# Smooth and Sparse Optimal Transport

arXiv: https://arxiv.org/abs/1710.06276 · HTML: https://mlsanity.com/paper/1710.06276/

## Key facts

- Cited by 83 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Oct 2017.

## Built on by (most cited first, 30 of 83)

- [Efficient and Modular Implicit Differentiation](https://mlsanity.com/paper/2105.15183/) (2021)
- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [Minibatch optimal transport distances; analysis and applications](https://mlsanity.com/paper/2101.01792/) (2021)
- [Wasserstein Adversarial Imitation Learning](https://mlsanity.com/paper/1906.08113/) (2019)
- [Score-based Generative Neural Networks for Large-Scale Optimal Transport](https://mlsanity.com/paper/2110.03237/) (2021)
- [Unbalanced Optimal Transport, from Theory to Numerics](https://mlsanity.com/paper/2211.08775/) (2022)
- [Sparsity-Constrained Optimal Transport](https://mlsanity.com/paper/2209.15466/) (2022)
- [Unbalanced Optimal Transport through Non-negative Penalized Linear Regression](https://mlsanity.com/paper/2106.04145/) (2021)
- [Scalable Optimal Transport Methods in Machine Learning: A Contemporary Survey](https://mlsanity.com/paper/2305.05080/) (2023)
- [Routers in Vision Mixture of Experts: An Empirical Study](https://mlsanity.com/paper/2401.15969/) (2024)
- [Sparse Needlets for Lighting Estimation with Spherical Transport Loss](https://mlsanity.com/paper/2106.13090/) (2021)
- [Learning Generalized Gumbel-max Causal Mechanisms](https://mlsanity.com/paper/2111.06888/) (2021)
- [On Unbalanced Optimal Transport: Gradient Methods, Sparsity and Approximation Error](https://mlsanity.com/paper/2202.03618/) (2022)
- [Unlocking Slot Attention by Changing Optimal Transport Costs](https://mlsanity.com/paper/2301.13197/) (2023)
- [Monge, Bregman and Occam: Interpretable Optimal Transport in High-Dimensions with Feature-Sparse Maps](https://mlsanity.com/paper/2302.04065/) (2023)
- [Estimating Barycenters of Distributions with Neural Optimal Transport](https://mlsanity.com/paper/2402.03828/) (2024)
- [Semi-Discrete Optimal Transport: Hardness, Regularization and Numerical Solution](https://mlsanity.com/paper/2103.06263/) (2021)
- [Convergence Rates for Regularized Optimal Transport via Quantization](https://mlsanity.com/paper/2208.14391/) (2022)
- [Scalable Optimal Transport in High Dimensions for Graph Distances, Embedding Alignment, and More](https://mlsanity.com/paper/2107.06876/) (2021)
- [Accelerating Sinkhorn Algorithm with Sparse Newton Iterations](https://mlsanity.com/paper/2401.12253/) (2024)
- [Accurate Neural Network Pruning Requires Rethinking Sparse Optimization](https://mlsanity.com/paper/2308.02060/) (2023)
- [A Fast and Accurate Splitting Method for Optimal Transport: Analysis and Implementation](https://mlsanity.com/paper/2110.11738/) (2021)
- [NeuroMax: Enhancing Neural Topic Modeling via Maximizing Mutual Information and Group Topic Regularization](https://mlsanity.com/paper/2409.19749/) (2024)
- [Robust Estimation under the Wasserstein Distance](https://mlsanity.com/paper/2302.01237/) (2023)
- [Structured Transforms Across Spaces with Cost-Regularized Optimal Transport](https://mlsanity.com/paper/2311.05788/) (2023)
- [Functional optimal transport: map estimation and domain adaptation for functional data](https://mlsanity.com/paper/2102.03895/) (2021)
- [Optimal transport weights for causal inference](https://mlsanity.com/paper/2109.01991/) (2021)
- [Efficient and Accurate Optimal Transport with Mirror Descent and Conjugate Gradients](https://mlsanity.com/paper/2307.08507/) (2023)
- [On Partial Optimal Transport: Revising the Infeasibility of Sinkhorn and Efficient Gradient Methods](https://mlsanity.com/paper/2312.13970/) (2023)
- [Differentiable Generalized Sliced Wasserstein Plans](https://mlsanity.com/paper/2505.22049/) (2025)

## Abstract (arXiv)

Entropic regularization is quickly emerging as a new standard in optimal transport (OT). It enables to cast the OT computation as a differentiable and unconstrained convex optimization problem, which can be efficiently solved using the Sinkhorn algorithm. However, entropy keeps the transportation plan strictly positive and therefore completely dense, unlike unregularized OT. This lack of sparsity can be problematic in applications where the transportation plan itself is of interest. In this paper, we explore regularizing the primal and dual OT formulations with a strongly convex term, which corresponds to relaxing the dual and primal constraints with smooth approximations. We show how to incorporate squared 2-norm and group lasso regularizations within that framework, leading to sparse and group-sparse transportation plans. On the theoretical side, we bound the approximation error introduced by regularizing the primal and dual formulations. Our results suggest that, for the regularized primal, the approximation error can often be smaller with squared 2-norm than with entropic regularization. We showcase our proposed framework on the task of color transfer.