Stochastic alternating projections

Illinois Journal of Mathematics

Volumn 54, Issue 3, 2010, Pages 963-979

  Diaconis, Persi ^a   Khare, Kshitij ^b   Saloff Coste, Laurent ^c  

^a STANFORD UNIVERSITY   (United States)

^b UNIVERSITY OF FLORIDA   (United States)

^c Department of Obstetrics Gynecology   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84861774271     PISSN: 00192082     EISSN: None     Source Type: Journal    
DOI: 10.1215/ijm/1336568522     Document Type: Review

References (35)

1. Y. Amit, Convergence properties of the Gibbs sampler for perturbations of Gaussians, Ann. Statist. 24 (1996), 122-140. MR 1389883
2. N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337-404. MR 0051437
3. K. Athreya, H. Doss and J. Sethuraman, On the convergence of the Markov chain simulation method, Ann. Statist. 24 (1996), 89-100. MR 1389881
4. H. Bauschke, Projection algorithms: Results and open problems, Inherently parallel algorithms in feasibility and optimization and their applications (Haifa 2000) (D. Butnariu, Y. Censor and S. Reich, eds.), North-Holland, Amserdam, 2001, pp. 11-22. MR 1853213
5. H. Bauschke, P. Combettes and S. Reich, The asymptotic behaviour of the composition of two resolvents, Nonlinear Anal. 60 (2005), 283-301. MR 2101879
6. H. Bauschke, P. Combettes and D. Luke, A strongly convergent reflection method for finding the projection onto the intersection of two closed convex sets in a Hilbert space, J. Approx. Theory 127 (2006), 178-192. MR 2058156
7. P. Berti, L. Pratelli and P. Rigo, Trivial intersections of σ-fields and Gibbs Sampling, Ann. Probab. 36 (2008), 2215-2234. MR 2478681
8. P. Berti, L. Pratelli and P. Rigo, Atomic intersection of σ-fields and some of its consequences, Probab. Theory Related Fields 148 (2010), 269-283. MR 2653229
9. A. Bufetov, Markov averaging and ergodic theorems for several operators, Topology, ergodic theory, real algebraic geometry, AMS Transl. Ser. 2, vol. 202, Amer. Math. Soc., Providence, RI, 2001, pp. 39-50. MR 1819180
10. D. L. Burkholder, Sufficiency in the undominated case, Ann. Math. Stat. 32 (1961), 1191-1200. MR 0131287
11. D. L. Burkholder, Successive conditional expectations of an integrable function, Ann. Math. Stat. 33 (1962), 887-893. MR 0143246
12. D. L. Burkholder, On the order structure of the set of sufficient subfields, Ann. Math. Stat. 33 (1962), 596-599. MR 0137227
13. D. L. Burkholder and Y. S. Chow, Iterates of conditional expectation operators, Proc. Amer. Math. Soc. 12 (1961), 490-495. MR 0142144
14. D. Butnariu, The expected projection method: Its behaviour and applications to linear operator equations and convex optimatization, J. Appl. Anal. 1 (2005), 93-108. MR 1390423
15. G. Casella and E. George, Explaining the Gibbs sampler, Amer. Statist. 46 (1992), 167-174. MR 1183069
16. G. Cohen, Iterates of a product of conditional expectation operators, J. Funct. Anal. 242 (2006), 658-668. MR 2274825
17. B. Delyon and F. Delyon, Generalization of von Neumann's spectral sets and integral representation of operators, Bull. Soc. Math. Fr. 127 (1999), 25-41. MR 1700467
18. F. Deutsch, The method of alternating orthogonal projections, Approximation Theory, Spline, Functions and Applications, Kluwer Academic, Dordrecht, 1992, pp. 105-121. MR 1165964
19. F. Deutsch, Best approximation in inner product spaces, Springer-Verlag, New York, 2001. MR 1823556
20. P. Diaconis, K. Khare and L. Saloff-Coste, Gibbs sampling, exponential families and orthogonal polynomials, Statist. Sci. 23 (2008), 151-178. MR 2446500
21. P. Diaconis, K. Khare and L. Saloff-Coste, Gibbs sampling, exponential families and coupling, Sankhya A 72 (2010), 136-169. MR 2658168
22. P. Diaconis and L. Saloff-Coste, Comparison techniques for reversible Markov chains, Ann. Appl. Probab. 3 (1993), 696-730. MR 1233621
23. I. Halperin, The product of projection operators, Acta Sci. Math. (Szeged) 23 (1962), 96-99. MR 0141978
24. G. Jones and J. Hobert, Honest exploration of intractable probability distributions via Markov chain Monte Carlo, Statist. Sci. 16 (2001), 312-334. MR 1888447
25. M. Kassabov, Subspace arrangements and property T, Groups Geom. Dyn. 5 (2011), 445-477.
26. H. J. Landau, An overview of time and frequency limiting, Fourier Techniques and Applications, Plenum, New York, 1985, pp. 201-220. MR 0819297
27. J. Liu, Monte Carlo strategies in scientific computing, Springer, New York, 2001. MR 1842342
28. S. P. Meyn and R. Tweedie, Markov chains and stochastic stability, 2nd ed., Cambridge Univ. Press, Cambridge, UK, 2009. MR 2509253
29. A. Netyanun and D. Solomon, Iterated products of projections on Hilbert spaces, American Mathematical Monthly 113 (2006), 644-648.
30. J. Neveu, Mathematical foundations of the calculus of probability, Holden-Day, San Francisco, 1965. MR 0198505
31. D. Ornstein, On the pointwise behaviour of iterates of a self adjoint operator, J. Math. Mech. 18 (1968), 473-477. MR 0236354
32. J. Rosenthal, Minorization conditions and convergence rates for Markov chain Monte Carlo, J. Amer. Statist. Assoc. 90 (1995), 558-566. MR 1340509
33. G. C. Rota, An "alternierende Verfahren" for general positive operators, Bull. Amer. Math. Soc. 68 (1962), 95-102. MR 0133847
34. H. A. Schwarz, Ueber einen Grenzũbergang durch alternirendes Verfahren, Vierteljahrsschrift der Naturforschenden Gessellschaft in Zurich 15 (1870), 272-286.
35. R. Zaharopol, On products of conditional expectation operators, Canad. Math. Bull. 33 (1990), 257-260. MR 1077092