Markov chain aggregation and its applications to combinatorial reaction networks

Optimal High-order Tensor SVD via Tensor-Train Orthogonal Iteration

This paper studies a general framework for high-order tensor SVD. We propose a new computationally efficient algorithm, tensor-train orthogonal iteration (TTOI), that aims to estimate the low tensor-train rank structure from the noisy high-order tensor observation. The proposed TTOI consists of initialization via TT-SVD [1] and new iterative backward/forward updates. We develop the general upper bound on estimation error for TTOI with the support of several new representation lemmas on tensor matricizations. By developing a matching information-theoretic lower bound, we also prove that TTOI achieves the minimax optimality under the spiked tensor model. The merits of the proposed TTOI are illustrated through applications to estimation and dimension reduction of high-order Markov processes, numerical studies, and a real data example on New York City taxi travel records. The software of the proposed algorithm is available online (https://github.com/Lili-Zheng-stat/TTOI).

ESTIMATION OF CELL LINEAGE TREES BY MAXIMUM-LIKELIHOOD PHYLOGENETICS

CRISPR technology has enabled cell lineage tracing for complex multicellular organisms through insertion-deletion mutations of synthetic genomic barcodes during organismal development. To reconstruct the cell lineage tree from the mutated barcodes, current approaches apply general-purpose computational tools that are agnostic to the mutation process and are unable to take full advantage of the data's structure. We propose a statistical model for the CRISPR mutation process and develop a procedure to estimate the resulting tree topology, branch lengths, and mutation parameters by iteratively applying penalized maximum likelihood estimation. By assuming the barcode evolves according to a molecular clock, our method infers relative ordering across parallel lineages, whereas existing techniques only infer ordering for nodes along the same lineage. When analyzing transgenic zebrafish data from McKenna, Findlay and Gagnon et al. (2016), we find that our method recapitulates known aspects of zebrafish development and the results are consistent across samples.

Approximate lumpability for Markovian agent-based models using local symmetries

We study a Markovian agent-based model (MABM) in this paper. Each agent is endowed with a local state that changes over time as the agent interacts with its neighbours. The neighbourhood structure is given by a graph. Recently, Simon, Taylor, and Kiss [40] used the automorphisms of the underlying graph to generate a lumpable partition of the joint state space, ensuring Markovianness of the lumped process for binary dynamics. However, many large random graphs tend to become asymmetric, rendering the automorphism-based lumping approach ineffective as a tool of model reduction. In order to mitigate this problem, we propose a lumping method based on a notion of local symmetry, which compares only local neighbourhoods of vertices. Since local symmetry only ensures approximate lumpability, we quantify the approximation error by means of the Kullback–Leibler divergence rate between the original Markov chain and aliftedMarkov chain. We prove the approximation error decreases monotonically. The connections to fibrations of graphs are also discussed.

Markov Chain Aggregation and Its Application to Rule-Based Modelling

Rule-based modelling allows to represent molecular interactions in a compact and natural way. The underlying molecular dynamics, by the laws of stochastic chemical kinetics, behaves as a continuous-time Markov chain. However, this Markov chain enumerates all possible reaction mixtures, rendering the analysis of the chain computationally demanding and often prohibitive in practice. We here describe how it is possible to efficiently find a smaller, aggregate chain, which preserves certain properties of the original one. Formal methods and lumpability notions are used to define algorithms for automated and efficient construction of such smaller chains (without ever constructing the original ones). We here illustrate the method on an example and we discuss the applicability of the method in the context of modelling large signaling pathways.