# Fast Dual Subgradient Optimization of the Integrated Transportation Distance Between Stochastic Kernels

arXiv: https://arxiv.org/abs/2312.01432 · HTML: https://mlsanity.com/paper/2312.01432/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 32 references that are in the map.
- First posted on arXiv 3 Dec 2023.

## Builds on (most cited first, 15 of 15)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://mlsanity.com/paper/1705.09634/) (2017)
- [Stochastic Optimization for Large-scale Optimal Transport](https://mlsanity.com/paper/1605.08527/) (2016)
- [Handbook of Convergence Theorems for (Stochastic) Gradient Methods](https://mlsanity.com/paper/2301.11235/) (2023)
- [An Improved Analysis of Stochastic Gradient Descent with Momentum](https://mlsanity.com/paper/2007.07989/) (2020)
- [Computational Optimal Transport: Complexity by Accelerated Gradient Descent Is Better Than by Sinkhorn's Algorithm](https://mlsanity.com/paper/1802.04367/) (2018)
- [A Wasserstein-type distance in the space of Gaussian Mixture Models](https://mlsanity.com/paper/1907.05254/) (2019)
- [On Efficient Optimal Transport: An Analysis of Greedy and Accelerated Mirror Descent Algorithms](https://mlsanity.com/paper/1901.06482/) (2019)
- [Sliced Wasserstein Distance for Learning Gaussian Mixture Models](https://mlsanity.com/paper/1711.05376/) (2017)
- [A Unified Analysis of Stochastic Momentum Methods for Deep Learning](https://mlsanity.com/paper/1808.10396/) (2018)
- [Multilevel Clustering via Wasserstein Means](https://mlsanity.com/paper/1706.03883/) (2017)
- [A Smoothed Dual Approach for Variational Wasserstein Problems](https://mlsanity.com/paper/1503.02533/) (2015)
- [Semi-dual Regularized Optimal Transport](https://mlsanity.com/paper/1811.05527/) (2018)
- [Decentralize and Randomize: Faster Algorithm for Wasserstein Barycenters](https://mlsanity.com/paper/1806.03915/) (2018)
- [An Integrated Transportation Distance Between Kernels and Approximate Dynamic Risk Evaluation in Markov Systems](https://mlsanity.com/paper/2311.06645/) (2023)

## Built on by (most cited first, 4 of 4)

- [Spec-LLaVA: Accelerating Vision-Language Models with Dynamic Tree-Based Speculative Decoding](https://mlsanity.com/paper/2509.11961/) (2025)
- [Comprehensive Overview of Artificial Intelligence Applications in Modern Industries](https://mlsanity.com/paper/2409.13059/) (2024)
- [Leveraging Optimal Transport for Distributed Two-Sample Testing: An Integrated Transportation Distance-based Framework](https://mlsanity.com/paper/2506.16047/) (2025)
- [TrafficKAN-GCN: Graph Convolutional-based Kolmogorov-Arnold Network for Traffic Flow Optimization](https://mlsanity.com/paper/2503.03276/) (2025)

## Abstract (arXiv)

A generalization of the Wasserstein metric, the integrated transportation distance, establishes a novel distance between probability kernels of Markov systems. This metric serves as the foundation for an efficient approximation technique, enabling the replacement of the original system's kernel with a kernel with a discrete support of limited cardinality. To facilitate practical implementation, we present a specialized dual algorithm capable of constructing these approximate kernels quickly and efficiently, without requiring computationally expensive matrix operations. Finally, we demonstrate the efficacy of our method through several illustrative examples, showcasing its utility in practical scenarios. This advancement offers new possibilities for the streamlined analysis and manipulation of stochastic systems represented by kernels.