A Matrix Chernoff Bound for Markov Chains and Its Application to Co-occurrence Matrices
This provides foundational theoretical guarantees for analyzing co-occurrence statistics in sequential data, which is crucial for graph representation learning and machine learning applications.
The paper tackles the problem of deriving concentration bounds for matrix-valued random variables sampled via Markov chains, proving a Chernoff-type bound that yields exponentially decreasing tail distributions for eigenvalues. It applies this to co-occurrence matrices, showing that a trajectory length of O(τ(log(n) + log(τ))/ε²) achieves an estimator with error ε, with experimental results confirming the theoretical convergence rate.
We prove a Chernoff-type bound for sums of matrix-valued random variables sampled via a regular (aperiodic and irreducible) finite Markov chain. Specially, consider a random walk on a regular Markov chain and a Hermitian matrix-valued function on its state space. Our result gives exponentially decreasing bounds on the tail distributions of the extreme eigenvalues of the sample mean matrix. Our proof is based on the matrix expander (regular undirected graph) Chernoff bound [Garg et al. STOC '18] and scalar Chernoff-Hoeffding bounds for Markov chains [Chung et al. STACS '12]. Our matrix Chernoff bound for Markov chains can be applied to analyze the behavior of co-occurrence statistics for sequential data, which have been common and important data signals in machine learning. We show that given a regular Markov chain with $n$ states and mixing time $τ$, we need a trajectory of length $O(τ(\log{(n)}+\log{(τ)})/ε^2)$ to achieve an estimator of the co-occurrence matrix with error bound $ε$. We conduct several experiments and the experimental results are consistent with the exponentially fast convergence rate from theoretical analysis. Our result gives the first bound on the convergence rate of the co-occurrence matrix and the first sample complexity analysis in graph representation learning.