Communication Versus Computation: Duality for Multiple-Access Channels and Source Coding

Zhu, Jingge; Lim, Sung Hoon; Gastpar, Michael

doi:10.1109/tit.2018.2849971

Cited by 7 publications

(6 citation statements)

References 14 publications

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“…Körner and Marton have shown the effectiveness of coding for function decoding instead of decoding the full information [16]. The study in [41] establishes duality results for the codes that can be leveraged both for computation and compression over MAC. The study in [42] devises nested codes for computation summations over MAC, and the framework in [43] applies linear nested codes to improve the achievable compute-forward region for MAC.…”

mentioning

confidence: 72%

“…The graph entropy approach for maximally coupled sources achieves a sum-rate given in (47), and hence does not provide an additional gain over the savings of the approach in Section III with a sum-rate achieving (46). However, for the setting with correlated sources, from (47) the rate requirement is half of the requirement in (41).…”

Section: Computing the M -Normmentioning

confidence: 99%

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Distributed Computation over MAC via Kolmogorov-Arnold Representation

Malak,

Tajer

2021

Preprint

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Kolmogorov's representation theorem provides a framework for decomposing any arbitrary realvalued, multivariate, and continuous function into a two-layer nested superposition of a finite number of functions. The functions at these two layers, are referred to as the inner and outer functions with the key property that the design of the inner functions is independent of that of the original function of interest to be computed. This brings modularity and universality to the design of the inner function, and subsequently, a part of computation. This paper capitalizes on such modularity and universality in functional representation to propose two frameworks for distributed computation over the additive multiple access channels (MACs). In the first framework, each source encodes the inner representations and sends them over the additive MAC. Subsequently, the receiver computes the outer functions to compute the function of interest. Transmitting the values of the inner functions instead of the messages directly leads to compression gains. In the second approach, in order to further increase the compression rate, the framework aims to also bring computing the outer functions to the source sites. Specifically, each source employs a graph-coloring-based approach to perform joint functional compression of the inner and the outer functions, which may attain further compression savings over the former. These modular encoding schemes provide an exact representation in the asymptotic regime and the non-asymptotic regime. Contrasting these with the baseline model where sources directly transmit data over MAC, we observe gains. To showcase the gains of these two frameworks and their discrepancies, they are applied to a number of commonly used computations in distributed systems, e.g., computing products, m -norms, polynomial functions, extremum values of functions, and affine transformations. I. INTRODUCTION Communication networks are being increasingly deployed for computation purposes. Examples include over-the-air function computation in wireless networks [1], private-key encryption [2], communication load versus distributed computation latency trade-offs [3]-[5], as well as storage, computation, communication trade-offs [6], [7], coded matrix multiplication [8], [9]

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mentioning

confidence: 72%

Section: Computing the M -Normmentioning

confidence: 99%

Distributed Computation over MAC via Kolmogorov-Arnold Representation

Malak,

Tajer

2021

Preprint

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“…The definition of computation problem varies in the literature. Some studies, including lattice codes [6], nested coset codes [5], [23], focus on recovering a desired linear combination of physical codewords X n j , j = [1 : k]. When the encoding mapping from message to codeword is linear, these two definitions can be used interchangeably.…”

