Interval-Passing Algorithm for Non-Negative Measurement Matrices: Performance and Reconstruction Analysis

Abstract

We consider the Interval-Passing Algorithm (IPA), an iterative reconstruction algorithm for reconstruction of non-negative sparse real-valued signals from noise-free measurements. We first generalize the IPA by relaxing the original constraint that the measurement matrix must be binary. The new algorithm operates on any non-negative sparse measurement matrix. We give a performance comparison of the generalized IPA with the reconstruction algorithms based on 1) linear programming and 2) verification decoding. Then we identify signals not recoverable by the IPA on a given measurement matrix, and show that these signals are related to stopping sets responsible to failures of iterative decoding algorithms on the binary erasure channel (BEC). Contrary to the results of the iterative decoding on the BEC, the smallest stopping set of a measurement matrix is not the smallest configuration on which the IPA fails. We analyze the recovery of sparse signals on subsets of stopping sets via the IPA and provide sufficient conditions on the exact recovery of sparse signals. Reconstruction performance of the IPA using the IEEE 802.16e LDPC codes as measurement matrices are given to show the effect of stopping sets in the performance of the IPA.

Interval-Passing Algorithm for Chemical Mixture Estimation

“…However we know that if the graphical representation of M contains some specific topologies (small size stopping sets), the IPA will fail in some cases [8]. Proof: The proof is identical to [8,Theorem 1].…”

“…However we know that if the graphical representation of M contains some specific topologies (small size stopping sets), the IPA will fail in some cases [8]. Proof: The proof is identical to [8,Theorem 1]. This theorem provides limitations on the reconstruction of chemical mixtures using the IPA, and a link with the analysis of the failures of LDPC codes under iterative decoding on the binary erasure channel.…”

“…The first class of message-passing algorithms, such as the approximate message-passing algorithm (AMP) presented in [13], are iterative thresholding algorithms. Other messagepassing algorithms exist in the literature such as list decoding and multiple-basis belief propagation [14], sparse matching pursuit [15], verification decoding [16] or interval-passing [8] that we present in the next section.…”

“…The IPA has been analyzed in [8]. Notably, it was shown that the IPA is a good complexity/performance trade-off between LP reconstruction and simple verification decoding [16].…”

“…Such algorithms include iterative algorithms reviewed in the survey provided in [7,Chapter 8]. In this letter we use a low-complexity iterative algorithm, called the interval-passing algorithm (IPA) [8]. The IPA uses sparse measurement matrices and is well suited for the chemical mixture problem as it is designed for non-negative sparse vectors.…”