Nous étudions deux systèmes basés sur des sommes de variables aléatoires de Bernoulli valant 1 avec égale probabilité et faiblement dépendantes. Nous montrons qu’une seule étape de la méthode de suppression d’interférences SD-PIC, utilisée dans la troisième génération de télécommunication mobile CDMA, permet déjà d’augmenter considérablement le nombre d’utilisateurs supporté par le système. Nous considérons également une variante du modèle neuronal de Hopfield. Nous montrons que cette variante, proposée par Amari et Yanai [2], admet une capacité de stockage supérieure au modèle original. Les deux situations conduisent à l’étude des déviations modérées d’une somme de variables aléatoires de Bernoulli faiblement corrélées. Nous montrons un principe de déviations modérées pour une telle somme convenablement normalisée.
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 [2] 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.
Mots clés : moderate deviations, large deviations, neural networks, storage capacity, Hopfield model, code division multiple access (CDMA) systems, parallel interference cancellation
@article{PS_2009__13__343_0, author = {L\"owe, Matthias and Vermet, Franck}, title = {Capacity bounds for the {CDMA} system and a neural network : a moderate deviations approach}, journal = {ESAIM: Probability and Statistics}, pages = {343--362}, publisher = {EDP-Sciences}, volume = {13}, year = {2009}, doi = {10.1051/ps:2008016}, mrnumber = {2528088}, language = {en}, url = {http://www.numdam.org/articles/10.1051/ps:2008016/} }
TY - JOUR AU - Löwe, Matthias AU - Vermet, Franck TI - Capacity bounds for the CDMA system and a neural network : a moderate deviations approach JO - ESAIM: Probability and Statistics PY - 2009 SP - 343 EP - 362 VL - 13 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2008016/ DO - 10.1051/ps:2008016 LA - en ID - PS_2009__13__343_0 ER -
%0 Journal Article %A Löwe, Matthias %A Vermet, Franck %T Capacity bounds for the CDMA system and a neural network : a moderate deviations approach %J ESAIM: Probability and Statistics %D 2009 %P 343-362 %V 13 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps:2008016/ %R 10.1051/ps:2008016 %G en %F PS_2009__13__343_0
Löwe, Matthias; Vermet, Franck. Capacity bounds for the CDMA system and a neural network : a moderate deviations approach. ESAIM: Probability and Statistics, Tome 13 (2009), pp. 343-362. doi : 10.1051/ps:2008016. http://www.numdam.org/articles/10.1051/ps:2008016/
[1] Multistage detection in asynchronous code division multiple acces communications. IEEE Trans. Commun. 38 (1990) 509-519.
and ,[2] Auto-associative memory with two-stage dynamics of nonmonotonic neurons. IEEE Trans. Neural Networks 7 (1996) 803-815.
and ,[3] Direct-sequence spread-spectrum multiple-access communications with random signature sequences: A large deviations analysis. IEEE Trans. Inform. Theory 37 (1991) 514-527.
and ,[4] Contribution to the theory of generalized inverses. J. SIAM 11 (1963) 667-699. | MR | Zbl
and ,[5] Sharp upper bounds for perfect retrieval in the Hopfield model. J. Appl. Probab. 36 (1999) 941-950. | MR | Zbl
,[6] Statistical mechanics of disordered system: A mathematical perspective. Cambridge Series in Statistical and Probabilistic Mathematics 18. Cambridge University Press (2006). | MR | Zbl
,[7] 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). | MR | Zbl
and ,[8] Analysis of adaptive multistage interference cancellation for CDMA using an improved Gaussian approximation. IEEE Trans. Commun. 44 (1996) 1308-1329.
and ,[9] Analysis of DS-CDMA parallel interference cancellation with phase and timing errors. IEEE JSAC 14 (1996) 1522-1535.
, , and ,[10] Storage capacity of non-monotonic neurons. Neural Networks 12 (1999) 1377-1389.
,[11] A Central Limit Theorem for Generalized Multilinear Forms. J. Multiv. Anal. 34 (1990) 275-289. | MR | Zbl
,[12] Information storage and retrieval in spin-glass like neural networks. J. Phys. Lett. 46 (1985) L359-L365.
