We consider the problem of distributed computation of a target function over a multiple-access channel. If the target and channel functions are matched (i.e., compute the same function), significant performance gains can be obtained by jointly designing the computation and communication tasks. However, in most situations there is mismatch between these two functions. In this work, we analyze the impact of this mismatch on the performance gains achievable with joint computation and communication designs over separation-based designs. We show that for most pairs of target and channel functions there is no such gain, and separation of computation and communication is optimal.
There is some work touching on the aspect of structural mismatch between the target and the channel functions. In @cite_13 , an example was given in which the mismatch between a linear target function with integer coefficients and a linear channel function with real coefficients can significantly reduce efficiency. In @cite_7 , it was conjectured that, for computation of finite-field addition over a real-addition channel, there could be a gap between the cut-set bound and the computation rate. In @cite_18 , mismatched computation when the network performs linear finite-field operations was studied. To the best of our knowledge, a systematic study of channel and computation mismatch is initiated in this work.
@cite_18 @cite_13 @cite_7
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We consider multiple non-colocated sources communicating over a network to a common sink. We assume that the network operation is fixed, and its end result is to convey a fixed linear deterministic transformation of the source data to the sink. This linear transformation is known both at the sources and at the sink. We are interested in the problem of function computation over such networks. We design communication protocols that can perform computation without modifying the network operation, by appropriately selecting the codebook that the sources employ to map their measurements to the data they send over the network. We analyze the asymptotic behavior of compute-and-forward relay networks in the regime of high signal-to-noise ratios. We consider a section of such a network consisting of K transmitters and K relays. The aim of the relays is to reliably decode an invertible function of the messages sent by the transmitters. An upper bound on the capacity of this system can be obtained by allowing full cooperation among the transmitters and among the relays, transforming the network into a K times K multiple-input multiple-output (MIMO) channel. The number of degrees of freedom of compute-and-forward is hence at most K. In this paper, we analyze the degrees of freedom achieved by the lattice coding implementation of compute-and-forward proposed recently by Nazer and Gastpar. We show that this lattice implementation achieves at most 2 (1+1 K) 2 degrees of freedom, thus exhibiting a very different asymptotic behavior than the MIMO upper bound. This raises the question if this gap of the lattice implementation to the MIMO upper bound is inherent to compute-and-forward in general. We answer this question in the negative by proposing a novel compute-and-forward implementation achieving K degrees of freedom. The following network computing problem is considered. Source nodes in a directed acyclic network generate independent messages and a single receiver node computes a target function f of the messages. The objective is to maximize the average number of times f can be computed per network usage, i.e., the “computing capacity”. The network coding problem for a single-receiver network is a special case of the network computing problem in which all of the source messages must be reproduced at the receiver. For network coding with a single receiver, routing is known to achieve the capacity by achieving the network min-cut upper bound. We extend the definition of min-cut to the network computing problem and show that the min-cut is still an upper bound on the maximum achievable rate and is tight for computing (using coding) any target function in multi-edge tree networks. It is also tight for computing linear target functions in any network. We also study the bound's tightness for different classes of target functions. In particular, we give a lower bound on the computing capacity in terms of the Steiner tree packing number and a different bound for symmetric functions. We also show that for certain networks and target functions, the computing capacity can be less than an arbitrarily small fraction of the min-cut bound.