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Reed-Muller Codes Polarize

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Reed-Muller Codes Polarize

Annual IEEE Symposium on Foundations of Computer Science

作者： Emmanuel Abbe Min Ye Mathematics Institute and the School of Computer and Communication Sciences at EPFL Switzerland Program in Applied and Computational Mathematics in Princeton University USA Department of Electrical Engineering Princeton University USA

Reed-Muller (RM) codes were introduced in 1954 and have long been conjectured to achieve Shannon's capacity on symmetric channels. The activity on this conjecture has recently been revived with the emergence of polar codes. RM codes and polar codes are generated by the same matrix G_m= [1/1 0/1] ^⊗m but using different subset of rows. RM codes select simply rows having largest weights. Polar codes select instead rows having the largest conditional mutual information proceeding top to down in G_m; while this is a more elaborate and channel-dependent rule, the top-to-down ordering allows Arikan to show that the conditional mutual information polarizes, and this gives directly a capacity-achieving code on any symmetric channel. RM codes are yet to be proved to have such a property, despite the recent success for the erasure channel. In this paper, we connect RM codes to polarization theory. We show that proceeding in the RM code ordering, i.e., not top-to-down but from the lightest to the heaviest rows in G_m, the conditional mutual information again polarizes. We further demonstrate that it does so faster than for polar codes. This implies that G_m contains another code, different than the polar code and called here the twin-RM code, that is provably capacity-achieving on any symmetric channel. This gives in particular a necessary condition for RM codes to achieve capacity on symmetric channels. It further gives a sufficient condition if the rows with largest conditional mutual information correspond to the heaviest rows, i.e., if the twin-RM code is the RM code. We demonstrate here that the two codes are at least similar and give further evidence that they are indeed the same.

关键词： Polar codes Mutual information Channel capacity Decoding Entropy Boolean functions

Generating function for projected entangled-pair states

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arXiv 2023年

作者： Tu, Wei-Lin Vanderstraeten, Laurens Schuch, Norbert Lee, Hyun-Yong Kawashima, Naoki Chen, Ji-Yao Faculty of Science and Technology Keio University 3-14-1 Hiyoshi Kohoku-ku Yokohama223-8522 Japan Center for Nonlinear Phenomena and Complex Systems Université Libre de Bruxelles Belgium University of Vienna Faculty of Physics Boltzmanngasse 5 Wien1090 Austria University of Vienna Faculty of Mathematics Oskar-Morgenstern-Platz 1 Wien1090 Austria Division of Display and Semiconductor Physics Korea University Sejong30019 Korea Republic of Department of Applied Physics Graduate School Korea University Sejong30019 Korea Republic of Interdisciplinary Program in E·ICT-Culture-Sports Convergence Korea University Sejong30019 Korea Republic of Institute for Solid State Physics The University of Tokyo Chiba Kashiwa277-8581 Japan Trans-scale Quantum Science Institute The University of Tokyo Bunkyo Tokyo113-0033 Japan Center for Neutron Science and Technology Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices School of Physics Sun Yat-sen University Guangzhou510275 China

Diagrammatic summation is a common bottleneck in modern applications of projected entangled-pair states, especially in computing low-energy excitations of a two-dimensional quantum many-body system. To solve this problem, here we extend the generating function approach for tensor network diagrammatic summation, a scheme previously proposed in the context of matrix product states. Taking the form of a one-particle excitation, we show that the excited state can be computed efficiently in the generating function formalism, which can further be used in evaluating the dynamical structure factor of the system. Our benchmark results for the spin-1/2 transverse-field Ising model and Heisenberg model on the square lattice provide a desirable accuracy, showing good agreement with known results. We then study the spin-1/2 J1-J2 model on the same lattice and investigate the dynamical properties of the putative gapless spin liquid phase. We conclude with a discussion on generalizations to multi-particle excitations. © 2023, CC BY.

