Two salient limitations have lengthy hindered the relevance of optimum transport strategies to machine studying. First, the computational value of normal sample-based solvers (when used on batches of samples) is prohibitive. Second, the mass conservation constraint makes OT solvers too inflexible in observe: as a result of they have to match textit{all} factors from each measures, their output could be closely influenced by outliers. A flurry of latest works has addressed these computational and modeling limitations. Nonetheless it has resulted in two separate strains of strategies: Whereas the computational outlook was a lot improved by entropic regularization, more moderen linear-time textit{low-rank} solvers maintain the promise to scale up OT additional. By way of modeling flexibility, the rigidity of mass conservation has been eased for entropic regularized OT because of unbalanced variants of OT that may penalize couplings whose marginals deviate from these specified by the supply and goal distributions. The purpose of this paper is to merge these two strains, low-rank and unbalanced, to realize the promise of solvers which might be each scalable and versatile. We suggest customized algorithms to implement these extensions for the linear OT downside and its fused-Gromov-Wasserstein generalization, and exhibit their sensible relevance to difficult spatial transcriptomics matching issues. These algorithms are applied within the ott-jax toolbox.