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Abstract We study optimal transport for stationary stochastic processes taking values in finite spaces. In order to reflect the stationarity of the underlying processes, we restrict attention to stationary couplings, also known as joinings. The resulting optimal joining problem captures differences in the long-run average behavior of the processes of interest. We introduce estimators of both optimal joinings and the optimal joining cost, and establish consistency of the estimators under mild conditions. Furthermore, under stronger mixing assumptions we establish finite-sample error rates for the estimated optimal joining cost that extend the best known results in the iid case. We also extend the consistency and rate analysis to an entropy-penalized version of the optimal joining problem. Finally, we validate our convergence results empirically as well as demonstrate the computational advantage of the entropic problem in a simulation experiment. 

We describe and study a transport-based procedure called network optimal transition coupling (NetOTC) for the comparison and alignment of two networks. The networks of interest may be directed or undirected, weighted or unweighted, and may have distinct vertex sets of different sizes. Given two networks and a cost function relating their vertices, NetOTC finds a transition coupling of their associated random walks having minimum expected cost. The minimizing cost quantifies the difference between the networks, while the optimal transport plan itself provides alignments of both the vertices and the edges of the two networks. Coupling of the full random walks, rather than their marginal distributions, ensures that NetOTC captures local and global information about the networks and preserves edges. NetOTC has no free parameters and does not rely on randomization. We investigate a number of theoretical properties of NetOTC and present experiments establishing its empirical performance. 

Despite the widespread exploration and availability of parcellations for the functional connectome, parcellations designed for the structural connectome are comparatively limited. Current research suggests that there may be no single “correct” parcellation and that the human brain is intrinsically a multiresolution entity. In this work, we propose the Continuous Structural Connectivitity-based, Nested (CoCoNest) family of parcellations—a fully data-driven, multiresolution family of parcellations derived from structural connectome data. The CoCoNest family is created using agglomerative (bottom-up) clustering and error-complexity pruning, which strikes a balance between the complexity of each parcellation and how well it preserves patterns in vertex-level, high-resolution connectivity data. We draw on a comprehensive battery of internal and external evaluation metrics to show that the CoCoNest family is competitive with or outperforms widely used parcellations in the literature. Additionally, we show how the CoCoNest family can serve as an exploratory tool for researchers to investigate the multiresolution organization of the structural connectome. 

Despite the widespread exploration and availability of parcellations for the functional connectome, parcellations designed for the structural connectome are comparatively limited. Current research suggests that there may be no single “correct” parcellation and that the human brain is intrinsically a multiresolution entity. In this work, we propose the Continuous Structural Connectivitity-based, Nested (CoCoNest) family of parcellations—a fully datadriven, multiresolution family of parcellations derived from structural connectome data. The CoCoNest family is created using agglomerative (bottom-up) clustering and error-complexity pruning, which strikes a balance between the complexity of each parcellation and how well it preserves patterns in vertex-level, high-resolution connectivity data. We draw on a comprehensive battery of internal and external evaluation metrics to show that the CoCoNest family is competitive with or outperforms widely used parcellations in the literature. Additionally, we show how the CoCoNest family can serve as an exploratory tool for researchers to investigate the multiresolution organization of the structural connectome 

Datasets in which measurements of two (or more) types are obtained from a common set of samples arise in many scientific applications. A common problem in the exploratory analysis of such data is to identify groups of features of different data types that are strongly associated. A bimodule is a pair (A,B) of feature sets from two data types such that the aggregate cross-correlation between the features in A and those in B is large. A bimodule (A,B) is stable if A coincides with the set of features that have significant aggregate correlation with the features in B, and vice-versa. This paper proposes an iterative-testing based bimodule search procedure (BSP) to identify stable bimodules. Compared to existing methods for detecting cross-correlated features, BSP was the best at recovering true bimodules with sufficient signal, while limiting the false discoveries. In addition, we applied BSP to the problem of expression quantitative trait loci (eQTL) analysis using data from the GTEx consortium. BSP identified several thousand SNP-gene bimodules. While many of the individual SNP-gene pairs appearing in the discovered bimodules were identified by standard eQTL methods, the discovered bimodules revealed genomic subnetworks that appeared to be biologically meaningful and worthy of further scientific investigation.