Faster Unbalanced Optimal Transport: Translation invariant Sinkhorn and 1-D Frank-Wolfe
Proposes translation invariant Sinkhorn and 1-D Frank-Wolfe algorithms for faster unbalanced optimal transport, demonstrating over 2x speedup in experiments.
Key Findings
Methodology
This work begins by analyzing the slow convergence of unbalanced optimal transport (UOT) with traditional Sinkhorn algorithms, pinpointing the lack of a global normalization of iterates as the root cause. To address this, the authors introduce a translation invariant dual formulation, defining the dual functions Gε and Hε, which incorporate a translation parameter λ optimized via convex maximization. This approach effectively removes the sensitivity to potential shifts, leading to accelerated convergence. For one-dimensional UOT, a Frank-Wolfe based solver is developed, where each iteration reduces to solving a linearized balanced OT problem in 1-D, achievable in linear time by leveraging sorted data. The method is further extended to compute UOT barycenters by formulating a convex multi-marginal problem that exploits the geometric structure of the support points, enabling efficient iterative solutions. Extensive numerical experiments validate the faster convergence and computational efficiency of these methods across synthetic and real datasets.
Key Results
- On synthetic 1-D data, the translation invariant Sinkhorn algorithms (Gε and Hε variants) outperform standard Sinkhorn by over 2x in convergence speed, especially when ε is much smaller than ρ. In real single-cell datasets, the proposed methods reduce total computation time by approximately 40%. The Frank-Wolfe solver for 1-D UOT achieves linear complexity, handling large datasets efficiently. The UOT barycenter algorithm converges rapidly, with computation times halved compared to classical multi-marginal approaches. Empirical results show that the Hε variant converges faster than Fε and Gε, with improvements of over 30% in convergence rate, confirming the theoretical guarantees.
- Furthermore, the introduction of the translation invariant dual functions Gε and Hε significantly enhances the stability and robustness of the algorithms. The experimental results across various regularization parameters, data sizes, and dimensions demonstrate consistent speedups and accuracy improvements, establishing these methods as practical tools for large-scale non-balanced optimal transport problems.
Significance
This research addresses a fundamental bottleneck in non-balanced optimal transport algorithms, providing a theoretically grounded and practically efficient solution. By removing the sensitivity to potential shifts, the proposed translation invariant framework ensures faster convergence, making UOT feasible for large-scale and real-time applications in machine learning, image processing, and computational biology. The linear-time solver for 1-D problems opens new avenues for high-dimensional extensions, while the barycenter computation facilitates advanced data aggregation tasks. These innovations significantly expand the applicability of OT methods, enabling more robust and scalable data matching, domain adaptation, and generative modeling, especially in scenarios with complex mass variations and outliers. Overall, this work bridges the gap between theoretical optimal transport and practical large-scale deployment.
Technical Contribution
The key technical innovation lies in the formulation of translation invariant dual functions Gε and Hε, which incorporate a maximization over a translation parameter λ, ensuring the dual problem's invariance under potential shifts. This leads to algorithms with improved convergence rates, as proven theoretically and verified empirically. For the 1-D case, the authors leverage the monotonicity of optimal transport plans, reducing each iteration to a linear time problem solvable via sorted data and simple linear optimization. The Frank-Wolfe method applied to the invariant dual functional H0 provides an efficient, projection-free approach with linear complexity, suitable for large datasets. The extension to barycenters involves solving a convex multi-marginal problem with a cost function derived from the geometric support points, enabling fast iterative updates. These methods collectively push the state-of-the-art in scalable, robust non-balanced OT algorithms.
Novelty
This work is pioneering in introducing the concept of translation invariance into the dual formulation of UOT, directly addressing the slow convergence caused by potential shifts. Unlike previous approaches relying solely on regularization or heuristic normalization, the proposed framework explicitly optimizes over the translation parameter, leading to provable acceleration. The linear-time Frank-Wolfe solver for 1-D UOT is a novel contribution, exploiting the structure of the problem to achieve computational efficiency. Additionally, the extension to barycenter computation in one dimension using convex multi-marginal formulations represents a significant advancement, enabling fast approximation of complex distribution averages. These innovations collectively constitute a new paradigm for scalable, stable non-balanced OT algorithms.
Limitations
- While the proposed translation invariant algorithms perform well in low-dimensional settings, their extension to high-dimensional spaces remains challenging due to increased computational complexity in optimizing the translation parameters and potential numerical instability. The Frank-Wolfe solver's efficiency heavily relies on the monotonicity and structure of 1-D problems, limiting its direct applicability in higher dimensions without significant modifications. Additionally, the algorithms' performance under extremely small ε values or highly irregular data distributions has not been fully explored, raising questions about robustness in edge cases. Future work is needed to adapt these methods for broader classes of problems and to analyze their theoretical convergence guarantees in more complex scenarios.
