Witsenhausen’s counterexample as Assisted Interference Suppression

International Journal of Systems, Control and Communications

Volumn 2, Issue 1-3, 2010, Pages 197-237

  Grover, Pulkit ^a   Sahai, Anant ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

distributed control; information theory; Witsenhausen s counterexample

Indexed keywords

EID: 77955678502     PISSN: 17559340     EISSN: 17559359     Source Type: Journal    
DOI: 10.1504/IJSCC.2010.031164     Document Type: Article

References (35)

1. Avestimehr, A. (2008) Wireless Network Information Flow: A Deterministic Approach, PhD Thesis, UC Berkeley, Berkeley, CA.
2. Avestimehr, A., Diggavi, S. and Tse, D. (2007) ‘A deterministic approach to wireless relay networks’, Proc. Allerton Conference on Communications, Control and Computing, Monticello, IL, USA.
3. Baglietto, M., Parisini, T. and Zoppoli, R. (1997) ‘Nonlinear approximations for the solution of team optimal control problems’, Proceedings of the IEEE Conference on Decision and Control CDC, San Diego, CA, USA, pp.4592–4594.
4. Bansal, R. and Basar, T. (1987) ‘Stochastic teams with nonclassical information revisited: When is an affine control optimal?’, IEEE Transactions on Automatic Control, Vol. 32, pp.554–559.
5. Boyd, S. and Vandenberghe, L. (2004) Convex Optimization, Cambridge University Press, Cambridge, UK.
6. Costa, M. (1983) ‘Writing on dirty paper’, IEEE Trans. Inform. Theory, Vol. 29, pp.439–441.
7. Cover, T.M. and Thomas, J.A. (1991) Elements of Information Theory, 1st ed., Wiley, New York.
8. Dembo, A. and Zeitouni, O. (1998) Large Deviations Techniques and Applications, Springer, New York, NY, USA.
9. Devroye, N., Mitran, P. and Tarokh, V. (2006) ‘Achievable rates in cognitive radio channels’, IEEE Trans. Inform. Theory, Vol. 52, No. 5, pp.1813–1827.
10. Durrett, R. (2005) Probability: Theory and Examples, 1st ed., Brooks/Cole, Belmont, CA.
11. Etkin, R., Tse, D. and Wang, H. (2007) ‘Gaussian interference channel capacity to within one bit’, Submitted to IEEE Transactions on Information Theory, Vol. 54, No. 12, pp.5534–5562.
12. Gallager, R.G. (1971) Information Theory and Reliable Communication, John Wiley, New York, NY.
13. Gamal, A.E. and Cover, T. (1980) ‘Multiple user information theory’, IEEE Trans. Inform. Theory, Vol. 68, pp.1466–1483.
14. Ho, Y.C. and Chang, T. (1980) ‘Another look at the nonclassical information structure problem’, IEEE Transactions on Automatic Control, Vol. 25, No. 3, pp.537–540.
15. Ho, Y.C., Kastner, M.P. and Wong, E. (1978) ‘Teams, signaling, and information theory’, IEEE Trans. Automat. Contr., Vol. 23, pp.305–312.
16. Jovicic, A. and Viswanath, P. (2006) ‘Cognitive radio: an information-theoretic perspective’, Proceedings of the 2006 International Symposium on Information Theory, Seattle, WA, pp.2413–2417.
17. Kay, S.M. (1998) Fundamentals of Statistical Signal Processing: Detection Theory, Vol. 2, Prientice-Hall PTR, New Jersey, USA.
18. Kim, Y.H., Sutivong, A. and Cover, T.M. (2008) ‘State amplification’, IEEE Trans. Inform. Theory, Vol. 54, pp.1850–1859.
19. Kotagiri, S. and Laneman, J. (2008) ‘Multiaccess channels with state known to some encoders and independent messages’, EURASIP Journal on Wireless Communications and Networking, Vol. 2008, pp.1–14.
20. Lee, J.T., Lau, E. and Ho, Y.C.L. (2001) ‘The Witsenhausen counterexample: a hierarchical search approach for nonconvex optimization problems’, IEEE Trans. Automat. Contr., Vol. 46, No. 3, pp.382–397.
21. Martins, N.C. (2006) ‘Witsenhausen’s counter example holds in the presence of side information’, Proceedings of the 45th IEEE Conference on Decision and Control CDC, San Diego, CA, USA, pp.1111–1116.
22. Merhav, N. and Shamai, S. (2007) ‘Information rates subject to state masking’, IEEE Trans. Inform. Theory, Vol. 53, pp.2254–2261.
23. Mitter, S.K. and Sahai, A. (1999) ‘Information and control: witsenhausen revisited’, in Yamamoto, Y. and Hara, S. (Eds.): Learning, Control and Hybrid Systems: Lecture Notes in Control and Information Sciences, Springer, New York, NY, Vol. 241, pp.281–293.
24. Papadimitriou, C.H. and Tsitsiklis, J.N. (1986) ‘Intractable problems in control theory’, SIAM Journal on Control and Optimization, Vol. 24, pp.639–654.
25. Rotkowitz, M. (2006) ‘Linear controllers are uniformly optimal for the witsenhausen counterexample’, Proceedings of the 45th IEEE Conference on Decision and Control CDC, San Diego, CA, USA, pp.553–558.
26. Simon, M. (2002) Probability Distributions Involving Gaussian Random Variables, Springer, New York, NY, USA.
27. Tatikonda, S. (2000) Control under Communication Constraints, PhD Thesis, Massachusetts Institute of Technology, Cambridge, MA.
28. Witsenhausen, H.S. (1968) ‘A counterexample in stochastic optimum control’, SIAM Journal on Control, Vol. 6, pp.131–147.
29. Witsenhausen, H.S. (1971) ‘Separation of estimation and control for discrete time systems’, Proceedings of the IEEE, Vol. 59, pp.1557–1566.
30. Tatikonda used a similar long-vector formulation with parallel dynamics but joint control to show that the sequential-rate distortion function was indeed the right choice to bound performance in the special case when only the state is penalised (Tatikonda, 2000).
31. Witsenhausen himself states “There does not appear to exist any counterexample involving fewer variables…” than the one presented in Witsenhausen (1968).
32. The common randomness can be viewed as allowing averaging between different strategies at different times. As we use long vector lengths, such performance can in fact be asymptotically attained by cutting the vector into segments and using different strategies on different segments. Thus, all asymptotic results in this paper continue to hold even without the common-randomness assumption.
33. At large power, the rates scale logarithmically in power. Therefore, in this regime, characterising the rate within a constant number of bits is equivalent to characterising the power required to achieve a specified rate within a constant factor. So the results in this work are similar in spirit to these recent results in information theory.
34. The deterministic-model perspective on amplification and attenuation is that they both perform a bit-shift operation on the interference.
35. For the purpose of simplifying the exposition, the proof actually includes a few exponentially-rate cases when this is not done.