Data Assimilation for Sign-indefinite Priors: A generalization of Sinkhorn's algorithm

TL;DR

This paper extends Sinkhorn's algorithm to handle sign-indefinite datasets, enabling calibration of multi-dimensional arrays with specified marginals, which broadens its applicability in statistics and machine learning.

Contribution

It introduces a generalized algorithm for updating sign-indefinite arrays to match marginals, extending the classical Sinkhorn algorithm to signed data.

Findings

The algorithm converges for sign-indefinite priors.

Reduces to classical Sinkhorn when data is positive.

Applicable to calibration in statistics and machine learning.

Abstract

The purpose of this work is to develop a framework to calibrate signed datasets so as to be consistent with specified marginals by suitably extending the Schr\"odinger-Fortet-Sinkhorn paradigm. Specifically, we seek to revise sign-indefinite multi-dimensional arrays in a way that the updated values agree with specified marginals. Our approach follows the rationale in Schr\"odinger's problem, aimed at updating a "prior" probability measure to agree with marginal distributions. The celebrated Sinkhorn's algorithm (established earlier by R.\ Fortet) that solves Schr\"odinger's problem found early applications in calibrating contingency tables in statistics and, more recently, multi-marginal problems in machine learning and optimal transport. Herein, we postulate a sign-indefinite prior in the form of a multi-dimensional array, and propose an optimization problem to suitably update this prior to ensure consistency with given marginals. The resulting algorithm generalizes the Sinkhorn algorithm in that it amounts to iterative scaling of the entries of the array along different coordinate directions. The scaling is multiplicative but also, in contrast to Sinkhorn, inverse-multiplicative depending on the sign of the entries. Our algorithm reduces to the classical Sinkhorn algorithm when the entries of the prior are positive.