A Tutorial on the Spectral Theory of Markov Chains

TL;DR

This tutorial offers an accessible introduction to the spectral properties of Markov chains, emphasizing their connections to graphs, eigenvalues, and applications in machine learning, with a focus on intuition over formal proofs.

Contribution

It compiles known results on Markov chain spectral theory, introduces new concepts, and emphasizes intuitive understanding for a broad audience.

Findings

Eigenvalues characterize Markov chain convergence.

Spectral properties relate to graph structures and random walks.

Applications in machine learning and data mining are discussed.

Abstract

Markov chains are a class of probabilistic models that have achieved widespread application in the quantitative sciences. This is in part due to their versatility, but is compounded by the ease with which they can be probed analytically. This tutorial provides an in-depth introduction to Markov chains, and explores their connection to graphs and random walks. We utilize tools from linear algebra and graph theory to describe the transition matrices of different types of Markov chains, with a particular focus on exploring properties of the eigenvalues and eigenvectors corresponding to these matrices. The results presented are relevant to a number of methods in machine learning and data mining, which we describe at various stages. Rather than being a novel academic study in its own right, this text presents a collection of known results, together with some new concepts. Moreover, the tutorial focuses on offering intuition to readers rather than formal understanding, and only assumes basic exposure to concepts from linear algebra and probability theory. It is therefore accessible to students and researchers from a wide variety of disciplines.