关键词： Excited states

Two-Stage Maximum Score Estimator

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arXiv 2020年

作者： Gao, Wayne Yuan Xu, Sheng Xu, Kan Department of Economics University of Pennsylvania 133 S 36th St. PhiladelphiaPA19104 United States The Program in Applied and Computational Mathematics Princeton University Fine Hall Washington Road PrincetonNJ08544 United States

This paper considers the asymptotic theory of a semiparametric M-estimator that is generally applicable to models that satisfy a monotonicity condition in one or several parametric indexes. We call this estimator the two-stage maximum score (TSMS) estimator, since our estimator involves a first-stage nonparametric regression when applied to the binary choice model of Manski (1975, 1985). We characterize the asymptotic distribution of the TSMS estimator, which features phase transitions depending on the dimension of the first-stage estimation. Effectively, the first-stage nonparametric estimator serves as an imperfect smoothing function on a non-smooth criterion function, leading to the pivotality of the first-stage estimation error with respect to the second-stage convergence rate and asymptotic distribution. Copyright © 2020, The Authors. All rights reserved.

关键词： Functions

Concurrency and reachability in treelike temporal networks

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Physical Review E 2019年第6期100卷 062305-062305页

作者： Eun Lee Scott Emmons Ryan Gibson James Moody Peter J. Mucha Carolina Center for Interdisciplinary Applied Mathematics Department of Mathematics University of North Carolina at Chapel Hill Chapel Hill North Carolina 27599 USA Duke Network Analysis Center and Department of Sociology Duke University Durham North Carolina 27708 USA Department of Applied Physical Sciences University of North Carolina at Chapel Hill Chapel Hill North Carolina 27599 USA

Network properties govern the rate and extent of various spreading processes, from simple contagions to complex cascades. Recently, the analysis of spreading processes has been extended from static networks to temporal networks, where nodes and links appear and disappear. We focus on the effects of accessibility, whether there is a temporally consistent path from one node to another, and reachability, the density of the corresponding accessibility graph representation of the temporal network. The level of reachability thus inherently limits the possible extent of any spreading process on the temporal network. We study reachability in terms of the overall levels of temporal concurrency between edges and the structural cohesion of the network agglomerating over all edges. We use simulation results and develop heterogeneous mean-field model predictions for random networks to better quantify how the properties of the underlying temporal network regulate reachability.

关键词： Epidemic

Climate, food and humans predict communities of mammals in the United States

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Diversity and Distributions 2024年第9期30卷 e13900-e13900页

