  We consider the problem of distributed lossy linear function computation in a
tree network. We examine two cases: (i) data aggregation (only one sink node
computes) and (ii) consensus (all nodes compute the same function). By
quantifying the accumulation of information loss in distributed computing, we
obtain fundamental limits on network computation rate as a function of
incremental distortions (and hence incremental loss of information) along the
edges of the network. The above characterization, based on quantifying
distortion accumulation, offers an improvement over classical cut-set type
techniques which are based on overall distortions instead of incremental
distortions. This quantification of information loss qualitatively resembles
information dissipation in cascaded channels [1]. Surprisingly, this
accumulation effect of distortion happens even at infinite blocklength.
Combining this observation with an inequality on the dominance of mean-square
quantities over relative-entropy quantities, we obtain outer bounds on the rate
distortion function that are tighter than classical cut-set bounds by a
difference which can be arbitrarily large in both data aggregation and
consensus. We also obtain inner bounds on the optimal rate using random
Gaussian coding, which differ from the outer bounds by $\mathcal{O}(\sqrt{D})$,
where $D$ is the overall distortion. The obtained inner and outer bounds can
provide insights on rate (bit) allocations for both the data aggregation
problem and the consensus problem. We show that for tree networks, the rate
allocation results have a mathematical structure similar to classical reverse
water-filling for parallel Gaussian sources.
