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Fundamental Limits of Weak Recovery with Applications to Phase Retrieval
TLDR
It is proved that—in the high-dimensional limit—a sharp phase transition takes place, and the threshold of the proposed spectral method is located in the regime of vanishingly small noise.
Reed–Muller Codes Achieve Capacity on Erasure Channels
TLDR
This work shows that symmetry alone implies near-optimal performance in any sequence of linear codes where the blocklengths are strictly increasing, the code rates converge, and the permutation group of each code is doubly transitive.
Unified scaling of polar codes: Error exponent, scaling exponent, moderate deviations, and error floors
TLDR
A unified framework to characterize the relationship between R, N, P<sub>e</sub>, and W is developed and it is proved that polar codes are not affected by error floors.
On the Decoding of Polar Codes on Permuted Factor Graphs
TLDR
For the 5G polar code of length 1024, the error-correction performance of the proposed decoder is more than 0.25 dB better than that of the BP decoder with the same number of random permutations at the frame error rate of 0.0001.
How to achieve the capacity of asymmetric channels
TLDR
It is proved that any suitable source code can be combined with any suitable channel code in order to provide optimal schemes for asymmetric channels, and both polar codes and spatially coupled codes fulfill the required conditions.
Global Convergence of Deep Networks with One Wide Layer Followed by Pyramidal Topology
TLDR
It is proved that, for deep networks, a single layer of width following the input layer suffices to ensure a similar guarantee of global minimum for over-parameterized neural networks.
Achieving Marton’s Region for Broadcast Channels Using Polar Codes
TLDR
Polar coding schemes for the two-user discrete memoryless broadcast channel (DM-BC) are presented which achieve Marton's region with both common and private messages and can be adopted to many other multiterminal scenarios whenever there polar indices need to be aligned.
From Polar to Reed-Muller Codes: A Technique to Improve the Finite-Length Performance
TLDR
Empirically the performance of polar codes at finite block lengths is boosted by moving along the family C-inter even under low-complexity decoding schemes such as, for instance, belief propagation or successive cancellation list decoder.
Construction of polar codes with sublinear complexity
TLDR
The proof technique consists in reducing the construction problem to the problem of computing the maximum cardinality of an antichain for a suitable partially ordered set and it is shown that, if the partial order is taken, one can construct a polar code by computing the reliability of roughly N/ log3/2 N synthetic channels.
Analysis of a Two-Layer Neural Network via Displacement Convexity
TLDR
It is proved that, in the limit in which the number of neurons diverges, the evolution of gradient descent converges to a Wasserstein gradient flow in the space of probability distributions over $\Omega$, which exhibits a special property known as displacement convexity.
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