Information-theoretic coordinate subset and partition selection of multivariate Markov chains via submodular optimization

We study the problem of optimally projecting the transition matrix of a finite ergodic multivariate Markov chain onto a lower-dimensional state space, as well as the problem of finding an optimal partition of coordinates such that the factorized Markov chain gives minimal information loss compared to the original multivariate chain. Specifically, we seek to construct a Markov chain that optimizes various information-theoretic criteria under cardinality constraints. These criteria include entropy rate, information-theoretic distance to factorizability, independence, and stationarity. We formulate these tasks as best subset or partition selection problems over multivariate Markov chains and leverage the (k-)submodular (or (k-)supermodular) structures of the objective functions to develop efficient greedy-based algorithms with theoretical guarantees. Along the way, we introduce a generalized version of the distorted greedy algorithm, which may be of independent interest. Finally, we illustrate the theory and algorithms through extensive numerical experiments with publicly available code on multivariate Markov chains associated with the Bernoulli–Laplace and Curie–Weiss models.

💡 Research Summary

The paper addresses the problem of reducing the dimensionality of a finite‑state, ergodic multivariate Markov chain by selecting a subset of its coordinates or by partitioning the coordinates into groups, while preserving as much information as possible. The authors formulate several information‑theoretic objectives: (i) maximizing the entropy rate of the projected chain, (ii) minimizing the Kullback‑Leibler (KL) divergence between the projected chain and its stationary distribution (distance to stationarity), (iii) minimizing the KL‑based distance to independence, and (iv) minimizing the KL‑based distance to factorizability (i.e., the loss incurred when approximating the full chain by the tensor product of the projected and its complement).

A central theoretical contribution is the identification of submodular or supermodular structure in each objective. The entropy‑rate map (S\mapsto H(P(S))) and the factorizability‑distance map (S\mapsto D(P|P(S)\otimes P(-S))) are shown to be submodular, reflecting diminishing returns: adding a coordinate to a small set yields a larger marginal gain than adding it to a larger set. Conversely, the distance‑to‑independence map (S\mapsto I(P(S))) is monotone increasing and supermodular, while its complement (S\mapsto I(P(-S))) is monotone decreasing and also supermodular. These properties enable the use of classic greedy algorithms that guarantee a ((1-1/e)) approximation for monotone submodular maximization under a cardinality constraint.

Beyond ordinary set selection, the authors consider the more general k‑submodular setting where the decision variable is a partition of the coordinates into at most (k) labeled groups. They extend the “distorted greedy” algorithm (originally devised for monotone submodular functions) to this k‑submodular context, providing theoretical approximation guarantees under cardinality‑type constraints on each group. This generalized algorithm is of independent interest for any k‑submodular maximization problem.

Algorithmically, each iteration requires computing marginal gains of the chosen objective, which can be done by accessing the relevant rows/columns of the transition matrix; thus the overall complexity scales linearly with the number of selected coordinates (or groups) and the dimension (d). The approach does not rely on spectral information (eigenvalues/eigenvectors), making it applicable when the transition matrix is high‑dimensional but its entries are known or have a tractable structure.

Empirical validation is performed on two well‑studied models. In the Bernoulli–Laplace diffusion model, the greedy entropy‑rate maximizer correctly identifies the most “mixing” coordinates, while the independence‑distance minimizer finds subsets that are nearly independent. In the Curie–Weiss spin model, the authors use the selected subset to construct a novel MCMC sampler that conditions on the identified core spins; the resulting sampler exhibits substantially faster mixing than a standard Gibbs sampler, especially near the critical temperature. These experiments demonstrate that the proposed combinatorial optimization framework can lead to practical algorithmic improvements in sampling and simulation.

The paper also discusses extensions to other stochastic systems such as hidden Markov models or multivariate Poisson processes, suggesting that the submodular‑optimization viewpoint can be a general tool for model reduction, dimensionality reduction, and efficient simulation design. In summary, by revealing submodular and supermodular structures in natural information‑theoretic criteria for multivariate Markov chains, the authors provide both rigorous approximation guarantees and practical algorithms for selecting optimal coordinate subsets or partitions, thereby contributing a valuable bridge between information theory, stochastic processes, and combinatorial optimization.