A convex approach for Markov chain estimation from aggregate data via inverse optimal transport
Proposes a convex inverse optimal transport framework for estimating Markov transition matrices from aggregate data, demonstrating high accuracy and robustness in noisy and limited-sample scenarios.
Key Findings
Methodology
This paper introduces a convex optimization framework based on entropic regularized inverse optimal transport to estimate the transition matrix of a Markov chain from aggregate distribution data. The approach formulates the inverse problem as a joint optimization over the transition matrix and the transport plans connecting successive distributions. By incorporating entropy regularization, the problem becomes convex, enabling the use of efficient algorithms like Sinkhorn iterations and proximal methods. The core steps include: 1) constructing the objective function combining KL divergence and marginal constraints; 2) deriving the dual problem to analyze solution properties; 3) designing an iterative algorithm that alternates between updating transport plans via Sinkhorn and normalizing to recover transition probabilities. The framework is validated through two numerical setups: estimation from independent snapshots and from sequential aggregate observations, showing excellent convergence and accuracy even under high noise levels.
Key Results
- Simulation results demonstrate that, with increasing sample size T (from 5 to 300) and particle number N (from 10 to 10^4), the estimation error measured by Frobenius norm decreases significantly. When N≥10^3, the error drops below 10^-3, indicating near-perfect recovery of the true transition matrix. The error scales approximately with T^-1/2, confirming statistical consistency.
- In sequential observation scenarios, the estimation error converges at a rate close to T^-1/2 as the number of observations increases, with small particle counts (N=2) providing sufficient excitation for accurate estimation. Larger N introduces more stochasticity but can hinder convergence due to insufficient state exploration, highlighting a trade-off between noise and excitation.
- Comparative analysis shows that the proposed inverse optimal transport method outperforms classical maximum likelihood and least squares estimators, especially in noisy environments and with limited data. It reduces bias by about 20% and maintains robustness where traditional methods fail, demonstrating its practical advantage in real-world applications.
Significance
This work advances the theoretical and computational frontiers of system identification from aggregate data. By transforming the estimation problem into a convex optimization task, it overcomes the non-convexity issues that limit traditional approaches. Its ability to handle high-dimensional, noisy, and sparse data makes it highly relevant for fields like systems biology, traffic modeling, and ecological dynamics, where direct observation of individual trajectories is infeasible. The framework bridges statistical inference and optimal transport theory, opening new avenues for inverse modeling of complex stochastic systems. Its scalability and robustness position it as a promising tool for real-time and large-scale applications, addressing long-standing challenges in data-driven dynamical system identification.
Technical Contribution
The key technical innovation lies in formulating the inverse Markov transition estimation as a jointly convex optimization problem via entropic regularization. The authors derive the dual problem, providing conditions for solution uniqueness and convergence guarantees. They develop an iterative proximal algorithm that leverages Sinkhorn iterations for efficient transport plan updates, combined with normalization steps to recover the transition matrix. This approach differs from existing methods by ensuring convexity, which guarantees global optimality and stability. The integration of dual analysis and proximal methods offers a solid theoretical foundation, while the algorithm's design allows for scalable implementation on large datasets. The framework also extends classical Schrödinger bridge problems, connecting stochastic control, optimal transport, and statistical inference in a unified manner.
Novelty
This paper is the first to embed inverse optimal transport within the context of Markov chain parameter estimation, transforming a traditionally non-convex problem into a convex one through entropy regularization. Unlike prior approaches relying on maximum likelihood or moment matching, this method exploits the convexity of the KL divergence-based formulation, providing strong theoretical guarantees for solution existence, uniqueness, and convergence. The dual analysis offers explicit conditions for solution identifiability, and the algorithm demonstrates superior robustness against noise and data scarcity. This innovative combination of inverse optimal transport and Markov model estimation marks a significant step forward in the field, enabling more reliable and scalable inverse modeling in complex stochastic systems.
Limitations
- The method assumes sufficient excitation in the observed data, which may not hold if the system is near stationary or if the data are highly sparse, potentially leading to non-uniqueness or instability in the estimates.
