# Recovery Bounds on Class-Based Optimal Transport: A Sum-of-Norms Regularization Framework

## TL;DR

This paper introduces a new convex optimal transport framework with sum-of-norms regularization that effectively recovers class structures, offers scalable algorithms, and enhances robustness and data structure preservation.

## Contribution

It proposes a novel sum-of-norms regularized convex OT model with theoretical recovery guarantees and an efficient proximal algorithm.

## Key findings

- Improved class structure preservation in data
- Enhanced robustness to data geometry
- Scalable computation of OT plans

## Abstract

We develop a novel theoretical framework for understating OT schemes respecting a class structure. For this purpose, we propose a convex OT program with a sum-of-norms regularization term, which provably recovers the underlying class structure under geometric assumptions. Furthermore, we derive an accelerated proximal algorithm with a closed-form projection and proximal operator scheme, thereby affording a more scalable algorithm for computing optimal transport plans. We provide a novel argument for the uniqueness of the optimum even in the absence of strong convexity. Our experiments show that the new regularizer not only results in a better preservation of the class structure in the data but also yields additional robustness to the data geometry, compared to previous regularizers.