, and ,[13] Collective computational properties of neural networks: New learning mechanisms. Phys. Rev. A 34 (1986) 4217-4228. | MR
, and ,[14] A large deviation principle for -variate von Mises-statistics and -statistics. J. Theoret. Probab. 8 (1995) 807-824. | MR | Zbl
and ,[15] Moderate deviations for i.i.d. random variables. ESAIM: PS 7 (2003) 209-218. | Numdam | MR | Zbl
and ,[16] A simple, accurate method to calculate spread spectrum multiple-access error probabilities. IEEE Trans. Commun. 40 (1992) 461-464. | Zbl
,[17] Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. USA 79 (1982) 2554-2558. | MR
,[18] Multiuser demodulation for DS-CDMA systems in fading channels, Ph.D. thesis, University of Oulu, Finland, 1998.
,[19] Associative recall of memory without errors. Phys. Rev. A 35 (1987) 380-392.
and ,[20] A novel technique for DS-CDMA system performance evaluation. VTC'99 spring, Houston, USA (1999).
, , and ,[21] On the capacity of a neuron with a nonmonotone output function. Network 2 (1991) 237-243. | MR
,[22] Brownian intersection local times: Upper tail asymptotics and thick points. Ann. Probab. 30 (2002) 1605-1656. | MR | Zbl
and ,[23] Advanced receivers for wideband CDMA systems, Ph.D. thesis, University of Oulu, Finland, 1999.
,[24] Error probabilities for binary direct sequence spread-spectrum communications with random signature sequences. IEEE Trans. Commun. COM-35 (1987) 87-98.
and ,[25] Bit-to-bit-error dependence in slotted DS/SSMA packet systems with random signature sequences. IEEE Trans. Commun. COM-37 (1989) 1052-1061.
and ,[26] On the storage capacity of Hopfield models with weakly correlated patterns. Ann. Appl. Probab. 8 (1999) 1216-1250. | Zbl
,[27] The storage capacity of the Hopfield model and moderate deviations. Statist. Probab. Lett. 75 (2005) 237-248. | MR | Zbl
and ,[28] The Capacity of -state Potts neural networks with parallel retrieval dynamics. Statist. Probab. Lett. 77 (2007) 1505-1514. | MR | Zbl
and ,[29] Mathematical aspects of spin glasses and neural networks, in A. Bovier and P. Picco (Eds.). Progress in Probability, Birkhäuser, Boston (1998). | MR | Zbl
[30] The capacity of the Hopfield associative memory. IEEE Trans. Inform. Theory 33 (1987) 461-482. | MR | Zbl
, , and ,[31] Generalized inverse of matrices and its applications. Wiley, New York (1971). | MR | Zbl
and ,[32] Associative memory with nonmonotone dynamics. Neural Networks 6 (1993) 115-126.
,[33] Analysis and improvement of the dynamics of autocorrelation associative memory. Trans. Inst. Electron. Inform. Commun. Eng. Jpn J73-D-II (1990) 232-242.
, and ,[34] Retrieval process of an associative memory with nonmonotonic input-output function. IEEE Int. Conf. Neural Networks 1 (1993) 353-358.
and ,[35] Memory capacities of local rules for synaptic modification. Concepts Neurosci. 2 (1991) 97-128.
,[36] Exactly soluble model of a spin-glas. Sov. J. Low Temp. Phys. 3 (1977) 378-383.
and ,[37] Thermodynamic formalism of neural computing; Nonlinear Phenomena of Complex Systems, volume 2, pp. 86-146. Kluwer Acad. Publ., Dordrecht (1996). | MR | Zbl
,[38] Artificial neural networks. A review from Physical and Mathematical point of view. Ann. Inst. H. Poincaré, Section A 64 (1996) 289-307. | Numdam | MR | Zbl
,[39] CDMA for wireless personal communications. Artech House (1996).
,[40] Théorie asymptotique des processus aléatoires faiblement dépendants. Springer (Ed.), Paris (2000). | MR | Zbl
,[41] Calculating error probabilities for DS CDMA systems: When not to use the Gaussian approximation. IEEE Globecom 3 (1996) 1744-1749.
and ,[42] Improving the performance of third-generation wireless communication systems. Adv. Appl. Probab. 36 (2004) 1046-1084. | MR | Zbl
and ,[43] Large deviations for code division multiple access systems. SIAM J. Appl. Math. 62 (2002) 1044-1065. | MR | Zbl
, and ,[44] 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. | MR
, and ,[45] Étude asymptotique d'un réseau neuronal : le modèle de mémoire associative de Hopfield, Ph.D. thesis, University of Rennes 1, France, 1994.
,Cité par Sources :