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

• Published in: Automatica (Oxford) Vol. 177; p. 112283
• Main Authors: Dong, Anqi, Georgiou, Tryphon T., Tannenbaum, Allen
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.07.2025
• Subjects: Multi-marginal Schrödinger bridges; Negative probabilities; Sinkhorn algorithm; Sinkhorn algorithm; Multi-marginal Schrödinger bridges; Negative probabilities
• ISSN: 0005-1098; 1873-2836
• DOI: 10.1016/j.automatica.2025.112283

We deal with a hitherto unstudied problem that is motivated by the need to model promotion/inhibition contributions of relations, as in gene regulatory networks, to explain observed empirical distributions. There is no known pre-existing work focusing on this problem which, in essence, consists in calibrating a sign-indefinite data set representing prior information on the joint strength of promotion/inhibition, to restore consistency with specified observed marginals. We address the problem in the general setting of relations between n objects, and then specialize to binary relations to compare with a possible adhoc alternative following a Metropolis–Hastings rationale. Thus, our paper begins by addressing the general problem to find a multi-dimensional array (representing relative contribution of the n factors) to match specified marginals; the salient feature is that the sign (promotion/inhibition) of entries should match that of the prior. We develop an approach that is based on minimizing a suitable entropy functional subject to the constraints. Our approach follows verbatim Schrödinger’s rationale (Schrödinger, 1931, 1932) that underlies the celebrated Sinkhorn–Knopp algorithm in statistics used to reconcile contingency tables (of positive/frequency data sets) with marginal data (Knight, 2008). The resulting algorithm for our case of sign-indefinite prior generalizes the Sinkhorn–Knopp 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–Knopp, inverse-multiplicative depending on the sign of entries. When the entries of the prior are non-negative, our algorithm reduces to the classical Sinkhorn–Knopp algorithm which, in recent years, has become the computational workhorse of optimal transport (Peyré and Cuturi, 2019). Interestingly, the extension of the Sinkhorn–Knopp algorithm herein is perhaps the only example of a coordinate gradient ascent algorithm, other than the classical Sinkhorn, that has a non-trivial closed-form solution.