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Studying the mechanisms underlying the genotype-phenotype association is crucial in genetics. Gene expression studies have deepened our understanding of the genotype  →  expression  →  phenotype mechanisms. However, traditional expression quantitative trait loci (eQTL) methods often overlook the critical role of gene co-expression networks in translating genotype into phenotype. This gap highlights the need for more powerful statistical methods to analyze genotype  →  network  →  phenotype mechanism. Here, we develop a network-based method, called spectral network quantitative trait loci analysis (snQTL), to map quantitative trait loci affecting gene co-expression networks. Our approach tests the association between genotypes and joint differential networks of gene co-expression via a tensor-based spectral statistics, thereby overcoming the ubiquitous multiple testing challenges in existing methods. We demonstrate the effectiveness of snQTL in the analysis of three-spined stickleback Gasterosteus aculeatus data. Compared to conventional methods, our method snQTL uncovers chromosomal regions affecting gene co-expression networks, including one strong candidate gene that would have been missed by traditional eQTL analyses. Our framework suggests the limitation of current approaches and offers a powerful network-based tool for functional loci discoveries. 

We consider the problem of structured tensor denoising in the presence of unknown permutations. Such data problems arise commonly in recommendation systems, neuroimaging, community detection, and multiway comparison applications. Here, we develop a general family of smooth tensor models up to arbitrary index permutations; the model incorporates the popular tensor block models and Lipschitz hypergraphon models as special cases. We show that a constrained least-squares estimator in the block-wise polynomial family achieves the minimax error bound. A phase transition phenomenon is revealed with respect to the smoothness threshold needed for optimal recovery. In particular, we find that a polynomial of degree up to (𝑚−2)⁢(𝑚+1)/2 is sufficient for accurate recovery of order-m tensors, whereas higher degrees exhibit no further benefits. This phenomenon reveals the intrinsic distinction for smooth tensor estimation problems with and without unknown permutations. Furthermore, we provide an efficient polynomial-time Borda count algorithm that provably achieves the optimal rate under monotonicity assumptions. The efficacy of our procedure is demonstrated through both simulations and Chicago crime data analysis. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work. 

We consider the problem of multiway clustering in the presence of unknown degree​ ​heterogeneity. Such data problems arise commonly in applications such as recommendation systems, neuroimaging, community detection, and hypergraph partitions​ ​in social networks. The allowance of degree heterogeneity provides great flexibility​ ​in clustering models, but the extra complexity poses significant challenges in both​ ​statistics and computation. Here, we develop a degree-corrected tensor block model​ ​with estimation accuracy guarantees. We present the phase transition of clustering​ ​performance based on the notion of angle separability, and we characterize three​ ​signal-to-noise regimes corresponding to different statistical-computational behaviors.​ ​In particular, we demonstrate that an intrinsic statistical-to-computational gap emerges​ ​only for tensors of order three or greater.​ ​Further, we develop an efficient polynomial time algorithm that provably achieves exact​ ​clustering under mild signal conditions. The​ ​efficacy of our procedure is demonstrated​ ​through both simulations and analyses of​ ​Peru Legislation dataset.