This paper investigates universal polar coding schemes. In particular, a notion of ordering (called convolutional path) is introduced between probability distributions to determine when a polar compression (or communication) scheme designed for one distribution can also succeed for another one. The original polar decoding algorithm is also generalized to an algorithm allowing to learn information about the source distribution using the idea of checkers. These tools are used to construct a universal compression algorithm for binary sources, operating at the lowest achievable rate (entropy), with low complexity and with guaranteed small error probability. In a second part of the paper, the problem of sketching high dimensional discrete signals which are sparse is approached via the polarization technique. It is shown that the number of measurements required for perfect recovery is competitive with the $O(k \log (n/k))$ bound (with optimal constant for binary signals), meanwhile affording a deterministic low complexity measurement matrix.
We propose application-layer coding schemes to recover lost data in delay-sensitive uplink (sensor-to-gateway) communications in the Internet of Things. Built on an approach that combines retransmissions and forward erasure correction, the proposed schemes' salient features include low computational complexity and the ability to exploit sporadic receiver feedback for efficient data recovery. Reduced complexity is achieved by keeping the number of coded transmissions as low as possible and by devising a mechanism to compute the optimal degree of a coded packet in O(1). Our major contributions a…
We give a unified treatment of some inequalities that are used in the proofs of channel polarization theorems involving a binary-input discrete memoryless channel.
We review connections between coding-theoretic objects and sparse learning problems. In particular, we show how seemingly different combinatorial objects such as error-correcting codes, combinatorial designs, spherical codes, compressed sensing matrices and group testing designs can be obtained from one another. The reductions enable one to translate upper and lower bounds on the parameters attainable by one object to another. We survey some of the well-known reductions in a unified presentation, and bring some existing gaps to attention. New reductions are also introduced; in particular, we b…
In this paper, we propose a Polar coding scheme for parallel Gaussian channels. The encoder knows the sum rate of the parallel channels but does not know the rate of any channel. By using the nesting property of Polar code, we design a coding/decoding scheme to achieve the sum rates.
This paper considers a source, which employs random linear coding (RLC) to encode a message, a legitimate destination, which can recover the message if it gathers a sufficient number of coded packets, and an eavesdropper. The probability of the eavesdropper accumulating enough coded packets to recover the message, known as the intercept probability, has been studied in the literature. In our work, the eavesdropper does not abandon its efforts to obtain the source message if RLC decoding has been unsuccessful; instead, it employs partial packet recovery (PPR) offline in an effort to repair erro…
Blind Compressed Sensing (BCS) is an extension of Compressed Sensing (CS) where the optimal sparsifying dictionary is assumed to be unknown and subject to estimation (in addition to the CS sparse coefficients). Since the emergence of BCS, dictionary learning, a.k.a. sparse coding, has been studied as a matrix factorization problem where its sample complexity, uniqueness and identifiability have been addressed thoroughly. However, in spite of the strong connections between BCS and sparse coding, recent results from the sparse coding problem area have not been exploited within the context of BCS…
We show that polar codes can be used to achieve the rate-distortion functions in the problem of hierarchical source coding also known as the successive refinement problem. We also analyze the distributed version of this problem, constructing a polar coding scheme that achieves the rate distortion functions for successive refinement with side information.
Arıkan's polar coding, is by now a well studied technique that allows achieving the symmetric capacity of binary input memoryless channels with low complexity encoding and decoding, provided that the polar decoding architecture is used and the decoding metric is matched to the true channel. In this paper, we analyze communication rates that are achievable when the polar coding/decoding architecture is used with the decoder using an incorrect model of the channel. We define the `polar mismatched capacity' as an analogue of the classical mismatched capacity, give an expression for it, and derive…
A polar coding scheme for interference networks is introduced. The scheme combines Arikan's monotone chain rules for multiple-access channels and a method by Hassani and Urbanke to 'align' two incompatible polarization processes. It achieves the Han--Kobayashi inner bound for two-user interference channels and generalizes to interference networks.
