Hostname: page-component-77c89778f8-9q27g Total loading time: 0 Render date: 2024-07-23T22:31:41.831Z Has data issue: false hasContentIssue false
  • English
  • Français

Capacity bounds for the CDMA system and a neural network: a moderate deviations approach

Published online by Cambridge University Press:  21 July 2009

Matthias Löwe  and
Franck Vermet
Matthias Löwe
Affiliation:
Fachbereich Mathematik und Informatik, Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany; maloewe@math.uni-muenster.de
Franck Vermet
Affiliation:
Laboratoire de Mathématiques, UMR CNRS 6205, Université de Bretagne Occidentale, 6 avenue Victor Le Gorgeu CS 93837, 29238 Brest Cedex 3, France; Franck.Vermet@univ-brest.fr
Get access

Abstract

We study two systems that are based on sums of weakly dependent Bernoulli random variables that take values ± 1 with equal probabilities. We show that already one step of the so-called soft decision parallel interference cancellation, used in the third generation of mobile telecommunication CDMA, is able to considerably increase the number of users such a system can host. We also consider a variant of the well-known Hopfield model of neural networks. We show that this variant proposed by Amari and Yanai [CITE] has a larger storage capacity than the original model. Both situations lead to the question of the moderate deviations behavior of a sum of weakly dependent Bernoulli random variables. We prove a moderate deviations principle for such a sum on the appropriate scale.

