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For applications in concatenated coding for optical communications systems, we examine the encoding and soft-decoding of short spherical codes constructed as constant-energy shells of the Cartesian power of pulse amplitude modulation constellations. These are unions of permutation codes having the same average power. We construct a list decoder for permutation codes by adapting Murty's algorithm, which is then used to determine mutual information curves for these permutation codes. In the process, we discover a straightforward expression for determining the likelihood of large subcodes of permutation codes. We refer to these subcodes, obtained by all possible sign flips of a given permutation codeword, as orbits. We introduce a simple process, which we call orbit decoding with frozen symbols, that allows us to extract soft information from noisy permutation codewords. In a sample communication system with probabilistic amplitude shaping protected by a standard low-density parity-check code that employs short permutation codes, we demonstrate that orbit decoding with frozen symbols provides a gain of about 0.3 dB in signal-to-noise ratio compared to the traditional symbol-by-symbol decoding. By using spherical codes composed of unions of permutation codes, we can increase the input entropy compared to using permutation codes alone. In one scheme, we consider a union of a small number of permutation codes. In this case, orbit decoding with frozen symbols provides about 0.2 dB gain compared to the traditional method. In another scheme, we use all possible permutations to form a spherical code that exhibits a computationally feasible trellis representation. The soft information obtained using the BCJR algorithm outperforms the traditional symbol-by-symbol method by 0.1 dB.

Characterization of directly imaged exoplanets is one of the most eagerly anticipated science functions of the James Webb Space Telescope. MIRI, the mid-IR instrument has the capability to provide unique spatially resolved photometric data points in a spectral range never achieved so far for such objects. We aim to present the very first on-sky contrast measurements of the MIRI's coronagraphs. In addition to a classical Lyot coronagraph at the longest wavelength, this observing mode implements the concept of the four quadrant phase mask for the very first time in a space telescope. We observed single stars together with a series of reference stars to measure raw contrasts as they are delivered on the detector, as well as reference subtracted contrasts. MIRI's coronagraphs achieve raw contrasts greater than $10^3$ at the smallest angular separations (within $1''$) and about $10^5$ further out (beyond $5\sim6''$). Subtracting the residual diffracted light left unattenuated by the coronagraph has the potential to bring the final contrast down to the background and detector limited noise floor at most angular separations (a few times $10^4$ at less than $1''$). MIRI coronagraphs behave as expected from simulations. In particular the raw contrasts for all four coronagraphs are fully consistent with the diffractive model. Contrasts obtained with subtracting reference stars also meet expectations and are fully demonstrated for two four quadrant phase masks (F1065C and F1140C). The worst contrast, measured at F1550C, is very likely due to a variation of the phase aberrations at the primary mirror during the observations, and not an issue of the coronagraph itself. We did not perform reference star subtraction with the Lyot mask at F2300C, but we anticipate that it would bring the contrast down to the noise floor., Comment: submitted to A&A

We consider the controllability of large-scale linear networked dynamical systems when complete knowledge of network structure is unavailable and knowledge is limited to coarse summaries. We provide conditions under which average controllability of the fine-scale system can be well approximated by average controllability of the (synthesized, reduced-order) coarse-scale system. To this end, we require knowledge of some inherent parametric structure of the fine-scale network that makes this type of approximation possible. Therefore, we assume that the underlying fine-scale network is generated by the stochastic block model (SBM) -- often studied in community detection. We then provide an algorithm that directly estimates the average controllability of the fine-scale system using a coarse summary of SBM. Our analysis indicates the necessity of underlying structure (e.g., in-built communities) to be able to quantify accurately the controllability from coarsely characterized networked dynamics. We also compare our method to that of the reduced-order method and highlight the regimes where both can outperform each other. Finally, we provide simulations to confirm our theoretical results for different scalings of network size and density, and the parameter that captures how much community-structure is retained in the coarse summary.

Elasticity is offered by cloud service providers to exploit under-utilized computing resources. The low-cost elastic nodes can leave and join any time during the computation cycle. The possibility of elastic events occurring together with the problem of slow nodes, referred to as stragglers, increases the uncertainty of the system, leading to computation delay. Recent results have shown that coded computing can be used to reduce the negative effect of elasticity and stragglers. In this paper, we propose two hierarchical coded elastic computing schemes that can further speed up the system by exploiting stragglers and effectively allocating tasks among available nodes. In our simulations, our scheme realizes 45% improvement in average finishing time compared to the state-of-the-art coded elastic computing scheme.

