[Paper Review] Efficient Transition Probability Computation for Continuous-Time Branching Processes via Compressed Sensing

Jason Xu, Vladimir N. Minin|PubMed|Mar 9, 2015

Single-cell and spatial transcriptomics29 references3 citations

TL;DR

This paper proposes a compressed sensing framework (CSGF) that accelerates transition probability computation in continuous-time branching processes by exploiting sparsity in transition probabilities. By reformulating the generating function inversion problem as a sparse signal recovery task, CSGF reduces ODE evaluations logarithmically, achieving up to two orders of magnitude speedup while maintaining high accuracy (error < 10⁻³ in absolute terms).

ABSTRACT

Branching processes are a class of continuous-time Markov chains (CTMCs) with ubiquitous applications. A general difficulty in statistical inference under partially observed CTMC models arises in computing transition probabilities when the discrete state space is large or uncountable. Classical methods such as matrix exponentiation are infeasible for large or countably infinite state spaces, and sampling-based alternatives are computationally intensive, requiring integration over all possible hidden events. Recent work has successfully applied generating function techniques to computing transition probabilities for linear multi-type branching processes. While these techniques often require significantly fewer computations than matrix exponentiation, they also become prohibitive in applications with large populations. We propose a compressed sensing framework that significantly accelerates the generating function method, decreasing computational cost up to a logarithmic factor by only assuming the probability mass of transitions is sparse. We demonstrate accurate and efficient transition probability computations in branching process models for blood cell formation and evolution of self-replicating transposable elements in bacterial genomes.

Motivation & Objective

• Address the computational bottleneck in statistical inference for partially observed continuous-time Markov chains (CTMCs) with large or uncountable state spaces.
• Overcome the prohibitive cost of matrix exponentiation and sampling-based methods in high-dimensional branching processes.
• Leverage sparsity in transition probabilities to reduce computational burden in generating function-based transition probability estimation.
• Enable efficient likelihood-based inference in complex branching process models previously infeasible due to computational limits.
• Develop a scalable, accurate method applicable to real-world biological models such as hematopoiesis and transposable element evolution.

Proposed method

• Formulate the transition probability computation as a sparse signal recovery problem using compressed sensing.
• Represent the probability generating function (PGF) inversion as a linear system where only a small subset of entries are non-zero.
• Use randomized measurements to sample the PGF at strategic points, reducing the number of required ODE evaluations.
• Apply iterative optimization via projected gradient descent (PGD) to reconstruct the transition probability matrix from compressed measurements.
• Incorporate sparsity-promoting regularization (e.g., ℓ1-norm) to enhance recovery accuracy under sparse support assumptions.
• Use a C implementation of ODE solvers (deSolve) and a naive R-based PGD for computational efficiency and reproducibility.

Experimental results

Research questions

• RQ1 Can compressed sensing be effectively applied to accelerate transition probability computation in continuous-time branching processes?
• RQ2 Does the sparsity of transition probabilities in realistic biological models enable significant computational gains without sacrificing accuracy?
• RQ3 How does the CSGF method compare in runtime and accuracy to standard generating function inversion and matrix exponentiation?
• RQ4 To what extent can the number of required PGF evaluations be reduced using compressed sensing while preserving numerical fidelity?
• RQ5 Can the CSGF framework be generalized to other generating function-based inference tasks in branching processes?

Key findings

• CSGF reduces the number of required PGF evaluations to less than 0.1% of the standard method in the birth-death-shift model (N=4096), achieving a 100-fold speedup.
• In the hematopoiesis model, CSGF reduces computational cost to approximately 0.5% of the baseline method, with less than 6% of total time spent on PGF evaluations in the worst case.
• Maximum absolute error in transition probability recovery is on the order of 10⁻³, with relative error (ε_rel) at most 10⁻², indicating high numerical accuracy.
• The method maintains robust performance across different population sizes (N=2⁹ to N=4096), demonstrating scalability and stability.
• The CSGF framework is computationally efficient even with a naive R implementation of PGD, suggesting further gains with optimized code.
• Sparsity in transition probabilities is empirically validated in two biologically relevant models, supporting the method’s practical applicability.