Analyses of metagenomic datasets that are sequenced to a depth of billions or trillions of bases can uncover hundreds of microbial genomes, but naive assembly of these data is computationally intensive, requiring hundreds of gigabytes to terabytes of RAM. We present latent strain analysis (LSA), a scalable, de novo pre-assembly method that separates reads into biologically informed partitions and thereby enables assembly of individual genomes. LSA is implemented with a streaming calculation of unobserved variables that we call eigengenomes. Eigengenomes reflect covariance in the abundance of short, fixed-length sequences, or k-mers. As the abundance of each genome in a sample is reflected in the abundance of each k-mer in that genome, eigengenome analysis can be used to partition reads from different genomes. This partitioning can be done in fixed memory using tens of gigabytes of RAM, which makes assembly and downstream analyses of terabytes of data feasible on commodity hardware. Using LSA, we assemble partial and near-complete genomes of bacterial taxa present at relative abundances as low as 0.00001%. We also show that LSA is sensitive enough to separate reads from several strains of the same species.