arXiv subjects

Loic Brunel

Publications and source records attributed to Loic Brunel.

4 recordsLinked to original sources

arXiv2020

A lattice-based approach to the expressivity of deep ReLU neural networks

We present new families of continuous piecewise linear (CPWL) functions in Rn having a number of affine pieces growing exponentially in $n$. We show that these functions can be seen as the high-dimensional generalization of the triangle wave function used by Telgarsky in 2016. We prove that they can be computed by ReLU networks with quadratic depth and linear width in the space dimension. We also investigate the approximation error of one of these functions by shallower networks and prove a separation result. The main difference between our functions and other constructions is their practical interest: they arise in the scope of channel coding. Hence, computing such functions amounts to performing a decoding operation.

cs.LG↗

arXiv2019

On the CVP for the root lattices via folding with deep ReLU neural networks

Point lattices and their decoding via neural networks are considered in this paper. Lattice decoding in Rn, known as the closest vector problem (CVP), becomes a classification problem in the fundamental parallelotope with a piecewise linear function defining the boundary. Theoretical results are obtained by studying root lattices. We show how the number of pieces in the boundary function reduces dramatically with folding, from exponential to linear. This translates into a two-layer ReLU network requiring a number of neurons growing exponentially in n to solve the CVP, whereas this complexity becomes polynomial in n for a deep ReLU network.

cs.LG↗

arXiv2019

Neural Lattice Decoders

Lattice decoders constructed with neural networks are presented. Firstly, we show how the fundamental parallelotope is used as a compact set for the approximation by a neural lattice decoder. Secondly, we introduce the notion of Voronoi-reduced lattice basis. As a consequence, a first optimal neural lattice decoder is built from Boolean equations and the facets of the Voronoi cell. This decoder needs no learning. Finally, we present two neural decoders with learning. It is shown that L1 regularization and {\em a priori} information about the lattice structure lead to a simplification of the model.

cs.IT↗

arXiv2007

Space-time coding techniques with bit-interleaved coded modulations for MIMO block-fading channels

The space-time bit-interleaved coded modulation (ST-BICM) is an efficient technique to obtain high diversity and coding gain on a block-fading MIMO channel. Its maximum-likelihood (ML) performance is computed under ideal interleaving conditions, which enables a global optimization taking into account channel coding. Thanks to a diversity upperbound derived from the Singleton bound, an appropriate choice of the time dimension of the space-time coding is possible, which maximizes diversity while minimizing complexity. Based on the analysis, an optimized interleaver and a set of linear precoders, called dispersive nucleo algebraic (DNA) precoders are proposed. The proposed precoders have good performance with respect to the state of the art and exist for any number of transmit antennas and any time dimension. With turbo codes, they exhibit a frame error rate which does not increase with frame length.

cs.IT↗