On Bridging Mixture Distributions
Abstract
The construction and the stability of bridges between mixture probability measures are investigated. Given a collection of Markov transitions bridging the components of two mixtures, together with a coupling of the mixture labels, we construct a bridge between the mixtures on an extended state space, and we show that this bridge inherits the entropic optimality of the component bridges. We then turn to Gaussian mixtures bridged by Gaussian Schrödinger bridges, whose parameters are generally unknown and must be estimated. We prove a $2$-Wasserstein continuity theorem between the exact bridge and its plug-in approximation. The analysis rests on a representation of the Riccati fixed point map in inverse coordinates; this map is globally $1$-Lipschitz and requires no matrix inversion. These results are applied to three estimation schemes, with d the dimension and N the number of samples. For Gaussian mixtures estimated by the Expectation-Maximization algorithm, the squared $2$-Wasserstein error is of order (dlog N/N)1/2+d2log N/N with probability at least 1−10N−1. For single Gaussian marginals estimated by their sample moments, it is of order d2/N in expectation. For the regularized empirical Monge map based on an ε-inflation of the sample covariance, the mean squared error is of order d2{(1+ε−1)/N+ε2}; as soon as N is a sufficiently large multiple of d, it reduces to d2/N+d ε2, up to an exponentially small term. These estimates are illustrated numerically.
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BibTeX
@misc{moral2026bridging,
title = {{On Bridging Mixture Distributions}},
author = {Pierre del Moral and Ajay Jasra and Ke Zhao},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.13383},
eprint = {2608.13383},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05738926v1},
}
RIS
TY - UNPB
TI - On Bridging Mixture Distributions
AU - Pierre del Moral
AU - Ajay Jasra
AU - Ke Zhao
PY - 2026
DA - 2026/09/04
PB - HAL
DO - 10.48550/arxiv.2608.13383
UR - https://hal.science/hal-05738926v1
AB - The construction and the stability of bridges between mixture probability measures are investigated. Given a collection of Markov transitions bridging the components of two mixtures, together with a coupling of the mixture labels, we construct a bridge between the mixtures on an extended state space, and we show that this bridge inherits the entropic optimality of the component bridges. We then turn to Gaussian mixtures bridged by Gaussian Schrödinger bridges, whose parameters are generally unknown and must be estimated. We prove a $2$-Wasserstein continuity theorem between the exact bridge and its plug-in approximation. The analysis rests on a representation of the Riccati fixed point map in inverse coordinates; this map is globally $1$-Lipschitz and requires no matrix inversion. These results are applied to three estimation schemes, with d the dimension and N the number of samples. For Gaussian mixtures estimated by the Expectation-Maximization algorithm, the squared $2$-Wasserstein error is of order (dlog N/N)^1/2+d^2log N/N with probability at least 1−10N^−1. For single Gaussian marginals estimated by their sample moments, it is of order d^2/N in expectation. For the regularized empirical Monge map based on an ε-inflation of the sample covariance, the mean squared error is of order d^2{(1+ε^−1)/N+ε^2}; as soon as N is a sufficiently large multiple of d, it reduces to d^2/N+d ε^2, up to an exponentially small term. These estimates are illustrated numerically.
ER -