Resistive memories store information in a crossbar arrangement of two-terminal devices that can be programmed to patterns of high or low resistance. While extremely compact, this technology suffers from the "sneak-path" problem: certain information patterns cannot be recovered, as multiple low resistances in parallel make a high resistance indistinguishable from a low resistance. In this paper, a multi-layer device is considered, and the number of bits it can store is derived exactly and asymptotic bounds are developed. The information density of a series of isolated arrays with extreme aspect ratios is derived in the single- and multi-layer cases with and without peripheral selection circuitry. This density is shown to be non-zero in the limit, unlike that of the arrays with moderate aspect ratios previously considered. A simple encoding scheme that achieves capacity asymptotically is presented.

Coded distributed computing was recently introduced to mitigate the effect of stragglers on distributed computing. This paper combines ideas of approximate computing with coded computing to further accelerate computation. We propose successive approximation coding (SAC) techniques that realize a tradeoff between accuracy and speed, allowing the distributed computing system to produce approximations that increase in accuracy over time. If a sufficient number of compute nodes finish their tasks, SAC exactly recovers the desired computation. We theoretically provide design guidelines for our SAC techniques, and numerically show that SAC achieves a better accuracy-speed tradeoff in comparison with previous methods.

An increasing bottleneck in decentralized optimization is communication. Bigger models and growing datasets mean that decentralization of computation is important and that the amount of information exchanged is quickly growing. While compression techniques have been introduced to cope with the latter, none has considered leveraging the temporal correlations that exist in consecutive vector updates. An important example is distributed momentum-SGD where temporal correlation is enhanced by the low-pass-filtering effect of applying momentum. In this paper we design and analyze compression methods that exploit temporal correlation in systems both with and without error-feedback. Experiments with the ImageNet dataset demonstrate that our proposed methods offer significant reduction in the rate of communication at only a negligible increase in computation complexity. We further analyze the convergence of SGD when compression is applied with error-feedback. In the literature, convergence guarantees are developed only for compressors that provide error-bounds point-wise, i.e., for each input to the compressor. In contrast, many important codes (e.g. rate-distortion codes) provide error-bounds only in expectation and thus provide a more general guarantee. In this paper we prove the convergence of SGD under an expected error assumption by establishing a bound for the minimum gradient norm., Comment: This paper was presented in part at the 11th International Symposium on Topics in Coding (ISTC), Montreal, QC, Canada, August 2021, and the paper has been accepted for publication in the IEEE Journal on Selected Areas in Information Theory (JSAIT) Volume 2, Issue 3 (2021), https://ieeexplore.ieee.org/document/9511618

We leverage state-of-the-art machine learning methods and a decade's worth of archival data from CFHT to predict observatory image quality (IQ) from environmental conditions and observatory operating parameters. Specifically, we develop accurate and interpretable models of the complex dependence between data features and observed IQ for CFHT's wide-field camera, MegaCam. Our contributions are several-fold. First, we collect, collate and reprocess several disparate data sets gathered by CFHT scientists. Second, we predict probability distribution functions (PDFs) of IQ and achieve a mean absolute error of $\sim0.07''$ for the predicted medians. Third, we explore the data-driven actuation of the 12 dome "vents" installed in 2013-14 to accelerate the flushing of hot air from the dome. We leverage epistemic and aleatoric uncertainties in conjunction with probabilistic generative modeling to identify candidate vent adjustments that are in-distribution (ID); for the optimal configuration for each ID sample, we predict the reduction in required observing time to achieve a fixed SNR. On average, the reduction is $\sim12\%$. Finally, we rank input features by their Shapley values to identify the most predictive variables for each observation. Our long-term goal is to construct reliable and real-time models that can forecast optimal observatory operating parameters to optimize IQ. We can then feed such forecasts into scheduling protocols and predictive maintenance routines. We anticipate that such approaches will become standard in automating observatory operations and maintenance by the time CFHT's successor, the Maunakea Spectroscopic Explorer, is installed in the next decade., Comment: 25 pages and 12 figures (main) + 7 pages and 8 figures (2 appendices). Accepted for publication in MNRAS

We study the problem of community recovery from coarse measurements of a graph. In contrast to the problem of community recovery of a fully observed graph, one often encounters situations when measurements of a graph are made at low-resolution, each measurement integrating across multiple graph nodes. Such low-resolution measurements effectively induce a coarse graph with its own communities. Our objective is to develop conditions on the graph structure, the quantity, and properties of measurements, under which we can recover the community organization in this coarse graph. In this paper, we build on the stochastic block model by mathematically formalizing the coarsening process, and characterizing its impact on the community members and connections. Through this novel setup and modeling, we characterize an error bound for community recovery. The error bound yields simple and closed-form asymptotic conditions to achieve the perfect recovery of the coarse graph communities.