作者： Kays, Roland Snider, Matthew H. Hess, George Cove, Michael V. Jensen, Alex Shamon, Hila McShea, William J. Rooney, Brigit Allen, Maximilian L. Pekins, Charles E. Wilmers, Christopher C. Pendergast, Mary E. Green, Austin M. Suraci, Justin Leslie, Matthew S. Nasrallah, Sophie Farkas, Dan Jordan, Mark Grigione, Melissa LaScaleia, Michael C. Davis, Miranda L. Hansen, Chris Millspaugh, Josh Lewis, Jesse S. Havrda, Michael Long, Robert Remine, Kathryn R. Jaspers, Kodi J. Lafferty, Diana J. R. Hubbard, Tru Studds, Colin E. Barthelmess, Erika L. Andy, Katherine Romero, Andrea Hawkins, Melissa T. R. Lombardi, Jason V. Sergeyev, Maksim Fisher-Reid, M. Caitlin Rentz, Michael S. Nagy, Christopher Davenport, Jon M. Rega-Brodsky, Christine C. Appel, Cara L. Lesmeister, Damon B. Giery, Sean T. Whittier, Christopher A. Alston, Jesse M. Sutherland, Chris Rota, Christopher Murphy, Thomas Lee, Thomas E. Mortelliti, Alessio Bergman, Dylan L. Compton, Justin A. Gerber, Brian D. Burr, Jess Rezendes, Kylie DeGregorio, Brett A. Wehr, Nathaniel H. Benson, John F. Jachowski, David S. Gray, Morgan Beyer, Dean E. Belant, Jerrold L. Horan, Robert V. Lonsinger, Robert C. Kuhn, Kellie M. Hasstedt, Steven C. M. Zimova, Marketa Moore, Sophie M. Herrera, Daniel J. Fritts, Sarah Edelman, Andrew J. Flaherty, Elizabeth A. Petroelje, Tyler R. Neiswenter, Sean A. Risch, Derek R. Iannarilli, Fabiola van der Merwe, Marius Maher, Sean P. Farris, Zach J. Webb, Stephen L. Mason, David S. Lashley, Marcus A. Wilson, Andrew M. Vanek, John P. Wehr, Samuel R. Conner, L. Mike Beasley, James C. Bontrager, Helen L. Baruzzi, Carolina Ellis-Felege, Susan N. Proctor, Mike D. Schipper, Jan Weiss, Katherine C. B. Darracq, Andrea K. Barr, Evan G. Alexander, Peter D. Şekercioğlu, Çağan H. Bogan, Daniel A. Schalk, Christopher M. Fantle-Lepczyk, Jean E. Lepczyk, Christopher A. LaPoint, Scott Whipple, Laura S. Rowe, Helen Ivy Mullen, Kayleigh Bird, Tori Zorn, Adam Brandt, LaRoy Lathrop, Richard G. McCain, Craig Crupi, Anthony P. Clark, James Parsons, Arielle Department of Forestry and Environmental Resources North Carolina State University Raleigh NC United States North Carolina Museum of Natural Sciences Raleigh NC United States Department of Forestry and Environmental Conservation Clemson University Clemson SC United States Smithsonian's National Zoo and Conservation Biology Institute Front Royal VA United States Illinois Natural History Survey Prairie Research Institute University of Illinois Champaign IL United States Fort Hood Natural Resources Management Branch United States Army Garrison Fort Hood TX United States Environmental Studies Department Center for Integrated Spatial Research University of California Santa Cruz Santa Cruz CA United States Science Research Initiative University of Utah Salt Lake City UT United States Sageland Collaborative Salt Lake City UT United States Conservation Science Partners Inc. Truckee CA United States National Climate Adaptation Science Center United States Geological Survey Reston VA United States Biology Department Swarthmore College Swarthmore PA United States Entomology Department Cornell University Ithaca NY United States Information Technology Department Pace University Pleasantville NY United States Seattle University Department of Biology Seattle WA United States Department of Biology Pace University Pleasantville NY United States Department of Ecology and Evolutionary Biology Storrs CT United States Wildlife Biology Program Department of Ecosystem and Conservation Sciences University of Montana Missoula MT United States Department of Forest and Environmental Biology SUNY-ESF Syracuse NY United States Arizona State University College of Integrative Sciences and Arts Mesa AZ United States University of Washington School of Environmental and Forest Sciences Seattle WA United States Woodland Park Zoo Seattle WA United States Department of Biology Wildlife Ecology and Conservation Science Lab Northern Michigan University Marquette M

Aim: The assembly of species into communities and ecoregions is the result of interacting factors that affect plant and animal distribution and abundance at biogeographic scales. Here, we empirically derive ecoregions for mammals to test whether human disturbance has become more important than climate and habitat resources in structuring communities. Location: Conterminous United States. Time Period: 2010–2021. Major Taxa Studied: Twenty-five species of mammals. Methods: We analysed data from 25 mammal species recorded by camera traps at 6645 locations across the conterminous United States in a joint modelling framework to estimate relative abundance of each species. We then used a clustering analysis to describe 8 broad and 16 narrow mammal communities. Results: Climate was the most important predictor of mammal abundance overall, while human population density and agriculture were less important, with mixed effects across species. Seed production by forests also predicted mammal abundance, especially hard-mast tree species. The mammal community maps are similar to those of plants, with an east–west split driven by different dominant species of deer and squirrels. Communities vary along gradients of temperature in the east and precipitation in the west. Most fine-scale mammal community boundaries aligned with established plant ecoregions and were distinguished by the presence of regional specialists or shifts in relative abundance of widespread species. Maps of potential ecosystem services provided by these communities suggest high herbivory in the Rocky Mountains and eastern forests, high invertebrate predation in the subtropical south and greater predation pressure on large vertebrates in the west. Main Conclusions: Our results highlight the importance of climate to modern mammals and suggest that climate change will have strong impacts on these communities. Our new empirical approach to recognizing ecoregions has potential to be applied to expanded communities o

