Medical diagnostic testing can be made significantly more efficient using pooled testing protocols. These typically require a sparse infection signal and use either binary or real-valued entries of O(1). However, existing methods do not allow for inferring viral loads which span many orders of magnitude. We develop a message passing algorithm coupled with a PCR (Polymerase Chain Reaction) specific noise function to allow accurate inference of realistic viral load signals. This work is in the non-adaptive setting and could open the possibility of efficient screening where viral load determination is clinically important.

We study the space of functions computed by random-layered machines, including deep neural networks and Boolean circuits. Investigating the distribution of Boolean functions computed on the recurrent and layer-dependent architectures, we find that it is the same in both models. Depending on the initial conditions and computing elements used, we characterize the space of functions computed at the large depth limit and show that the macroscopic entropy of Boolean functions is either monotonically increasing or decreasing with the growing depth.

Mean field theory has been successfully used to analyze deep neura l networks (DNN) in the infinite size limit. Given the finite size of realistic D NN, we utilize the large deviation theory and path integral analysis to study the deviation of functions represented by DNN from their typical mean field solution s. The parameter perturbations investigated include weight sparsification (dilution) a nd binarization, which are commonly used in model simplification, for both ReLU and sign activation functions. We find that random networks with ReLU activation are m ore robust to parameter perturbations with respect to their counterparts wit h sign activation, which arguably is reflected in the simplicity of the functions they generate . Keywords: large deviation theory, path integral, deep neural networks, fu nction sensitivity 1. Introduction Learning machines realized by deep neural networks (DNN) have ac hieved impressive success in performing various machine learning tasks, such as spee ch recognition, image classification and natural language processing [1].

The problem of resource allocation in sparse graphs with real variables is studied using methods of statistical physics. An efficient distributed algorithm is devised on the basis of insight gained from the analysis and is examined using numerical simulations, showing excellent performance and full agreement with the theoretical results.

We analyze Gallager codes by employing a simple mean-field approximation that distorts the model geometry and preserves important interactions between sites. The method naturally recovers the probability propagation decoding algorithm as an extremization of a proper free-energy. We find a thermodynamic phase transition that coincides with information theoretical upper-bounds and explain the practical code performance in terms of the free-energy landscape.

We analyze Gallager codes by employing a simple mean-field approximation that distorts the model geometry and preserves important interactions between sites. The method naturally recovers the probability propagation decoding algorithm as an extremization of a proper free-energy. We find a thermodynamic phase transition that coincides with information theoretical upper-bounds and explain the practical code performance in terms of the free-energy landscape.

We analyze Gallager codes by employing a simple mean-field approximation thatdistorts the model geometry and preserves important interactions between sites. The method naturally recovers the probability propagation decodingalgorithm as an extremization of a proper free-energy. We find a thermodynamic phase transition that coincides with information theoreticalupper-bounds and explain the practical code performance in terms of the free-energy landscape.

The performance of regular and irregular Gallager-type errorcorrecting codeis investigated via methods of statistical physics. The transmitted codeword comprises products of the original message bitsselected by two randomly-constructed sparse matrices; the number of nonzero row/column elements in these matrices constitutes a family of codes. We show that Shannon's channel capacity may be saturated in equilibrium for many of the regular codes while slightly lower performance is obtained for others which may be of higher practical relevance. Decoding aspects are considered byemploying the TAP approach which is identical to the commonly used belief-propagation-based decoding. We show that irregular codes may saturate Shannon's capacity but with improved dynamical properties. 1 Introduction The ever increasing information transmission in the modern world is based on reliably communicatingmessages through noisy transmission channels; these can be telephone lines, deep space, magnetic storing media etc. Error-correcting codes play a significant role in correcting errors incurred during transmission; this is carried out by encoding the message prior to transmission and decoding the corrupted received code-word for retrieving the original message.

The performance of regular and irregular Gallager-type errorcorrecting code is investigated via methods of statistical physics. The transmitted codeword comprises products of the original message bits selected by two randomly-constructed sparse matrices; the number of nonzero row/column elements in these matrices constitutes a family of codes. We show that Shannon's channel capacity may be saturated in equilibrium for many of the regular codes while slightly lower performance is obtained for others which may be of higher practical relevance. Decoding aspects are considered by employing the TAP approach which is identical to the commonly used belief-propagation-based decoding. We show that irregular codes may saturate Shannon's capacity but with improved dynamical properties. 1 Introduction The ever increasing information transmission in the modern world is based on reliably communicating messages through noisy transmission channels; these can be telephone lines, deep space, magnetic storing media etc. Error-correcting codes play a significant role in correcting errors incurred during transmission; this is carried out by encoding the message prior to transmission and decoding the corrupted received code-word for retrieving the original message.

We study the dynamics of supervised learning in layered neural networks, inthe regime where the size p of the training set is proportional to the number N of inputs. Here the local fields are no longer described by Gaussian distributions.