Friedrich-Alexander-Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡大学）；Vrije Universiteit Brussel（布鲁塞尔自由大学）；Technical University of Denmark（丹麦技术大学）
In support of applications involving multiview sources in distributed object recognition using lightweight cameras, we propose a new method for the distributed coding of sparse sources as visual descriptor histograms extracted from multiview images. The problem is challenging due to the computational and energy constraints at each camera as well as the limitations regarding inter-camera communication. Our approach addresses these challenges by exploiting the sparsity of the visual descriptor histograms as well as their intra- and inter-camera correlations. Our method couples distributed source…
In this paper, we propose a low-complexity, secrecy capacity achieving polar coding scheme for the cognitive interference channel with confidential messages (CICC) under the strong secrecy criterion. Existing polar coding schemes for interference channels rely on the use of polar codes for the multiple access channel, the code construction problem of which can be complicated. We show that the whole secrecy capacity region of the CICC can be achieved by simple point-to-point polar codes due to the cognitivity, and our proposed scheme requires the minimum rate of randomness at the encoder.
Existing polar coding schemes for the two-user interference channel follow the original idea of Han and Kobayashi, in which component messages are encoded independently and then mapped by some deterministic functions (i.e., homogeneous superposition coding). In this paper, we propose a new polar coding scheme for the interference channel based on the heterogeneous superposition coding approach of Chong, Motani and Garg. We prove that fully-joint decoding (the receivers simultaneously decode both senders' common messages and the intended sender's private message) in the Han-Kobayashi strategy c…
Exact recovery of a sparse solution for an underdetermined system of linear equations implies full search among all possible subsets of the dictionary, which is computationally intractable, while l1 minimization will do the job when a Restricted Isometry Property holds for the dictionary. Yet, practical sparse recovery algorithms may fail to recover the vector of coefficients even when the dictionary deviates from the RIP only slightly. To enjoy l1 minimization guarantees in a wider sense, a method based on a combination of full-search and l1 minimization is presented. The idea is based on par…
We provide novel coded computation strategies for distributed matrix-matrix products that outperform the recent "Polynomial code" constructions in recovery threshold, i.e., the required number of successful workers. When $m$-th fraction of each matrix can be stored in each worker node, Polynomial codes require $m^2$ successful workers, while our MatDot codes only require $2m-1$ successful workers, albeit at a higher communication cost from each worker to the fusion node. We also provide a systematic construction of MatDot codes. Further, we propose "PolyDot" coding that interpolates between Po…
A method to construct nonasymmetric distributed source coding (DSC) scheme using polar codes which can achieve any point on the dominant face of the Slepian-Wolf (SW) rate region for sources with uniform marginals is considered. In addition to nonasymmetric case, we also discuss and show explicitly how asymmetric and single source compression is done using successive cancellation (SC) polar decoder. We then present simulation results that exhibit the performance of the considered methods.
A polar coding scheme for fading channels is proposed in this paper. More specifically, the focus is Gaussian fading channel with a BPSK modulation technique, where the equivalent channel could be modeled as a binary symmetric channel with varying cross-over probabilities. To deal with variable channel states, a coding scheme of hierarchically utilizing polar codes is proposed. In particular, by observing the polarization of different binary symmetric channels over different fading blocks, each channel use corresponding to a different polarization is modeled as a binary erasure channel such th…
Polar coding was conceived originally as a technique for boosting the cutoff rate of sequential decoding, along the lines of earlier schemes of Pinsker and Massey. The key idea in boosting the cutoff rate is to take a vector channel (either given or artificially built), split it into multiple correlated subchannels, and employ a separate sequential decoder on each subchannel. Polar coding was originally designed to be a low-complexity recursive channel combining and splitting operation of this type, to be used as the inner code in a concatenated scheme with outer convolutional coding and seque…
-We develop a polar coding scheme for empirical coordination in a two-node network with a noisy link in which the input and output signals have to be coordinated with the source and the reconstruction. In the case of non-causal encoding and decoding, we show that polar codes achieve the best known inner bound for the empirical coordination region, provided that a vanishing rate of common randomness is available. This scheme provides a constructive alternative to random binning and coding proofs.