关键词： climate macroecology mammal communities masting species distribution models

Recursive projection-aggregation decoding of Reed-Muller codes

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arXiv 2019年

作者： Ye, Min Abbe, Emmanuel Mathematics Institute School of Computer and Communication Sciences EPFL Switzerland Program in Applied and Computational Mathematics Department of Electrical Engineering Princeton University Department of Electrical Engineering Princeton University United States

We propose a new class of efficient decoding algorithms for Reed-Muller (RM) codes over binary-input memoryless channels. The algorithms are based on projecting the code on its cosets, recursively decoding the projected codes (which are lower-order RM codes), and aggregating the reconstructions (e.g., using majority votes). We further provide extensions of the algorithms using list-decoding and code concatenation techniques. We run our main algorithm for AWGN channels and Binary Symmetric Channels at the short code length (≤ 1024) and low code rate (≤ 0.5) regime. Simulation results show that the new algorithm not only outperforms the previous decoding algorithms for RM codes, it also outperforms the optimal decoder for polar codes (SCL+CRC) with the same parameters by a wide margin. The performance of the new algorithm for RM codes in those regimes is in fact close to that of the maximal likelihood decoder. Finally, the new decoder naturally allows for parallel implementations. Copyright © 2019, The Authors. All rights reserved.

关键词： Decoding

Critical pore radius and transport properties of disordered hard- And overlapping-sphere models

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arXiv 2021年

作者： Klatt, Michael A. Ziff, Robert M. Torquato, Salvatore Department of Physics Princeton University PrincetonNJ08544 United States Institut für Theoretische Physik FAU Erlangen-Nürnberg Staudtstr. 7 Erlangen91058 Germany Center for the Study of Complex Systems Department of Chemical Engineering University of Michigan Ann ArborMI48109 United States Department of Physics Princeton University PrincetonNJ08544 United States Department of Chemistry Princeton Institute for the Science and Technology of Materials Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Transport properties of porous media are intimately linked to their pore-space microstructures. We quantify geometrical and topological descriptors of the pore space of certain disordered and ordered distributions of spheres, including pore-size functions and the critical pore radius δc. We focus on models of porous media derived from maximally random jammed sphere packings, overlapping spheres, equilibrium hard spheres, "quantizer" sphere packings, and crystalline sphere packings. For precise estimates of the percolation thresholds, we use a strict relation of the void percolation around sphere configurations to weighted bond percolation on the corresponding Voronoi networks. We use the Newman-Ziff algorithm to determine the percolation threshold using universal properties of the cluster size distribution. The critical pore radius δc is often used as the key characteristic length scale that determines the fluid permeability k. A recent study [Torquato. Adv. Wat. Resour. 140,103565 (2020)] suggested for porous media with a well-connected pore space an alternative estimate of k based on the second moment of the pore size hδ2i, which is easier to determine than δc. Here, we compare δc to the second moment of the pore size hδ2i, and indeed confirm that, for all porosities and all models considered, δc2 is to a good approximation proportional to hδ2i. However, unlike hδ2i, the permeability estimate based on δc2 does not predict the correct ranking of k for our models. Thus, we confirm hδ2i to be a promising candidate for convenient and reliable estimates of the fluid permeability for porous media with a well-connected pore space. Moreover, we compare the fluid permeability of our models with varying degrees of order, as measured by the τ order metric. We find that (effectively) hyperuniform models tend to have lower values of k than their nonhyperuniform counterparts. Our findings could facilitate the design of porous media with desirable transport properties via target

