# Fast Computation of Branching Process Transition Probabilities via ADMM

## TL;DR

This paper introduces an efficient ADMM-based algorithm for computing transition probabilities in branching processes, significantly reducing computational costs and enabling scalable analysis in biological modeling.

## Contribution

The paper presents a novel ADMM algorithm that leverages sparse optimization to compute branching process transition probabilities without matrix inversions, enhancing scalability.

## Key findings

- Reduces computational cost by orders of magnitude.
- Easily parallelizable and insensitive to tuning parameters.
- Demonstrates scalability in biological models of blood cell production and transposon evolution.

## Abstract

Branching processes are a class of continuous-time Markov chains (CTMCs) prevalent for modeling stochastic population dynamics in ecology, biology, epidemiology, and many other fields. The transient or finite-time behavior of these systems is fully characterized by their transition probabilities. However, computing them requires marginalizing over all paths between endpoint-conditioned values, which often poses a computational bottleneck. Leveraging recent results that connect generating function methods to a compressed sensing framework, we recast this task from the lens of sparse optimization. We propose a new solution method using variable splitting; in particular, we derive closed form updates in a highly efficient ADMM algorithm. Notably, no matrix products -- let alone inversions -- are required at any step. This reduces computational cost by orders of magnitude over existing methods, and the resulting algorithm is easily parallelizable and fairly insensitive to tuning parameters. A comparison to prior work is carried out in two applications to models of blood cell production and transposon evolution, showing that the proposed method is orders of magnitudes more scalable than existing work.