Linear Coding Schemes for the Distributed Computation of Subspaces

Let X₁, ..., X_m be a set of m statistically dependent sources over the common alphabet F_q, that are linearly independent when considered as functions over the sample space. We consider a distributed function computation setting in which the receiver is interested in the lossless computation of the elements of an s-dimensional subspace W spanned by the elements of the row vector [X₁, ..., X_m]Γ in which the (m × s) matrix Γ has rank s. A sequence of three increasingly refined approaches is presented, all based on linear encoders. The first approach uses a common matrix to encode all the sources and a Korner-Marton like receiver to directly compute W. The second improves upon the first by showing that it is often more efficient to compute a carefully chosen superspace U of W. The superspace is identified by showing that the joint distribution of the {X_i} induces a unique decomposition of the set of all linear combinations of the {X_i}, into a chain of subspaces identified by a normalized measure of entropy. This subspace chain also suggests a third approach, one that employs nested codes. For any joint distribution of the {X_i} and any W, the sum-rate of the nested code approach is no larger than that under the Slepian-Wolf (SW) approach. Under the SW approach, W is computed by first recovering each of the {X_i}. For a large class of joint distributions and subspaces W, the nested code approach is shown to improve upon SW. Additionally, a class of source distributions and subspaces are identified, for which the nested-code approach is sum-rate optimal.