A probabilistic shaping method for multi-level coding (MLC) is presented, where the transmitted symbols are forced to have a shaped non-uniform distribution. It is shown that shaping only a single bit-level suffices to compensate for most of the shaping loss on the fading channels. A polar code based implementation of the proposed scheme is presented, where shaping is performed by using a precoder at the transmitter without increasing the decoding complexity. Simulation results show that performance improvements can be obtained compared to BICM- and MLC-based polar coding without shaping on Ra…
The polarization phenomenon for a single source is extended to a framework with multiple correlated sources. It is shown in addition to extracting the randomness of the source, the polar transforms takes the original arbitrary dependencies to extremal dependencies. Polar coding schemes for the Slepian-Wolf problem and for secret key generations are then proposed based on this phenomenon. In particular, constructions of secret keys achieving the secrecy capacity and compression schemes achieving the Slepian-Wolf capacity region are obtained with a complexity of $O(n \log (n))$.
This work presents a polar coding scheme for fading channels, focusing primarily on fading binary symmetric and additive exponential noise channels. For fading binary symmetric channels, a hierarchical coding scheme is presented, utilizing polar coding both over channel uses and over fading blocks. The receiver uses its channel state information (CSI) to distinguish states, thus constructing an overlay erasure channel over the underlying fading channels. By using this scheme, the capacity of a fading binary symmetric channel is achieved without CSI at the transmitter. Noting that a fading AWGN…
Recently, a new class of error-control codes, the polar codes, have attracted much attention. The polar codes are the first known class of capacity-achieving codes for many important communication channels. In addition, polar codes have low-complexity encoding algorithms. Therefore, these codes are favorable choices for low-complexity devices, for example, in ubiquitous computing and sensor networks. However, the polar codes fall short in terms of finite-length error probabilities, compared with the state-of-the-art codes, such as the low-density parity-check codes. In this paper, in order to …
We study polar coding for stochastic processes with memory. For example, a process may be defined by the joint distribution of the input and output of a channel. The memory may be present in the channel, the input, or both. We show that $ψ$-mixing processes polarize under the standard Arıkan transform, under a mild condition. We further show that the rate of polarization of the \emph{low-entropy} synthetic channels is roughly $O(2^{-\sqrt{N}})$, where $N$ is the blocklength. That is, essentially the same rate as in the memoryless case.
We consider a generalization of the gradient coding framework where a dataset is divided across $n$ workers and each worker transmits to a master node one or more linear combinations of the gradients over its assigned data subsets. Unlike the conventional framework which requires the master node to recover the sum of the gradients over all the data subsets in the presence of straggler workers, we relax the goal to computing the sum of at least some $α$ fraction of the gradients. We begin by deriving a lower bound on the computation load of any scheme and also propose two strategies which achie…
In a series of papers N. Ma and P. Ishwar (2011-13) considered a range of distributed source coding problems that arise in the context of iterative computation of functions, characterizing the region of achievable communication rates. We consider the problems of interactive computation of functions by two terminals and interactive computation in a collocated network, showing that the rate regions for both these problems can be achieved using several rounds of polar-coded transmissions.
A pruned variant of polar coding is reinvented for all binary erasure channels. For small $\varepsilon>0$, we construct codes with block length $\varepsilon^{-5}$, code rate $\text{Capacity}-\varepsilon$, error probability $\varepsilon$, and encoding and decoding time complexity $O(N\log|\log\varepsilon|)$ per block, equivalently $O(\log|\log\varepsilon|)$ per information bit (Propositions 5 to 8). This result also follows if one applies systematic polar coding [Arıkan 10.1109/LCOMM.2011.061611.110862] with simplified successive cancelation decoding [Alamdar-Yazdi-Kschischang 10.1109/LCOMM.201…
This paper presents a polar coding scheme to achieve secrecy in block fading binary symmetric wiretap channels without the knowledge of instantaneous channel state information (CSI) at the transmitter. For this model, a coding scheme that hierarchically utilizes polar codes is presented. In particular, on polarization of different binary symmetric channels over different fading blocks, each channel use is modeled as an appropriate binary erasure channel over fading blocks. Polar codes are constructed for both coding over channel uses for each fading block and coding over fading blocks for cert…
A framework is proposed that allows for a joint description and optimization of both binary polar coding and the multilevel coding (MLC) approach for $2^m$-ary digital pulse-amplitude modulation (PAM). The conceptual equivalence of polar coding and multilevel coding is pointed out in detail. Based on a novel characterization of the channel polarization phenomenon, rules for the optimal choice of the bit labeling in this coded modulation scheme employing polar codes are developed. Simulation results for the AWGN channel are included.