Distributed optimization is widely deployed in practice to solve a broad range of problems. In a typical asynchronous scheme, workers calculate gradients with respect to out-of-date optimization parameters while the master uses stale (i.e., delayed) gradients to update the parameters. While using stale gradients can slow the convergence, asynchronous methods speed up the overall optimization with respect to wall clock time by allowing more frequent updates and reducing idling times. In this paper, we present a variable per-epoch minibatch scheme called Anytime Minibatch with Delayed Gradients (AMB-DG). In AMB-DG, workers compute gradients in epochs of a fixed time while the master uses stale gradients to update the optimization parameters. We analyze AMB-DG in terms of its regret bound and convergence rate. We prove that for convex smooth objective functions, AMB-DG achieves the optimal regret bound and convergence rate. We compare the performance of AMB-DG with that of Anytime Minibatch (AMB) which is similar to AMB-DG but does not use stale gradients. In AMB, workers stay idle after each gradient transmission to the master until they receive the updated parameters from the master while in AMB-DG workers never idle. We also extend AMB-DG to the fully distributed setting. We compare AMB-DG with AMB when the communication delay is long and observe that AMB-DG converges faster than AMB in wall clock time. We also compare the performance of AMB-DG with the state-of-the-art fixed minibatch approach that uses delayed gradients. We run our experiments on a real distributed system and observe that AMB-DG converges more than two times., Comment: Accepted for publication in the IEEE Transaction on Signal and Information Processing over Networks

Intelligent scheduling of the sequence of scientific exposures taken at ground-based astronomical observatories is massively challenging. Observing time is over-subscribed and atmospheric conditions are constantly changing. We propose to guide observatory scheduling using machine learning. Leveraging a 15-year archive of exposures, environmental, and operating conditions logged by the Canada-France-Hawaii Telescope, we construct a probabilistic data-driven model that accurately predicts image quality. We demonstrate that, by optimizing the opening and closing of twelve vents placed on the dome of the telescope, we can reduce dome-induced turbulence and improve telescope image quality by (0.05-0.2 arc-seconds). This translates to a reduction in exposure time (and hence cost) of $\sim 10-15\%$. Our study is the first step toward data-based optimization of the multi-million dollar operations of current and next-generation telescopes., Comment: 4 pages, 3 figures. Accepted to Machine Learning and the Physical Sciences Workshop at the 34th Conference on Neural Information Processing Systems (NeurIPS)

Data-driven graph learning models a network by determining the strength of connections between its nodes. The data refers to a graph signal which associates a value with each graph node. Existing graph learning methods either use simplified models for the graph signal, or they are prohibitively expensive in terms of computational and memory requirements. This is particularly true when the number of nodes is high or there are temporal changes in the network. In order to consider richer models with a reasonable computational tractability, we introduce a graph learning method based on representation learning on graphs. Representation learning generates an embedding for each graph node, taking the information from neighbouring nodes into account. Our graph learning method further modifies the embeddings to compute the graph similarity matrix. In this work, graph learning is used to examine brain networks for brain state identification. We infer time-varying brain graphs from an extensive dataset of intracranial electroencephalographic (iEEG) signals from ten patients. We then apply the graphs as input to a classifier to distinguish seizure vs. non-seizure brain states. Using the binary classification metric of area under the receiver operating characteristic curve (AUC), this approach yields an average of 9.13 percent improvement when compared to two widely used brain network modeling methods.

