Trans-acting expression quantitative trait loci (trans-eQTLs) account for ≥70% expression heritability and could therefore facilitate uncovering mechanisms underlying the origination of complex diseases. Identifying trans-eQTLs is challenging because of small effect sizes, tissue specificity, and a severe multiple-testing burden. Tejaas predicts trans-eQTLs by performing L2-regularized "reverse" multiple regression of each SNP on all genes, aggregating evidence from many small trans-effects while being unaffected by the strong expression correlations. Combined with a novel unsupervised k-nearest neighbor method to remove confounders, Tejaas predicts 18851 unique trans-eQTLs across 49 tissues from GTEx. They are enriched in open chromatin, enhancers, and other regulatory regions. Many overlap with disease-associated SNPs, pointing to tissue-specific transcriptional regulation mechanisms.