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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| Vydáno v: | IEEE journal on selected areas in communications Ročník 31; číslo 4; s. 678 - 690 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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01.04.2013
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| 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. |
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| 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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| Cites_doi | 10.1109/CDC.2009.5400474 10.1109/TIT.2010.2103852 10.1109/TIT.2011.2169532 10.1109/TIT.1982.1056560 10.1109/Allerton.2011.6120304 10.1109/TIT.2009.2032853 10.1109/18.915643 10.1561/0100000028 10.1109/ISIT.2011.6033872 10.1109/TIT.1979.1056022 10.1109/TIT.1987.1057272 10.1109/TIT.2010.2090225 10.1109/TIT.2010.2095070 10.1109/TIT.1983.1056669 10.1109/TIT.1973.1055037 10.1109/TIT.2010.2050835 |
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| StartPage | 678 |
| 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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