Motivated by the emerging area of graph signal processing (GSP), we introduce a novel method to draw inference from spatiotemporal signals. Data acquisition in different locations over time is common in sensor networks, for diverse applications ranging from object tracking in wireless networks to medical uses such as electroencephalography (EEG) signal processing. In this paper we leverage novel techniques of GSP to develop a hierarchical feature extraction approach by mapping the data onto a series of spatiotemporal graphs. Such a model maps signals onto vertices of a graph and the time-space dependencies among signals are modeled by the edge weights. Signal components acquired from different locations and time often have complicated functional dependencies. Accordingly, their corresponding graph weights are learned from data and used in two ways. First, they are used as a part of the embedding related to the topology of graph, such as density. Second, they provide the connectivities of the base graph for extracting higher level GSP-based features. The latter include the energies of the signal's graph Fourier transform in different frequency bands. We test our approach on the intracranial EEG (iEEG) data set of the Kaggle epileptic seizure detection contest. In comparison to the winning code, the results show a slight net improvement and up to 6 percent improvement in per subject analysis, while the number of features are decreased by 75 percent on average.

In this paper we propose a new framework for distributed source coding of structured sources, such as sparse signals. Our framework capitalizes on recent advances in the theory of linear inverse problems and signal representations using incoherent projections. Our approach acquires and quantizes incoherent linear measurements of the signal, which are represented as separate bitplanes. Each bitplane is coded using a distributed source code of the appropriate rate, and transmitted. The decoder, starts from the least significant biplane and, using a prediction of the signal as side information, iteratively recovers each bitplane based on the source prediction and the signal, assuming all the previous bitplanes of lower significance have already been recovered. We provide theoretical results guiding the rate selection, relying only on the least squares prediction error of the source. This is in contrast to existing approaches which rely on difficult-to-estimate information-theoretic metrics to set the rate. We validate our approach using simulations on remote-sensing multispectral images, comparing them with existing approaches of similar complexity., Comment: 16 pages, 8 figures

This document summarizes how far the Nancy Grace Roman Space Telescope Coronagraph Instrument (Roman CGI) will go toward demonstrating high-contrast imaging and spectroscopic requirements for potential future exoplanet direct imaging missions, illustrated by the HabEx and LUVOIR concepts. The assessment is made for two levels of assumed CGI performance: (i) current best estimate (CBE) as of August 2020, based on laboratory results and realistic end-to-end simulations with JPL-standard Model Uncertainty Factors (MUFs); (ii) CGI design specifications inherited from Phase B requirements. We find that the predicted performance (CBE) of many CGI subsystems compares favorably with the needs of future missions, despite providing more modest point source detection limits than future missions. This is essentially due to the challenging pupil of the Roman Space Telescope; this pupil pushes the coronagraph masks sensitivities to misalignments to be commensurate with future missions. In particular, CGI will demonstrate active low-order wavefront control and photon counting capabilities at levels of performance either higher than, or comparable to, the needs of future missions., Comment: 10 pages, 3 tables. Revised version v2: added some co-authors

Distributed optimization is vital in solving large-scale machine learning problems. A widely-shared feature of distributed optimization techniques is the requirement that all nodes complete their assigned tasks in each computational epoch before the system can proceed to the next epoch. In such settings, slow nodes, called stragglers, can greatly slow progress. To mitigate the impact of stragglers, we propose an online distributed optimization method called Anytime Minibatch. In this approach, all nodes are given a fixed time to compute the gradients of as many data samples as possible. The result is a variable per-node minibatch size. Workers then get a fixed communication time to average their minibatch gradients via several rounds of consensus, which are then used to update primal variables via dual averaging. Anytime Minibatch prevents stragglers from holding up the system without wasting the work that stragglers can complete. We present a convergence analysis and analyze the wall time performance. Our numerical results show that our approach is up to 1.5 times faster in Amazon EC2 and it is up to five times faster when there is greater variability in compute node performance., Comment: International Conference on Learning Representations (ICLR), May 2019, New Orleans, LA, USA

In distributed computing systems slow working nodes, known as stragglers, can greatly extend finishing times. Coded computing is a technique that enables straggler-resistant computation. Most coded computing techniques presented to date provide robustness by ensuring that the time to finish depends only on a set of the fastest nodes. However, while stragglers do compute less work than non-stragglers, in real-world commercial cloud computing systems (e.g., Amazon's Elastic Compute Cloud (EC2)) the distinction is often a soft one. In this paper, we develop hierarchical coded computing that exploits the work completed by all nodes, both fast and slow, automatically integrating the potential contribution of each. We first present a conceptual framework to represent the division of work amongst nodes in coded matrix multiplication as a cuboid partitioning problem. This framework allows us to unify existing methods and motivates new techniques. We then develop three methods of hierarchical coded computing that we term bit-interleaved coded computation (BICC), multilevel coded computation (MLCC), and hybrid hierarchical coded computation (HHCC). In this paradigm, each worker is tasked with completing a sequence (a hierarchy) of ordered subtasks. The sequence of subtasks, and the complexity of each, is designed so that partial work completed by stragglers can be used, rather than ignored. We note that our methods can be used in conjunction with any coded computing method. We illustrate this by showing how we can use our methods to accelerate all previously developed coded computing techniques by enabling them to exploit stragglers. Under a widely studied statistical model of completion time, our approach realizes a $66\%$ improvement in the expected finishing time. On Amazon EC2, the gain was $27\%$ when stragglers are simulated.