关键词： Porous materials

The impact of cosmic ray feedback during the epoch of reionisation

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arXiv 2025年

作者： Farcy, Marion Rosdahl, Joakim Dubois, Yohan Blaizot, Jérémy Martin-Alvarez, Sergio Haehnelt, Martin Kimm, Taysun Teyssier, Romain Institute for Physics Laboratory for Galaxy Evolution and Spectral Modelling EPFL Observatoire de Sauverny Chemin Pegasi 51 Versoix1290 Switzerland Centre de Recherche Astrophysique de Lyon CNRS UMR 5574 Univ. Lyon Ens de Lyon 9 avenue Charles André Saint-Genis-LavalF-69230 France Institut d’Astrophysique de Paris CNRS UMR 7095 Sorbonne Université 98 bis bd Arago Paris75014 France Stanford University StanfordCA94305 United States Kavli Institute for Cosmology Institute of Astronomy Madingley Road CambridgeCB3 0HA United Kingdom Department of Astronomy Yonsei University 50 Yonsei-ro Seodaemun-gu Seoul03722 Korea Republic of Department of Astrophysical Sciences Princeton University Peyton Hall PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University Fine Hall Washington Road PrincetonNJ08544-1000 United States

Galaxies form and evolve via a multitude of complex physics. In this work, we investigate the role of cosmic ray (CR) feedback in galaxy evolution and reionisation, by examining its impact on the escape of ionising radiation from galaxies. For this purpose, we present two sphinx cosmological radiation-magneto-hydrodynamics simulations, allowing for the first time a study of the impact of CR feedback on thousands of resolved galaxies during the Epoch of Reionisation (EoR). The simulations differ in their feedback prescriptions: one adopts a calibrated strong supernova (SN) feedback, while the other simulation reduces the strength of SN feedback and includes CR feedback instead. We show that both comparably regulate star formation, reasonably match observations of high-redshift UV luminosity functions, and produce a similar amount of hydrogen ionising photons. In contrast to the model with strong SN feedback, the model with CRs lead to incomplete reionisation, which is in strong disagreement with observational estimates of the reionisation history. This is due to CR feedback shaping the ISM differently, filling with gas the low density cavities otherwise carved by SN explosions. As a result, this reduces the escape of ionising photons, at any halo mass, and primarily in the close vicinity of the stars. Our study indicates that CR feedback regulates galaxy growth during the EoR, but negatively affects reionisation, a tension which paves the way for further exploration and refinement of existing galaxy formation and feedback models. Such improvements are crucial in order to capture and understand the process of reionisation and the underlying evolution of galaxies through cosmic time. © 2025, CC BY.

关键词： Cosmic rays

Localization transition in three-dimensional stealthy hyperuniform disordered systems

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arXiv 2021年

作者： Sgrignuoli, F. Torquato, S. Dal Negro, L. Department of Electrical and Computer Engineering & Photonics Center Boston University 8 Saint Mary’s Street BostonMA02215 United States Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials Program in Applied and Computational Mathematics Princeton University PrincetonNJ08540 United States Division of Material Science and Engineering Boston University 15 Saint Mary’s Street BrooklineMA02446 United States Department of Physics Boston University 590 Commonwealth Avenue BostonMA02215 United States

The purpose of this work is to understand the fundamental connection between structural correlations and light localization in three-dimensional (3D) open scattering systems of finite size. We numerically investigate the transport of vector electromagnetic waves scattered by resonant electric dipoles spatially arranged in 3D space by stealthy hyperuniform disordered point patterns. Three-dimensional stealthy hyperuniform disordered systems (3D-SHDS) are engineered with different structural correlation properties determined by their degree of stealthiness χ. Such fine control of exotic states of amorphous matter enables the systematic design of optical media that interpolate in a tunable fashion between uncorrelated random structures and crystalline materials. By solving the electromagnetic multiple scattering problem using Green’s matrix spectral method, we establish a transport phase diagram that demonstrates a clear delocalization-localization transition beyond a critical scattering density that depends on χ. The transition is characterized by studying the Thouless number and the spectral statistics of the scattering resonances. In particular, by tuning the χ parameter, we demonstrate large spectral gaps and suppressed sub-radiant proximity resonances, facilitating light localization. Moreover, consistently with previous studies, our results show a region of the transport phase diagram where the investigated scattering systems become transparent. Our work provides a systematic description of the transport and localization properties of light in stealthy hyperuniform structures and motivates the engineering of novel photonic systems with enhanced light-matter interactions for applications to both classical and quantum devices. Copyright © 2021, The Authors. All rights reserved.

关键词： Phase diagrams