A new Sinkhorn algorithm with Deletion and Insertion operations

TL;DR

This paper introduces a novel Sinkhorn algorithm variant that incorporates deletion and insertion operations, enabling the estimation of epsilon-assignments between sets of different sizes within a differentiable, iterative framework suitable for neural network integration.

Contribution

The paper presents a new Sinkhorn algorithm with deletion and insertion capabilities for epsilon-assignments, handling set size discrepancies and supporting differentiable, backpropagation-compatible computations.

Findings

Handles sets of different sizes naturally.

Supports differentiable, iterative computation.

Enables integration into neural network training.

Abstract

This technical report is devoted to the continuous estimation of an epsilon-assignment. Roughly speaking, an epsilon assignment between two sets V1 and V2 may be understood as a bijective mapping between a sub part of V1 and a sub part of V2 . The remaining elements of V1 (not included in this mapping) are mapped onto an epsilon pseudo element of V2 . We say that such elements are deleted. Conversely, the remaining elements of V2 correspond to the image of the epsilon pseudo element of V1. We say that these elements are inserted. As a result our method provides a result similar to the one of the Sinkhorn algorithm with the additional ability to reject some elements which are either inserted or deleted. It thus naturally handles sets V1 and V2 of different sizes and decides mappings/insertions/deletions in a unified way. Our algorithms are iterative and differentiable and may thus be easily inserted within a backpropagation based learning framework such as artificial neural networks.