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Abstract Many genetic models (including models for epistatic effects as well as genetic-by-environment) involve covariance structures that are Hadamard products of lower rank matrices. Implementing these models require factorizing large Hadamard product matrices. The available algorithms for factorization do not scale well for big data, making the use of some of these models not feasible with large sample sizes. Here, based on properties of Hadamard products and (related) Kronecker products we propose an algorithm that produces an approximate decomposition that is orders of magnitude faster than the standard eigenvalue decomposition. In this article, we describe the algorithm, show how it can be used to factorize large Hadamard product matrices, present benchmarks, and illustrate the use of the method by presenting an analysis of data from the northern testing locations of the G×E project from the Genomes-to-Fields Initiative (n∼60,000). We implemented the proposed algorithm in the open-source ‘tensorEVD’ R-package. 

Abstract Genotype-by-environment (G×E) interactions can significantly affect crop performance and stability. Investigating G×E requires extensive data sets with diverse cultivars tested over multiple locations and years. The Genomes-to-Fields (G2F) Initiative has tested maize hybrids in more than 130 year-locations in North America since 2014. Here, we curate and expand this data set by generating environmental covariates (using a crop model) for each of the trials. The resulting data set includes DNA genotypes and environmental data linked to more than 70,000 phenotypic records of grain yield and flowering traits for more than 4000 hybrids. We show how this valuable data set can serve as a benchmark in agricultural modeling and prediction, paving the way for countless G×E investigations in maize. We use multivariate analyses to characterize the data set’s genetic and environmental structure, study the association of key environmental factors with traits, and provide benchmarks using genomic prediction models.