOVE.

Adaptive Estimation in Structured Factor Models with Applications to Overlapping Clustering