In this paper, we propose CodedSketch, as a distributed straggler-resistant scheme to compute an approximation of the multiplication of two massive matrices. The objective is to reduce the recovery threshold, defined as the total number of worker nodes that we need to wait for to be able to recover the final result. To exploit the fact that only an approximated result is required, in reducing the recovery threshold, some sorts of pre-compression are required. However, compression inherently involves some randomness that would lose the structure of the matrices. On the other hand, considering t…
In this paper, we first propose an universal polar coding scheme for parallel Gaussian channels with non-binary inputs. It is assumed that the encoder knows only the sum capacity of M parallel channels instead of the capacity of any single channel. By decomposing each parallel channel into T = [log2r] sub channels, we therefore obtain MT binary sub-channels. A super polar coding scheme that across all sub-channels is then proposed. This scheme can achieve the sum capacity when the block length is sufficiently large. We have also discussed the applications of parallel polar coding design for bo…
A polar coding scheme is proposed for the Wiretap Broadcast Channel with two legitimate receivers and one eavesdropper. We consider a model in which the transmitter wishes to send different confidential (and non-confidential) information to each legitimate receiver. First, we compare two inner-bounds on the achievable region of this model from the literature, and then we design a polar coding scheme that achieves the stronger one. In the proposed scheme, the encoding uses polar-based Marton's coding, where one inner-layer must be reliably decoded by both legitimate receivers, and each receiver…
A framework is proposed that allows for a joint description and optimization of both binary polar coding and $2^m$-ary digital pulse-amplitude modulation (PAM) schemes such as multilevel coding (MLC) and bit-interleaved coded modulation (BICM). The conceptual equivalence of polar coding and multilevel coding is pointed out in detail. Based on a novel characterization of the channel polarization phenomenon, rules for the optimal choice of the labeling in coded modulation schemes employing polar codes are developed. Simulation results regarding the error performance of the proposed schemes on th…
We consider the problem of massive matrix multiplication, which underlies many data analytic applications, in a large-scale distributed system comprising a group of worker nodes. We target the stragglers' delay performance bottleneck, which is due to the unpredictable latency in waiting for slowest nodes (or stragglers) to finish their tasks. We propose a novel coding strategy, named \emph{entangled polynomial code}, for designing the intermediate computations at the worker nodes in order to minimize the recovery threshold (i.e., the number of workers that we need to wait for in order to compu…
Information-theoretic work for wiretap channels is mostly based on random coding schemes. Designing practical coding schemes to achieve information-theoretic security is an important problem. By applying the two recently developed techniques for polar codes, we propose a polar coding scheme to achieve the secrecy capacity of the general wiretap channel.
