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We consider the problem of distributed lossy linear function computation in atree network. We examine two cases: (i) data aggregation (only one sink nodecomputes) and (ii) consensus (all nodes compute the same function). Byquantifying the accumulation of information loss in distributed computing, weobtain fundamental limits on network computation rate as a function ofincremental distortions (and hence incremental loss of information) along theedges of the network. The above characterization, based on quantifyingdistortion accumulation, offers an improvement over classical cut-set typetechniques which are based on overall distortions instead of incrementaldistortions. This quantification of information loss qualitatively resemblesinformation dissipation in cascaded channels [1]. Surprisingly, thisaccumulation effect of distortion happens even at infinite blocklength.Combining this observation with an inequality on the dominance of mean-squarequantities over relative-entropy quantities, we obtain outer bounds on the ratedistortion function that are tighter than classical cut-set bounds by adifference which can be arbitrarily large in both data aggregation andconsensus. We also obtain inner bounds on the optimal rate using randomGaussian coding, which differ from the outer bounds by $\mathcal{O}(\sqrt{D})$,where $D$ is the overall distortion. The obtained inner and outer bounds canprovide insights on rate (bit) allocations for both the data aggregationproblem and the consensus problem. We show that for tree networks, the rateallocation results have a mathematical structure similar to classical reversewater-filling for parallel Gaussian sources.

Rate Distortion for Lossy In-network Function Computation: Information Dissipation and Sequential Reverse Water-Filling