Future Work
Future research will focus on extending the translation invariant framework to high-dimensional and nonlinear settings, possibly through embedding techniques or kernel methods. Integrating these algorithms with deep neural networks could enable end-to-end learning of complex data distributions with mass variations. Developing adaptive schemes for selecting regularization parameters and translation updates will further improve robustness and convergence speed. Additionally, exploring applications in large-scale image and video processing, as well as in biological data analysis, will demonstrate the practical impact of these methods. Theoretical work on convergence guarantees in non-convex or non-monotonic scenarios will also be a priority, aiming to broaden the scope and reliability of the proposed algorithms.
AI Executive Summary
Optimal transport (OT) has become a cornerstone in machine learning and data analysis, providing a principled way to compare probability distributions. Traditional balanced OT algorithms, such as Sinkhorn, have achieved remarkable success due to their computational efficiency and parallelizability. However, when extended to unbalanced optimal transport (UOT), which allows for mass creation and destruction, these algorithms often suffer from slow convergence, especially in the presence of small regularization parameters. This bottleneck limits the scalability and practical deployment of UOT in large-scale applications like single-cell genomics, image registration, and domain adaptation.
Recognizing this challenge, the authors of this paper introduce a novel framework based on the concept of translation invariance. They observe that the dual potentials in UOT are sensitive to arbitrary shifts, which hampers convergence. To address this, they formulate a translation invariant dual problem by explicitly optimizing over a translation parameter λ, leading to the definitions of the dual functions Gε and Hε. These functions are designed to be invariant under potential shifts, effectively removing the root cause of slow convergence. The authors develop accelerated algorithms—Gε-Sinkhorn and Hε-Sinkhorn—that alternate between maximizing over the dual variables and the translation parameter, achieving faster convergence rates validated both theoretically and empirically.
In addition to the general multi-dimensional case, the paper makes a significant contribution by focusing on the one-dimensional UOT problem. Leveraging the monotonicity of optimal transport plans in 1-D, the authors propose a Frank-Wolfe based solver that reduces each iteration to solving a simple linear problem in sorted data, resulting in an O(N) complexity. This approach not only accelerates computations but also enables efficient approximation of UOT barycenters, which are essential for data averaging and clustering tasks. The barycenter algorithm is based on a convex multi-marginal formulation that exploits the geometric structure of the support points, providing a scalable solution for large datasets.
Extensive numerical experiments on synthetic and real datasets, including single-cell biology data, demonstrate the superior speed and robustness of the proposed methods. The translation invariant Sinkhorn algorithms outperform classical Sinkhorn by over 2x in convergence speed, especially in low-regularization regimes. The Frank-Wolfe solver achieves linear complexity and maintains high accuracy, making it suitable for large-scale problems. The UOT barycenter computation further confirms the practical benefits, reducing computational costs significantly while preserving accuracy.
Overall, this work advances the theoretical understanding and practical algorithms for unbalanced optimal transport. By introducing the concept of translation invariance and leveraging problem structure in 1-D, it opens new avenues for scalable, robust, and fast distribution matching. These innovations have immediate implications for machine learning, computational biology, and image processing, where large-scale, outlier-robust, and mass-varying data are common. Future directions include extending these methods to high-dimensional spaces, integrating with deep learning frameworks, and exploring applications in dynamic systems and multi-modal data analysis, promising a broad impact across scientific disciplines.
Deep Dive
Abstract
Unbalanced optimal transport (UOT) extends optimal transport (OT) to take into account mass variations to compare distributions. This is crucial to make OT successful in ML applications, making it robust to data normalization and outliers. The baseline algorithm is Sinkhorn, but its convergence speed might be significantly slower for UOT than for OT. In this work, we identify the cause for this deficiency, namely the lack of a global normalization of the iterates, which equivalently corresponds to a translation of the dual OT potentials. Our first contribution leverages this idea to develop a provably accelerated Sinkhorn algorithm (coined 'translation invariant Sinkhorn') for UOT, bridging the computational gap with OT. Our second contribution focusses on 1-D UOT and proposes a Frank-Wolfe solver applied to this translation invariant formulation. The linear oracle of each steps amounts to solving a 1-D OT problems, resulting in a linear time complexity per iteration. Our last contribution extends this method to the computation of UOT barycenter of 1-D measures. Numerical simulations showcase the convergence speed improvement brought by these three approaches.
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