Tree Structure Based Analyses on Compressive Sensing for Binary Sparse Sources

Jingjing Fu1,2, Zhouchen Lin2, Bing Zeng1, Feng Wu2

1. Hong Kong University of Science and Technology, Kowloon, Hong Kong 2. Microsoft Research Asia, Beijing, China

jjfu@ust.hk, zhoulin@microsoft.com, eezeng@ust.hk, fengwu@microsoft.com

Compressive sensing theory asserts that the sparse signals can be recovered from a small amount of measurements under incoherent sampling. To specify, let signal with , where has only non-zero elements and . is called as -sparse with respect to the transform . Random measurements are generated by

, where . is a randomly sampling matrix, and the number of mea-surement . In the compressing sensing theory, the recovery of from can be achieved with probability close to one by solving the following convex optimization.

In this paper we propose a new approach to theoretically analyze compressive sensing directly from the randomly sampling matrix instead of a certain recovery algorithm. For simplifying our analyses, we consider as a binary sparse source with independent and identical distribution , where the transform is omitted as an identity matrix. We set the probability 1 and 0 , where . For a binary source, we assume that the random sampling matrix as binary and that one measure-ment always randomly samples elements in , that is, only entries are non-zero in every row vector of . Thereby, one measurement equals the sum of S selected elements and is evaluated in the range [0, S]. Since both the source and the measurement are bi-nary, the numeric relation complexity of the tree structure is reduced and furthermore the recovery of an element node can be deduced from its related measurements. Taking any-one of source bits (e.g. ), we can constitute a tree by parsing the randomly sampling matrix, where the selected source bit as the root. In the rest of tree, measurement nodes and source nodes are connected alternatively according to .

For convenient analysis, we reform the tree structure in a statistical way to yield a reg-ular tree structure. Simply speaking, we assume each element is sampled the same times

, where / 1 1 / . With the regular tree, we formulate the probability that one source root node cannot be recovered from randomly sampling measurements as the un-recovery probability . We deduce that the un-recovery probability of element node in (2k-1)-th layer can be derived from that in (2k+1)-th layer. Accordingly, the un-recovery probability of source node can be formulated layer by layer.

We define the case that 0 as successful recovery, otherwise as unsuccessful re-covery. After further analyses upon the tree structure, we derive two conditions are re-quired for successful recovery of compressive sensing.

a) The average times that a source node is sampled must be no less than two. b) 1 1 , for 0,1 .

Where 1 , it is the probability that of the value of the measurement is nei-ther zero nor S.

2010 Data Compression Conference

1068-0314/10 $26.00 © 2010 IEEE

DOI 10.1109/DCC.2010.60

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