# A Deterministic Algorithm for the Capacity of Finite-State Channels

## TL;DR

This paper introduces two modified gradient ascent algorithms to efficiently compute the capacity of finite-state channels with Markovian inputs, achieving polynomial or exponential accuracy under certain conditions.

## Contribution

It presents novel algorithms for capacity computation of finite-state channels, including cases with and without strong concavity, with proven convergence and accuracy guarantees.

## Key findings

- First algorithm achieves polynomial accuracy in polynomial time.
- Special cases allow exponential accuracy in polynomial time.
- Second algorithm ensures local convergence when strong concavity is absent.

## Abstract

We propose two modified versions of the classical gradient ascent method to compute the capacity of finite-state channels with Markovian inputs. For the case that the channel mutual information is strongly concave in a parameter taking values in a compact convex subset of some Euclidean space, our first algorithm proves to achieve polynomial accuracy in polynomial time and, moreover, for some special families of finite-state channels our algorithm can achieve exponential accuracy in polynomial time under some technical conditions. For the case that the channel mutual information may not be strongly concave, our second algorithm proves to be at least locally convergent.