Singular value distribution of dense random matrices with block Markovian dependence

Abstract

A block Markov chain is a Markov chain whose state space can be partitioned into a finite number of clusters such that the transition probabilities only depend on the clusters. Block Markov chains thus serve as a model for Markov chains with communities. This paper establishes limiting laws for the singular value distributions of the empirical transition matrix and empirical frequency matrix associated to a sample path of the block Markov chain whenever the length of the sample path is Θ(n 2 ) with n the size of the state space.The proof approach is split into two parts. First, we introduce a class of symmetric random matrices with dependence called approximately uncorrelated random matrices with variance profile. We establish their limiting eigenvalue distributions by means of the moment method. Second, we develop a coupling argument to show that this general-purpose result applies to block Markov chains.

Detection and Evaluation of Clusters within Sequential Data

“…We observe a good match between the theory and the bulk of the distribution for both N and L. Particularly interesting is that the peak near zero and the triangular tail in the interval [4,5]. The theoretical there matches the observed distribution for N / √ n. Such features would not be predicted in a simpler model without communities such as a matrix with i.i.d.…”

“…The proof of Proposition 2 is similar to the proof of [70, Theorem 1.2] which is there given below [70,Proposition 4.7]. The intermediate [70,Lemma 4.4(ii)] should however be replaced by Lemma 1 below, and the role of [70,Equation (22)] is taken over by Lemma 2 below. Lemma 1.…”

“…As evident from the proofs, studying the spectrum of BMC can yield sharp bounds for recoverability [2]. Spectral properties of the random matrices constructed from sample paths of BMCs were investigated further in [6], [5]. The more recent paper [5] proves convergence of a bulk of singular values to a limiting distribution in the dense regime = Θ(n 2 ); a result that we use and refine in our experiments (see Section 6.4).…”

“…Using similar means as in the animal movement data in this paper, GPS coordinate sequences for New York City taxi trips are investigated in [1], [8], [5]. For both examples the lowdimensional representation reveals insight into taxi customer and animal movement behavior, respectively.…”

“…The method is based on a result from [5] which describes the histogram of the singular values of N in the asymptotic regime n → ∞ under the condition that = Θ(n 2 ). The results in [6] can further be interpreted as the statement that the K nonzero singular values of E[ N ] correspond to the K largest singular values of N .…”