A polar coding scheme is proposed for the Wiretap Broadcast Channel with two legitimate receivers and one eavesdropper. We consider a model in which the transmitter wishes to send a private and a confidential message that must be reliably decoded by the receivers, and the confidential message must also be (strongly) secured from the eavesdropper. The coding scheme aims to use the optimal rate of randomness and does not make any assumption regarding the symmetry or degradedness of the channel. This paper extends previous work on polar codes for the wiretap channel by proposing a new chaining co…
We construct a channel coding scheme to achieve the capacity of any discrete memoryless channel based solely on the techniques of polar coding. In particular, we show how source polarization and randomness extraction via polarization can be employed to "shape" uniformly-distributed i.i.d. random variables into approximate i.i.d. random variables distributed ac- cording to the capacity-achieving distribution. We then combine this shaper with a variant of polar channel coding, constructed by the duality with source coding, to achieve the channel capacity. Our scheme inherits the low complexity e…
The long-haul communication systems can offer ultra high-speed data transfer rates but suffer from burst errors. The high-rate and high-performance staircase codes provide an efficient way for long-haul transmission. The staircase coding scheme is a concatenation structure, which provides the opportunity to improve the performance of high-rate polar codes. At the same time, the polar codes make the staircase structure more reliable. Thus, a high-rate polar-staircase coding scheme is proposed, where the systematic polar codes are applied as the component codes. The soft cancellation decoding of…
In this paper, we propose a polar coding based scheme for set reconciliation between two network nodes. The system is modeled as a well-known Slepian-Wolf setting induced by a fixed number of deletions. The set reconciliation process is divided into two phases: 1) a deletion polar code is employed to help one node to identify the possible deletion indices, which may be larger than the number of genuine deletions; 2) a lossless compression polar code is then designed to feedback those indices with minimum overhead. Our scheme can be viewed as a generalization of polar codes to some emerging net…
The adversarial wiretap channel (AWTC) model is a secure communication model in which adversary can directly read and write the transmitted bits in legitimate communication with fixed fractions. In this paper we propose a secure polar coding scheme to provide secure and reliable communication over the AWTC model. For the adversarial reading and writing action, we present a $ρ$ equivalent channel block and study its transformation under the channel polarization operation. We find that the generated channels are polarized in the sense that part of them are full-noise channels with probability al…
Polar codes, introduced recently by Arıkan, are the first family of codes known to achieve capacity of symmetric channels using a low complexity successive cancellation decoder. Although these codes, combined with successive cancellation, are optimal in this respect, their finite-length performance is not record breaking. We discuss several techniques through which their finite-length performance can be improved. We also study the performance of these codes in the context of source coding, both lossless and lossy, in the single-user context as well as for distributed applications.
We develop a low-complexity polar coding scheme for the discrete memoryless broadcast channel with confidential messages under strong secrecy and randomness constraints. Our scheme extends previous work by using an optimal rate of uniform randomness in the stochastic encoder, and avoiding assumptions regarding the symmetry or degraded nature of the channels. The price paid for these extensions is that the encoder and decoders are required to share a secret seed of negligible size and to increase the block length through chaining. We also highlight a close conceptual connection between the prop…
We construct a joint coordination-channel polar coding scheme for strong coordination of actions between two agents $\mathsf X$ and $\mathsf Y$, which communicate over a discrete memoryless channel (DMC) such that the joint distribution of actions follows a prescribed probability distribution. We show that polar codes are able to achieve our previously established inner bound to the strong noisy coordination capacity region and thus provide a constructive alternative to a random coding proof. Our polar coding scheme also offers a constructive solution to a channel simulation problem where a DM…
We consider lossy source compression of a binary symmetric source using polar codes and the low-complexity successive encoding algorithm. It was recently shown by Arikan that polar codes achieve the capacity of arbitrary symmetric binary-input discrete memoryless channels under a successive decoding strategy. We show the equivalent result for lossy source compression, i.e., we show that this combination achieves the rate-distortion bound for a binary symmetric source. We further show the optimality of polar codes for various problems including the binary Wyner-Ziv and the binary Gelfand-Pinske…
