Optimal transport (OT) and Gromov-Wasserstein (GW) alignment provide interpretable geometric frameworks for comparing, transforming, and aggregating heterogeneous datasets -- tasks ubiquitous in data science and machine learning. Because these frameworks are computationally expensive, large-scale applications often rely on closed-form solutions for Gaussian distributions under quadratic cost. This work provides a comprehensive treatment of Gaussian, quadratic cost OT and inner product GW (IGW) alignment, closing several gaps in the literature to broaden applicability. First, we treat the open problem of IGW alignment between uncentered Gaussians on separable Hilbert spaces by giving a closed-form expression up to a quadratic optimization over unitary operators, for which we derive tight analytic upper and lower bounds. If at least one Gaussian measure is centered, the solution reduces to a fully closed-form expression, which we further extend to an analytic solution for the IGW barycenter between centered Gaussians. We also present a reduction of Gaussian multimarginal OT with pairwise quadratic costs to a tractable optimization problem and provide an efficient algorithm to solve it using a rank-deficiency constraint. To demonstrate utility, we apply our results to knowledge distillation and heterogeneous clustering on synthetic and real-world datasets.

We study the complexity of approximating the multimarginal optimal transport (OT) problem, a generalization of the classical optimal transport distance, considered here between $m$ discrete probability distributions supported each on $n$ support points. First, we show that the multimarginal OT problem is not a minimum-cost flow problem when $m \geq 3$. This implies that many of the combinatorial algorithms developed for classical OT are not applicable to multimarginal OT, and therefore the standard interior-point algorithm bounds result in an intractable complexity bound of $\widetilde{\mathcal{O}}(n^{3m})$. Second, we propose and analyze two simple algorithms for approximating the multimarginal OT problem. The first algorithm, which we refer to as multimarginal Sinkhorn, improves upon previous multimarginal generalizations of the celebrated Sinkhorn algorithm. We show that it achieves a near-linear time complexity bound of $\widetilde{\mathcal{O}}(m^3 n^m / \varepsilon^2)$ for a tolerance $\varepsilon \in (0, 1)$. This matches the best known complexity bound for the Sinkhorn algorithm when $m = 2$ for approximating the classical OT distance. The second algorithm, which we refer to as multimarginal Randkhorn, accelerates the first algorithm by incorporating a randomized estimate sequence and achieves a complexity bound of $\widetilde{\mathcal{O}}(m^{8/3} n^{m+1/3}/\varepsilon)$. This improves on the complexity bound of the first algorithm by $1/\varepsilon$ and matches the best known complexity bound for the Randkhorn algorithm when $m=2$ for approximating the classical OT distance.