Intersection of a lattice with hyperplan

E.В Nachmani, Y.В BeвЂ™ery and D.В Burshtein,.

вЂњCommunication Algorithms via Deep Learning,вЂќДЄrXiv preprint arXiv:1805.09317, MayВ 2018.Д¬omplexity of lattice problems, a cryptographic perspective.

H.В Kim, Y.В Jiang, R.В Rana, S.В Kannan, S.В Oh, P.В Viswanath,.

K.В Hornik, вЂњApproximation Capabilities of Multilayer Feedforward Networks,вЂќ.

I.В Goodfellow, Y.В Bengio, and A.В Courville.

вЂњLeech constellations of Construction-A lattices,вЂќ G.В Cybenko, вЂњApproximations by superpositions of sigmoidal functions,вЂќ Springer-Verlag, New York, 3rd edition, 1999. Theory, vol.В 49, no.В 4, pp.В 1030-1037, AprilВ 2003.ДЄ course in computational algebraic number theory. вЂњLattice decoding for joint detection in direct-sequence CDMA systems,вЂќ

J.J.В Boutros, N.В diВ Pietro, and Y.-C.В Huang,.

Neural Network Learning: Theoretical Fundations.

E.В Agrell, T.В Eriksson, A.В Vardy, and K.В Zeger,.

Figure 5: Neural lattice decoder with learning, NLD3, with constraints and L1 regularization, applied to the lattice D 4. Its performance is still Maximum-Likelihood like the HLD.Д¬aseВ 2 is also quasi-optimal, the slight loss is due to an imperfect training.Д¬aseВ 3 generates an error-floor while caseВ 4 exhibits a great robustness.ДЄ similar behavior of NLD3 was observed when utilized on other point lattices. The second hidden layer shrank from five to two neurons. The NLD3 model in caseВ 1 simplifies the Boolean equation of z 1 to two terms only (to be compared to the five AND conditions above): Of course, learning does not lead to the same model weights in these cases.

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For casesВ 3 and 4, in order to further simplify the model structure, three edges were pruned with two different strategies between the first and the second hidden layers. The performance of NLD3 with D 4 is given in Figure 5.Д¬asesВ 1 and 2 correspond to the results of L1 regularization with respectively Heaviside and sigmoid activation functions. Figure 4: Decoders with learning, NLD2, without constraint. We state that NLD2 is a competitive decoding algorithm for non-dense lattices. The ratio log 2 ( W ) n = 0.78 (sub-linear).ДЇrom its complexity and its illustrated performance, Made up of three hidden layers with 30-50-50 neurons respectively and W = 6280 parameters. The NLD2 neural network is not suited to decoding dense lattices.ДЇor T 55, the NLD2 in caseВ 1 has three hidden layers with 50-100-100 neurons respectively. The ratio log 2 ( W ) n = 2.0 (supra linear). Its performance is very close to MLD but this model has W = 83200 parameters and is too complex relative to HLD for E 8. , the NLD2 has three hidden layers each with 200 neurons. The lattice kissing number ( П„ ( E 8 ) = 240 and П„ ( T 55 ) = 30). In all NLD2 models, the size of first hidden layer is taken to be of the same order of magnitude as The performance of NLD2 is shown on Figure 4 for E 8 ( n = 8) and the MIMO lattice T 55 ( n = 16) takenДЇrom. The discussion of the sample complexity of NLD2 is omitted due to lack of space. To be competitive, the number of parameters of NLD2 should grow slower than 2 n. Without any constraint on its architecture. NLD2 is a standard fully-connected feed-forward sigmoid network ДЄ version of this theorem can be stated as follows : Was proved by G.В Cybenko for sigmoid networks ДЄnd then generalized by K.В Hornik to multilayer feed-forward architectures. The assumptions of the Universal Approximation Theorem. In order to help a neural network solve or approximate the CVP, we force the decoder input to satisfy Vector ^ z, except for low-density lattices where message passing solves directly the coordinates of ^ x. Indeed, the majority of known lattice decoders search for the closest lattice point ^ x = ^ z G by looking in Z n for the best In other words, to our modest knowledge, it appears that feed-forward networksДo not have the capability of observing the entire space R n to infer the value of a variable that belongs to an infinite set the probability of a given event involving that variable.

Other learning models are trained to produce a good estimate for a real numberД¬haracterizing one variable, e.g. Neural network classifiers are trained to take the best decision about the value of a variable that belongs to a finite set,

Intersection of a lattice with hyperplan