Section: Discussionmentioning

confidence: 99%

Homologous codes for multiple access channels

Sen

Kim

2017

2017 IEEE International Symposium on Information Theory (ISIT)

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Building on recent development by Padakandla and Pradhan, and by Lim, Feng, Pastore, Nazer, and Gastpar, this paper studies the potential of structured nested coset coding as a complete replacement for random coding in network information theory. The roles of two techniques used in nested coset coding to generate nonuniform codewords, namely, shaping and channel transformation, are clarified and illustrated via the simple example of the two-sender multiple access channel. While individually deficient, the optimal combination of shaping and channel transformation is shown to achieve the same performance as traditional random codes for the general two-sender multiple access channel. The achievability proof of the capacity region is extended to the multiple access channels with more than two senders, and with one or more receivers. A quantization argument consistent with the construction of nested coset codes is presented to prove achievability for their Gaussian counterparts. These results open up new possibilities of utilizing nested coset codes with the same generator matrix for a broader class of applications. I. INTRODUCTIONRandom independently and identically distributed (i.i.d.) code ensembles play a fundamental role in network information theory, with most existing coding schemes built on them; see, for example, [1]- [3]. As shown by the classical example by Körner and Marton [4], however, using the same code at multiple users can achieve strictly better performance for some communication problems. Recent studies illustrate the benefit of such structured coding for computing linear combinations in [5]-[10], for the interference channels in [11]-[14], and for the multiple access channels with state information in [15]. Consequently, there has been a flurry of research activities on structured coding in network information theory, facilitated in part by several standalone workshops and tutorials at major conferences by leading researchers.Most of the existing results are based on lattice codes or linear codes on finite alphabets. Recently, Padakandla and Pradhan [15] brought a new dimension to the arsenal of structured coding by developing nested coset codes for network information theory; see also Miyake [16] for nested coset codes for point-to-point communication. In these nested coset coding schemes, a coset code of a rate higher than the target is first generated randomly. A codeword of a desired property (such as type or joint type) is then selected from a subset (a coset of a subcode). This construction is reminiscent of the multicoding scheme in Gelfand-Pinsker coding for channels with state and Marton coding for broadcast channels. But in a sense, nested coset coding is more fundamental in that the scheme at its core is relevant even for single-user communication. By a careful combination of individual and common parts of coset codes, the proposed coding scheme in [15] achieves rates for multiple access channels (MACs) with state beyond what can be achieved by existing random or structured coding sche...

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“…where ∼ CN (0, 2 ) denotes the independent AWGN with zero-mean and variance 2 at WD 's receiver. Note that, in (10), each WD ∈ N \ { ( )} is either a leaf WD or an intermediate WD. Therefore, for all ∈ K −ℓ K inter , we have…”

Section: ) Leaf Wd Transmissionmentioning

confidence: 99%

“…Recently, over-the-air (OTA) aggregation (a.k.a, over-the-air computation (AirComp)) methods [8]- [10] have been proposed to directly compute a nomographic function with only O (1) radio resource blocks, which is independent of the number of WDs and the number of input data files . By exploiting the signal superposition property of the wireless multiple access channel (MAC), different WDs can simultaneously transmit their data to the destination node over the same frequency band, and the destination node can then reconstruct the targeted nomographic function value from the received signal directly [11].…”

Section: Introductionmentioning

confidence: 99%

Multi-Level Over-the-Air Aggregation of Mobile Edge Computing over D2D Wireless Networks

Wang¹,

Lau²

2021

Preprint

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In this paper, we consider a wireless multihop device-to-device (D2D) based mobile edge computing (MEC) system, where the destination wireless device (WD) is scheduled to compute nomographic functions. Under the MapReduce framework and motivated by reducing communication resource overhead, we propose a new multi-level over-the-air (OTA) aggregation scheme for the destination WD to collect the individual partially aggregated intermediate values (IVAs) for reduction from multiple source WDs in the data shuffling phase. For OTA aggregation per level, the source WDs employ a channelinverse structure multiplied by their individual transmit coefficients in transmission over the same timefrequency resource blocks, and the destination WD finally uses a receive filtering factor to construct the aggregated IVA. Under this setup, we develop a unified transceiver design framework that minimizes the mean squared error (MSE) of the aggregated IVA at the destination WD subject to the source WDs' individual power constraints, by jointly optimizing the source WDs' individual transmit coefficients and the destination WD's receive filtering factor. The formulated power-constrained MSE minimization problem is non-convex. First, based on the primal decomposition method, we derive the closed-form solution under the special case of a common transmit coefficient. It shows that all the source WDs' common transmit is determined by the minimal transmit power budget among the source WDs. Next, for the general case, we transform the original problem into a quadratic fractional programming problem, and then develop a low-complexity algorithm to obtain the (near-) optimal solution by leveraging Dinkelbach's algorithm along with the Gaussian randomization method. Numerical results are provided to demonstrate the significant performance gains achieved by the proposed multi-level OTA aggregation scheme over various existing schemes.

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Communication Versus Computation: Duality for Multiple-Access Channels and Source Coding

Cited by 7 publications

References 14 publications

Distributed Computation over MAC via Kolmogorov-Arnold Representation

Distributed Computation over MAC via Kolmogorov-Arnold Representation

Homologous codes for multiple access channels

Multi-Level Over-the-Air Aggregation of Mobile Edge Computing over D2D Wireless Networks

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