Sinkhorn Algorithm for Sequentially Composed Optimal Transports (2412.03120v4)

Abstract: Sinkhorn algorithm is the de-facto standard approximation algorithm for optimal transport, which has been applied to a variety of applications, including image processing and natural language processing. In theory, the proof of its convergence follows from the convergence of the Sinkhorn--Knopp algorithm for the matrix scaling problem, and Altschuler et al. show that its worst-case time complexity is in near-linear time. Very recently, sequentially composed optimal transports were proposed by Watanabe and Isobe as a hierarchical extension of optimal transports. In this paper, we present an efficient approximation algorithm, namely Sinkhorn algorithm for sequentially composed optimal transports, for its entropic regularization. Furthermore, we present a theoretical analysis of the Sinkhorn algorithm, namely (i) its exponential convergence to the optimal solution with respect to the Hilbert pseudometric, and (ii) a worst-case complexity analysis for the case of one sequential composition.

• The paper introduces a novel Sinkhorn-type algorithm specifically designed for sequentially composed optimal transport (SeqOT).
• The proposed algorithm efficiently projects sequential and parallel constraints and demonstrates exponential convergence properties using the Hilbert metric.
• For single sequential compositions, the algorithm achieves improved computational complexity, notably transitioning from matrix-matrix to matrix-vector products for greater efficiency.

Essay: Sinkhorn Algorithm for Sequentially Composed Optimal Transports

In this preprint by Kazuki Watanabe and Noboru Isobe, the authors investigate the extension of the Sinkhorn algorithm to a new variant of the optimal transport (OT) problem, termed sequentially composed optimal transport (SeqOT). The advancement is predicated on the desire to explore hierarchical extensions of OT, which incorporate additional constraints in alignment with hierarchical reinforcement learning frameworks, thus broadening the traditional application range of OT.

The authors begin by revisiting the fundamental aspects of the Sinkhorn algorithm, a pivotal technique for finding approximate solutions to OT problems. By applying entropic regularization, this algorithm integrates nicely into computational practices leveraging its GPU-friendly structure, allowing expansive use in machine learning tasks. The theoretical underpinning provided by the convergence guarantee of the Sinkhorn-Knopp matrix scaling framework is crucial, ensuring computational robustness.

Contributions and Theoretical Developments

The paper's main contribution involves the formulation of a novel Sinkhorn-type algorithm tailored for SeqOT. The authors meticulously develop a new algorithmic iteration that respects both the sequential and parallel compositional constraints inherent in SeqOT, thereby extending Sinkhorn's applicability beyond the standard OT framework. The proposed algorithm achieves significant computational advancements over existing methods. Notably, it projects sequential adjacency and marginal constraint sets in an alternate manner, facilitating the reconciliation of sequential structures with parallel computational methodologies.

In their theoretical exploration, Watanabe and Isobe substantiate the exponential convergence properties of their algorithm concerning the Hilbert metric. This result is particularly significant as it underlines the rapid convergence behavior of the algorithm, hence optimizing computational efficiency. For cases involving a single sequential composition, the researchers conduct a deep complexity analysis, demonstrating that their approach demands fewer computational resources compared to previously established methods. It is emphasized that the computational gains stem from a shift from matrix-matrix products to more efficient matrix-vector products.

Numerical Results and Complexity Analysis

The authors achieve a breakthrough in computational complexity for SeqOT, particularly in cases with a single sequential composition. For a given error tolerance δ, the paper posits a complexity bound of $$O(m \log m (\|C^{(1)}\|_\infty + \|C^{(2)}\|_\infty) \delta^{-3})$$. This complexity is notably more efficient than existing approaches, which require at least $O(m^4)$ operations, primarily due to the innovative mechanism employed in the new Sinkhorn algorithm for matrix-vector interaction.

Implications and Future Prospects

The implications of this work stretch across both theoretical and practical domains. On a theoretical level, the extension of Sinkhorn algorithms to handle SeqOT challenges inspires further research into complex hierarchical planning problems across various fields, notably within machine learning and optimization. Practically, the computational improvements afforded by this approach pave the way for application in large-scale systems where hierarchical component interactions are essential.

Looking onward, the authors suggest potential avenues for further improvement and complexity analysis for cases wherein multiple sequential compositions are involved. Such advancements could reinforce the method's robustness, making SeqOT an even more compelling tool in the context of large-scale optimization tasks involving hierarchical decompositions.

In summary, the paper provides a well-reasoned extension of the Sinkhorn algorithm, backed by stringent theoretical proofs and complexity benchmarks, thus presenting a formidable tool for OT in hierarchical settings. This work will likely stimulate additional research efforts in the domain of compositional OT and optimal transportation within machine learning applications.