Keywords

Moderate deviationslarge deviationsneural networksstorage capacityHopfield modelcode division multiple access (CDMA) systemsparallel interference cancellation
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 13 , July 2009 , pp. 343 - 362
Copyright
© EDP Sciences, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aazhang, B. and Varanasi, M.K., Multistage detection in asynchronous code division multiple acces communications. IEEE Trans. Commun. 38 (1990) 509–519.
Amariand, S. and Yanai, H.-F., Auto-associative memory with two-stage dynamics of nonmonotonic neurons. IEEE Trans. Neural Networks 7 (1996) 803–815.
Bahr, R.K. and Sadowski, J.S., Direct-sequence spread-spectrum multiple-access communications with random signature sequences: A large deviations analysis. IEEE Trans. Inform. Theory 37 (1991) 514–527.
Ben-Israel, A. and Charnes, A., Contribution to the theory of generalized inverses. J. SIAM 11 (1963) 667–699.
Bovier, A., Sharp upper bounds for perfect retrieval in the Hopfield model. J. Appl. Probab. 36 (1999) 941–950. CrossRef
A. Bovier, Statistical mechanics of disordered system: A mathematical perspective. Cambridge Series in Statistical and Probabilistic Mathematics 18. Cambridge University Press (2006).
A. Bovier and V. Gayrard, Hopfield models as a generalized mean field model, preprint. In Mathematics of spin glasses and neural networks, A. Bovier and P. Picco (Eds.). Progress in Probability, Birkhäuser (1998).
Buehrer, R.M. and Woerner, B.D., Analysis of adaptive multistage interference cancellation for CDMA using an improved Gaussian approximation. IEEE Trans. Commun. 44 (1996) 1308–1329. CrossRef
Buehrer, R.M., Kaul, A., Striglis, S. and Woerner, B.D., Analysis of DS-CDMA parallel interference cancellation with phase and timing errors. IEEE JSAC 14 (1996) 1522–1535.
Crespi, B., Storage capacity of non-monotonic neurons. Neural Networks 12 (1999) 1377–1389. CrossRef
de Jong, P., Central Limit Theorem, A for Generalized Multilinear Forms. J. Multiv. Anal. 34 (1990) 275–289. CrossRef
Dreyfus, G., Guyon, I. and Personnaz, L., Information storage and retrieval in spin-glass like neural networks. J. Phys. Lett. 46 (1985) L359–L365.
Dreyfus, G., Guyon, I. and Personnaz, L., Collective computational properties of neural networks: New learning mechanisms. Phys. Rev. A 34 (1986) 4217–4228.
Eichelsbacher, P. and Löwe, M., A large deviation principle for m-variate von Mises-statistics and U-statistics. J. Theoret. Probab. 8 (1995) 807–824. CrossRef
Eichelsbacher, P. and Löwe, M., Moderate deviations for i.i.d. random variables. ESAIM: PS 7 (2003) 209–218. CrossRef
Holtzman, J.M., A simple, accurate method to calculate spread spectrum multiple-access error probabilities. IEEE Trans. Commun. 40 (1992) 461–464. CrossRef
Hopfield, J.J., Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. USA 79 (1982) 2554–2558. CrossRef
M. Juntti, Multiuser demodulation for DS-CDMA systems in fading channels, Ph.D. thesis, University of Oulu, Finland, 1998.
Kanter, I. and Sampolinski, H., Associative recall of memory without errors. Phys. Rev. A 35 (1987) 380–392. CrossRef
M.J. Klok, G. Hooghiemstra, T. Ojanperä and R. Prasad, A novel technique for DS-CDMA system performance evaluation. VTC'99 spring, Houston, USA (1999).
Kobayashi, K., On the capacity of a neuron with a nonmonotone output function. Network 2 (1991) 237–243. CrossRef
König, W. and Mörters, P., Brownian intersection local times: Upper tail asymptotics and thick points. Ann. Probab. 30 (2002) 1605–1656.
M. Latva-aho, Advanced receivers for wideband CDMA systems, Ph.D. thesis, University of Oulu, Finland, 1999.
J.S. Lehnert and M.B. Pursley, Error probabilities for binary direct sequence spread-spectrum communications with random signature sequences. IEEE Trans. Commun. COM-35 (1987) 87–98.
J.S. Lehnert and R.K. Morrow, Bit-to-bit-error dependence in slotted DS/SSMA packet systems with random signature sequences. IEEE Trans. Commun. COM-37 (1989) 1052–1061.
Löwe, M., On the storage capacity of Hopfield models with weakly correlated patterns. Ann. Appl. Probab. 8 (1999) 1216–1250.
Löwe, M. and Vermet, F., The storage capacity of the Hopfield model and moderate deviations. Statist. Probab. Lett. 75 (2005) 237–248. CrossRef
Löwe, M. and Vermet, F., The Capacity of q-state Potts neural networks with parallel retrieval dynamics. Statist. Probab. Lett. 77 (2007) 1505–1514. CrossRef
Mathematical aspects of spin glasses and neural networks, in A. Bovier and P. Picco (Eds.). Progress in Probability, Birkhäuser, Boston (1998).
McEliece, R., Posner, E., Rodemich, E. and Venkatesh, S., The capacity of the Hopfield associative memory. IEEE Trans. Inform. Theory 33 (1987) 461–482. CrossRef
S.K. Mitra and C.R. Rao, Generalized inverse of matrices and its applications. Wiley, New York (1971).
Morita, M., Associative memory with nonmonotone dynamics. Neural Networks 6 (1993) 115–126. CrossRef
M. Morita, S. Yoshizawa and K. Nakano, Analysis and improvement of the dynamics of autocorrelation associative memory. Trans. Inst. Electron. Inform. Commun. Eng. Jpn J73-D-II (1990) 232–242.
Nishimori, N. and Opris, I., Retrieval process of an associative memory with nonmonotonic input-output function. IEEE Int. Conf. Neural Networks 1 (1993) 353–358. CrossRef
Palm, G., Memory capacities of local rules for synaptic modification. Concepts Neurosci. 2 (1991) 97–128.
Pastur, L.A. and Figotin, A.L., Exactly soluble model of a spin-glas. Sov. J. Low Temp. Phys. 3 (1977) 378–383.
D. Petritis, Thermodynamic formalism of neural computing; Nonlinear Phenomena of Complex Systems, volume 2, pp. 86–146. Kluwer Acad. Publ., Dordrecht (1996).
P. Picco, Artificial neural networks. A review from Physical and Mathematical point of view. Ann. Inst. H. Poincaré, Section A 64 (1996) 289–307.
R. Prasad, CDMA for wireless personal communications. Artech House (1996).
E. Rio, Théorie asymptotique des processus aléatoires faiblement dépendants. Springer (Ed.), Paris (2000).
Sunay, M.O. and Mclane, P.J., Calculating error probabilities for DS CDMA systems: When not to use the Gaussian approximation. IEEE Globecom 3 (1996) 1744–1749.
van der Hofstad, R. and Klok, M.J., Improving the performance of third-generation wireless communication systems. Adv. Appl. Probab. 36 (2004) 1046–1084. CrossRef
van der Hofstad, R., Hooghiemstra, G. and Klok, M.J., Large deviations for code division multiple access systems. SIAM J. Appl. Math. 62 (2002) 1044–1065.
van der Hofstad, R., Löwe, M. and Vermet, F., The effect of system load on the existence of bit-errors in CDMA with and without parallel interference cancelation. IEEE Trans. Inform. Theory 52 (2006) 4733–4741. CrossRef
F. Vermet, Étude asymptotique d'un réseau neuronal : le modèle de mémoire associative de Hopfield, Ph.D. thesis, University of Rennes 1, France, 1994.