Abstract
We study the method of bounding the spectral gap of a reversible Markov chain by establishing canonical paths between the states. We provide natural examples where improved bounds can be obtained by allowing variable length functions on the edges. We give a simple heuristic for computing good length functions. Further generalization using multicommodity flow yields a bound which is an invariant of the Markov chain, and which can be computed at an arbitrary precision in polynomial time via semidefinite programming. We show that, for any reversible Markov chain on n states, this bound is off by a factor of order at most log2 n, and that this can be tight.
Based on DIMACS Tech Eeport 95-41, September 95. This work was partly done while the author was at the Massachusetts Institute of Technology, at the Institute for Mathematics and its Applications, at DIMACS and at XEROX Palo Alto Research Center.
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References
D. Aldous. Reversible Markov Chains and random walks on graphs. Book in preparation.
P. Diaconis and L. Saloff-Coste. Personal Communication, 1995.
P. Diaconis and L. Saloff-Coste. Comparison theorems for reversible Markov Chains. The Annals of Applied Probability, 3:696–730, 1993.
P. Diaconis and L. Saloff-Coste. What do we know about the Metropolis algorithm? In 27th Annual ACM Symposium on Theory of Computing, pages 112–129. ACM Press, 1995.
P. Diaconis and D. Stroock. Geometric bounds for eigenvalues of Markov Chains. The Annals of Applied Probability, 1:36–61, 1991.
J. Fill. Unpublished manuscript, July 1990.
J. Fill. Eigenvalue bounds on convergence to stationarity for nonreversible markov chains with an application to the exclusion process. The Annals of Applied Probability, 1:62–87, 1991.
M. R. Garey and D. S. Johnson. Computers and intractability: a guide to the theory of NP-completeness. Freeman and Company, San Fransisco, 1979.
M. Grötschel, L. Lovász, and A. Schrijver. The ellipsoid method and its consequences in combinatorial optimization. Combinatorica, 1:169–197, 1981.
M. Jerrum and A. Sinclair. Approximating the permanent. SIAM J. on Comput., 18:1149–1178, 1989.
N. Linial, E. London, and Y. Rabinovich. The geometry of graphs and some of its algorithmic applications. Combinatorica, 15:215–245, 1995.
L. Lovász and M. Simonovits. Random walks in a convex body and an improved volume algorithm. Random Strutures & Algorithms, 4(4):359–412, 1993.
E. Seneta. Non-negative matrices and Markov Chains. Springer-Verlag, 1981.
A. Sinclair. Improved bounds for mixing rates of Markov Chains and multicommodity flow. Combinatorics, Probability and Computing, 1:351–370, 1992.
A. Sinclair and M. Jerrum. Approximate counting, uniform generation, and rapidly mixing Markov Chains. Information and Computation, 82:93–113, 1989.
A. D. Sokal. Optimal Poincaré inequalities for the spectra of Markov Chains. Unpublished manuscript, September 1992.
L. Vandenberghe and S. Boyd. Semidefinite programming. SIAM Review, 1995. To appear.
G. A. Watson. Characterization of the subdiflerential of some matrix norms. Linear Algebra and Appl., 170:33–45, 1992.
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Kahale, N. (1996). A semidefinite bound for mixing rates of Markov chains. In: Cunningham, W.H., McCormick, S.T., Queyranne, M. (eds) Integer Programming and Combinatorial Optimization. IPCO 1996. Lecture Notes in Computer Science, vol 1084. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61310-2_15
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DOI: https://doi.org/10.1007/3-540-61310-2_15
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