The Wide Field Infrared Survey Telescope (WFIRST) Coronagraph Instrument (CGI) will be the first high-performance stellar coronagraph using active wavefront control for deep starlight suppression in space, providing unprecedented levels of contrast, spatial resolution, and sensitivity for astronomical observations in the optical. One science case enabled by the CGI will be taking images and(R~50)spectra of faint interplanetary dust structures present in the habitable zone of nearby sunlike stars (~10 pc) and within the snow-line of more distant ones(~20pc), down to dust density levels commensurate with that of the solar system zodiacal cloud. Reaching contrast levels below~10-7 for the first time, CGI will cross an important threshold in debris disks physics, accessing disks with low enough optical depths that their structure is dominated by transport phenomena than collisions. Hence, CGI results will be crucial for determining how exozodiacal dust grains are produced and transported in low-density disks around mature stars. Additionally, CGI will be able to measure the brightness level and constrain the degree of asymmetry of exozodiacal clouds around individual nearby sunlike stars in the optical, at the ~10x solar zodiacal emission level. This information will be extremely valuable for optimizing the observational strategy of possible future exo-Earth direct imaging missions, especially those planning to operate at optical wavelengths, such as Habitable Exoplanet Observatory (HabEx) and the Large Ultraviolet/Optical/Infrared Surveyor (LUVOIR)., Comment: 6 pages, 2 figures

Coded matrix multiplication is a technique to enable straggler-resistant multiplication of large matrices in distributed computing systems. In this paper, we first present a conceptual framework to represent the division of work amongst processors in coded matrix multiplication as a cuboid partitioning problem. This framework allows us to unify existing methods and motivates new techniques. Building on this framework, we apply the idea of hierarchical coding (Ferdinand & Draper, 2018) to coded matrix multiplication. The hierarchical scheme we develop is able to exploit the work completed by all processors (fast and slow), rather than ignoring the slow ones, even if the amount of work completed by stragglers is much less than that completed by the fastest workers. On Amazon EC2, we achieve a 37% improvement in average finishing time compared to non-hierarchical schemes.

We introduce a semiparametric approach to neighbor-based classification. We build off the recently proposed Boundary Trees algorithm by Mathy et al.(2015) which enables fast neighbor-based classification, regression and retrieval in large datasets. While boundary trees use an Euclidean measure of similarity, the Differentiable Boundary Tree algorithm by Zoran et al.(2017) was introduced to learn low-dimensional representations of complex input data, on which semantic similarity can be calculated to train boundary trees. As is pointed out by its authors, the differentiable boundary tree approach contains a few limitations that prevents it from scaling to large datasets. In this paper, we introduce Differentiable Boundary Sets, an algorithm that overcomes the computational issues of the differentiable boundary tree scheme and also improves its classification accuracy and data representability. Our algorithm is efficiently implementable with existing tools and offers a significant reduction in training time. We test and compare the algorithms on the well known MNIST handwritten digits dataset and the newer Fashion-MNIST dataset by Xiao et al.(2017)., Comment: 9 pages, 2 figures

Coronagraph instruments on future space telescopes will enable the direct detection and characterization of Earth-like exoplanets around Sun-like stars for the first time. The quest for the optimal optical coronagraph designs has made rapid progress in recent years thanks to the Segmented Coronagraph Design and Analysis (SCDA) initiative led by the Exoplanet Exploration Program at NASA's Jet Propulsion Laboratory. As a result, several types of high-performance designs have emerged that make use of dual deformable mirrors to (1) correct for optical aberrations and (2) suppress diffracted starlight from obstructions and discontinuities in the telescope pupil. However, the algorithms used to compute the optimal deformable mirror surface tend to be computationally intensive, prohibiting large scale design surveys. Here, we utilize the Fast Linearized Coronagraph Optimizer (FALCO), a tool that allows for rapid optimization of deformable mirror shapes, to explore trade-offs in coronagraph designs for obstructed and segmented space telescopes. We compare designs for representative shaped pupil Lyot and vortex coronagraphs, two of the most promising concepts for the LUVOIR space mission concept. We analyze the optical performance of each design, including their throughput and ability to passively suppress light from partially resolved stars in the presence of low-order aberrations. Our main result is that deformable mirror based apodization can sufficiently suppress diffraction from support struts and inter-segment gaps whose widths are on the order of $\sim$0.1% of the primary mirror diameter to detect Earth-sized planets within a few tens of milliarcseconds from the star., Comment: To appear in the Proceedings of the SPIE, vol. 10698

