A Note on Projection-Based Recovery of Clusters in Markov Chains
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Let T_0 be the transition matrix of a purely clustered Markov chain, i.e. a direct sum of k ≥ 2 irreducible stochastic matrices. Given a perturbation T(x) = T_0 + xE of T_0 such that T(x) is also stochastic, how small must x be in order for us to recover the indices of the direct summands of T_0? We give a simple algorithm based on the orthogonal projection matrix onto the left or right singular subspace corresponding to the k smallest singular values of I - T(x) which allows for exact recovery all clusters when x = O(σ_n - k/||E||_2√(n_1)) and approximate recovery of a single cluster when x = O(σ_n - k/||E||_2), where n_1 is the size of the largest cluster and σ_n - k the (k + 1)st smallest singular value of T_0.
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