Linear Coding Schemes for the Distributed Computation of Subspaces

Let X 1 , ..., 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...

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Published in:IEEE journal on selected areas in communications Vol. 31; no. 4; pp. 678 - 690
Main Authors: Lalitha, V., Prakash, N., Vinodh, K., Kumar, P. V., Pradhan, S. S.
Format: Journal Article
Language:English
Published: IEEE 01.04.2013
Subjects:
Decoding
Distributed function computation
Encoding
Entropy
Joints
linear encoders
nested codes
normalized entropy
Random variables
Receivers
source compression
Vectors
ISSN:0733-8716
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Abstract Let X 1 , ..., 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 [X 1 , ..., 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.
AbstractList Let X 1 , ..., 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 [X 1 , ..., 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.
Author Prakash, N.
Vinodh, K.
Pradhan, S. S.
Kumar, P. V.
Lalitha, V.
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SubjectTerms Decoding
Distributed function computation
Encoding
Entropy
Joints
linear encoders
nested codes
normalized entropy
Random variables
Receivers
source compression
Vectors
Title Linear Coding Schemes for the Distributed Computation of Subspaces
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Volume 31
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