We consider achieving the rates in the capacity region of a multi-level 3-receiver broadcast channel, in which the second receiver is degraded with respect to the first receiver, with degraded message sets. We propose a two-level chaining strategy based on polar codes that achieves the capacity region of the considered setting without time-sharing. We also look at a slight variation of this problem, where the first receiver only requires to decode its own private message and the other two receivers require to decode another private message common to them. We observe that the capacity region do…
It has been shown that an extension of the basic binary polar transformation also polarizes over finite fields. With it the direct encoding of q-ary sources and channels is a process that can be implemented with simple and efficient algorithms. However, direct polar decoding of q-ary sources and channels is more involved. In this paper we obtain a recursive equation for the likelihood ratio expressed as a LR vector. With it successive cancellation (SC) decoding is applied in a straightforward way. The complexity is quadratic in the order of the field, but the use of the LR vector introduces fa…
We propose two coding schemes for distributed matrix multiplication in the presence of stragglers. These coding schemes are adaptations of LT codes and Raptor codes to distributed matrix multiplication and are termed \emph{factored LT (FLT) codes} and \emph{factored Raptor (FR) codes}. Empirically, we show that FLT codes have near-optimal recovery thresholds when the number of worker nodes is very large, and that FR codes have excellent recovery thresholds while the number of worker nodes is moderately large. FLT and FR codes have better recovery thresholds when compared to Product codes and t…
Visible light communication (VLC) could provide short-range optical wireless communication together with illumination using LED lighting. However, conventional forward error correction (FEC) codes for reliable communication do not have the features for dimming support and flicker mitigation which are required in VLC for the main functionality of lighting. Therefore, auxiliary coding techniques are usually needed, which eventually reduce the coding efficiency and increase the complexity. In this paper, a polar codes-based FEC coding scheme for dimmable VLC is proposed to increase the coding eff…
It is shown that polar coding schemes achieve the known achievable rate regions for several multi-terminal communications problems including lossy distributed source coding, multiple access channels and multiple descriptions coding. The results are valid for arbitrary alphabet sizes (binary or nonbinary) and arbitrary distributions (symmetric or asymmetric).
In this paper, we propose a polar coding scheme for secure communication over the multiple-input, single-output, single-antenna eavesdropper (MISOSE) fading channel. We consider the case of block fading channels with known eavesdropper channel state information (CSI) and the case of fading channels with known eavesdropper channel distribution information (CDI). We use the artificial noise assisted secure precoding method to maximize the secrecy capacity in the first case, and to overcome the unawareness of the eavesdropper CSI in the second case. We show that our proposed scheme can provide bo…
The problem of communication over binary dirty paper (DP) using nested polar codes is considered. An improved scheme, focusing on low delay, short to moderate blocklength communication is proposed. Successive cancellation list (SCL) decoding with properly defined CRC is used for channel coding, and SCL encoding without CRC is used for source coding. The performance is compared to the best achievable rate of any coding scheme for binary DP using nested codes. A well known problem with nested polar codes for binary DP is the existence of frozen channel code bits that are not frozen in the source…
In this paper, we propose a new polar coding scheme for the unsourced, uncoordinated Gaussian random access channel. Our scheme is based on sparse spreading, treat interference as noise and successive interference cancellation (SIC). On the transmitters side, each user randomly picks a code-length and a transmit power from multiple choices according to some probability distribution to encode its message, and an interleaver to spread its encoded codeword bits across the entire transmission block. The encoding configuration of each user is transmitted by compressive sensing, similar to some prev…
In this paper, we consider the problem of polar coding for block fading channels, with emphasis on those with instantaneous channel state information (CSI) at neither the transmitter nor the receiver. Our approach is to decompose a block fading channel of $T_c$ symbols per coherent interval into $T_c$ binary-input sub-channels in a capacity-preserving way, and design a polar code for each of them. For the case when instantaneous CSI is available at the receiver, a random interleaver can be used to enable joint encoding and decoding of all sub-channels so as to enhance the finite length perform…
Guangzhou Institute of Technology, Xidian University（西安电子科技大学广州研究院）；Key Laboratory of Intelligent Computing and Signal Processing Ministry of Education, Anhui University（安徽大学智能计算与信号处理教育部重点实验室）
Explicit coding schemes are proposed to achieve the rate-distortion function of the Heegard-Berger problem using polar codes. Specifically, a nested polar code construction is employed to achieve the rate-distortion function for the doubly-symmetric binary sources when the side information may be absent. The nested structure contains two optimal polar codes for lossy source coding and channel coding, respectively. Moreover, a similar nested polar lattice construction is employed when the source and the side information are jointly Gaussian. The proposed polar lattice is constructed by nesting …