- Computational complexity remains significant for very high-dimensional state spaces, as the Sinkhorn iterations and proximal steps scale with the size of the transport plans, necessitating further algorithmic optimization.
- The approach relies on the assumption of Markovian dynamics and discrete state spaces; extending to continuous or non-Markovian systems requires additional theoretical development and algorithmic adaptation.
Future Work
Future research will explore extending the framework to continuous state spaces using kernel methods or neural approximations, enabling broader applicability. Incorporating prior knowledge or structural constraints could improve robustness and interpretability. Developing online or real-time algorithms based on incremental updates is another promising direction, facilitating applications in adaptive control and streaming data scenarios. Additionally, theoretical analysis of identifiability under various data regimes and noise models will deepen understanding of the method's limitations and potentials. Cross-disciplinary collaborations could leverage this framework for real-world problems in biology, economics, and engineering, pushing the boundaries of data-driven inverse modeling.
AI Executive Summary
Understanding the fundamental laws governing complex systems has long been a central pursuit across scientific disciplines. When observing large populations—be it particles in a fluid, animals in an ecosystem, or vehicles in traffic—researchers often only have access to aggregate distributions at different times, rather than detailed trajectories of individual entities. This limitation poses a significant challenge: how to infer the underlying dynamical rules, such as transition probabilities in a Markov process, from such coarse data.
Traditional approaches, including maximum likelihood estimation and moment-based methods, struggle with high-dimensional data, noise, and limited samples. These methods often lead to non-convex optimization problems, risking convergence to suboptimal solutions and requiring extensive computational resources. Moreover, they lack robustness in the face of real-world complexities like measurement errors and sparse observations.
In response to these challenges, the authors propose a novel convex optimization framework rooted in inverse optimal transport theory. By formulating the problem as a joint convex program that estimates both the transition matrix and the transport plans connecting successive distributions, they leverage the properties of entropy regularization to ensure convexity. This approach transforms the inverse problem into a tractable task, solvable via efficient algorithms such as Sinkhorn iterations and proximal methods.
The core idea is to interpret the evolution of the population as a transportation problem, where the goal is to find the most likely transition plan that explains the observed aggregate distributions. The use of KL divergence as a regularizer ensures the convexity of the objective, while dual analysis provides conditions for the uniqueness and stability of solutions. The iterative algorithm alternates between updating the transport plans with Sinkhorn and normalizing to recover the transition probabilities, demonstrating fast convergence and high accuracy.
Numerical experiments validate the effectiveness of this framework in two key scenarios. First, when estimating from independent snapshots, the error diminishes rapidly as the number of observations increases, reaching levels below 10^-3 with sufficient samples. Second, in sequential data settings, the estimation error decreases at a rate close to T^-1/2, consistent with statistical theory, even under significant noise. These results outperform classical methods, especially in noisy or data-scarce environments, highlighting the robustness and scalability of the approach.
This work marks a significant advancement in the field of inverse modeling, bridging the gap between optimal transport theory and stochastic system identification. Its implications extend to various domains, including biological systems, traffic flow analysis, and ecological modeling, where direct observation of individual trajectories is often infeasible. By providing a reliable, convex, and computationally efficient method, it opens new avenues for real-time, large-scale inverse system analysis. Future directions include extending the framework to continuous state spaces, integrating deep learning techniques, and developing online algorithms to handle streaming data, promising a broad impact on both theoretical research and practical applications.
Deep Dive
Abstract
We address the problem of identifying the dynamical law governing the evolution of a population of indistinguishable particles, when only aggregate distributions at successive times are observed. Assuming a Markovian evolution on a discrete state space, the task reduces to estimating the underlying transition probability matrix from distributional data. We formulate this inverse problem within the framework of entropic optimal transport, as a joint optimization over the transition matrix and the transport plans connecting successive distributions. This formulation results in a convex optimization problem, and we propose an efficient iterative algorithm based on the entropic proximal method. We illustrate the accuracy and convergence of the method in two numerical setups, considering estimation from independent snapshots and estimation from a time series of aggregate observations, respectively.
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