In cloud computing systems slow processing nodes, often referred to as "stragglers", can significantly extend the computation time. Recent results have shown that error correction coding can be used to reduce the effect of stragglers. In this work we introduce a scheme that, in addition to using error correction to distribute mixed jobs across nodes, is also able to exploit the work completed by all nodes, including stragglers. We first consider vector-matrix multiplication and apply maximum distance separable (MDS) codes to small blocks of sub-matrices. The worker nodes process blocks sequentially, working block-by-block, transmitting partial per-block results to the master as they are completed. Sub-blocking allows a more continuous completion process, which thereby allows us to exploit the work of a much broader spectrum of processors and reduces computation time. We then apply this technique to matrix-matrix multiplication using product code. In this case, we show that the order of computing sub-tasks is a new degree of design freedom that can be exploited to reduce computation time further. We propose a novel approach to analyze the finishing time, which is different from typical order statistics. Simulation results show that the expected computation time decreases by a factor of at least two in compared to previous methods.

We present the first scattered-light images of two debris disks around the F8 star HD 104860 and the F0V star HD 192758, respectively $\sim45$ and $\sim67$ pc away. We detected these systems in the F110W and F160W filters through our re-analysis of archival Hubble Space Telescope NICMOS data with modern starlight subtraction techniques. Our image of HD 104860 confirms the morphology previously observed by Herschel in thermal emission with a well-defined ring at radius $\sim114$ au inclined $\sim58$ degrees. Although the outer edge profile is consistent with dynamical evolution models, the sharp inner edge suggests sculpting by unseen perturbers. Our images of HD 192758 reveal a disk at radius $\sim95$ au inclined by $\sim59$ degrees, never resolved so far. These disks have low scattering albedos of 10% and 13% respectively, inconsistent with water ice grain compositions. They are reminiscent of several other disks with similar inclination and scattering albedos: Fomalhaut, HD 92945, HD 202628, and HD 207129. They are also very distinct from brighter disks in the same inclination bin, which point to different compositions between these two populations. Varying scattering albedo values can be explained by different grain porosities, chemical compositions, or grain size distributions, which may indicate distinct formation mechanisms or dynamical processes at work in these systems. Finally, these faint disks with large infrared excesses may be representative of an underlying population of systems with low albedo values. Searches with more sensitive instruments on HST or on the James Webb Space Telescope and using state-of-the art starlight-subtraction methods may help discover more of such faint systems., Comment: Accepted for publication in ApJ

We present a hardware-based implementation of Linear Program (LP) decoding for binary linear codes. LP decoding frames error-correction as an optimization problem. In contrast, variants of Belief Propagation (BP) decoding frame error-correction as a problem of graphical inference. LP decoding has several advantages over BP-based methods, including convergence guarantees and better error-rate performance in high-reliability channels. The latter makes LP decoding attractive for optical transport and storage applications. However, LP decoding, when implemented with general solvers, does not scale to large blocklengths and is not suitable for a parallelized implementation in hardware. It has been recently shown that the Alternating Direction Method of Multipliers (ADMM) can be applied to decompose the LP decoding problem. The result is a message-passing algorithm with a structure very similar to BP. We present new intuition for this decoding algorithm as well as for its major computational primitive: projection onto the parity polytope. Furthermore, we present results for a fixed-point Verilog implementation of ADMM-LP decoding. This implementation targets a Field-Programmable Gate Array (FPGA) platform to evaluate error-rate performance and estimate resource usage. We show that Frame Error Rate (FER) performance well within 0.5dB of double-precision implementations is possible with 10-bit messages. Finally, we outline a number of research opportunities that should be explored en-route to the realization of an Application Specific Integrated Circuit (ASIC) implementation capable of gigabit per second throughput.