In this work, we perform a complete failure analysis of the interval-passing algorithm (IPA) for compressed sensing. The IPA is an efficient iterative algorithm for reconstructing a $k$ -sparse nonnegative $n$ -dimensional real signal ${x}$ from a small number of linear measurements ${y}$ . In particular, we show that the IPA fails to recover ${x}$ from ${y}$ if and only if it fails to recover a corresponding binary vector of the same support, and also that only positions of nonzero values in the measurement matrix are of importance to the success of recovery. Based on this observation, we introduce termatiko sets and show that the IPA fails to fully recover ${x}$ if and only if the support of ${x}$ contains a nonempty termatiko set, thus giving a complete (graph-theoretic) description of the failing sets of the IPA. Two heuristics to locate small-size termatiko sets are presented. For binary column-regular measurement matrices with no 4-cycles, we provide a lower bound on the termatiko distance, defined as the smallest size of a nonempty termatiko set. For measurement matrices constructed from the parity-check matrices of array low-density parity-check codes, upper bounds on the termatiko distance equal to half the best known upper bound on the minimum distance are provided for column-weight at most 7, while for column-weight 3, the exact termatiko distance and its corresponding multiplicity are provided. Next, we show that adding redundant rows to the measurement matrix does not create new termatiko sets, but rather potentially removes termatiko sets and thus improves performance. An algorithm is provided to efficiently search for such redundant rows. Finally, we present numerical results for different specific measurement matrices and also for protograph-based ensembles of measurement matrices, as well as simulation results of IPA performance, showing the influence of small-size termatiko sets.

中文翻译：

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压缩感知区间传递算法失效分析

在这项工作中，我们对用于压缩感知的间隔传递算法 (IPA) 进行了完整的故障分析。IPA 是一种有效的迭代算法，用于重建 $千$ -稀疏非负 $n$ 维实信号 ${x}$ 来自少量的线性测量 ${y}$ . 特别是，我们表明 IPA 无法恢复 ${x}$ 从 ${y}$ 当且仅当它无法恢复相同支持的相应二元向量，并且只有测量矩阵中非零值的位置对恢复成功很重要。基于这一观察，我们引入Termatiko 套 并表明 IPA 未能完全恢复 ${x}$ 当且仅当支持 ${x}$ 包含一个非空的 termatiko 集，从而给出了 IPA 失败集的完整（图论）描述。提供了两种用于定位小型 Termatiko 集的启发式方法。对于没有 4 循环的二元列正则测量矩阵，我们提供了一个下界距离，定义为非空 termatiko 集的最小大小。对于从阵列低密度奇偶校验码的奇偶校验矩阵构造的测量矩阵，为列权重提供最多等于最小距离的最佳已知上限的一半的 termatiko 距离的上限，而对于列权重 3，提供了精确的 termatiko 距离及其相应的多重性。接下来，我们展示了向测量矩阵添加冗余行不会创建新的 termatiko 集，而是潜在地删除了 termatiko 集，从而提高了性能。提供了一种算法来有效地搜索此类冗余行。最后，我们展示了不同特定测量矩阵以及基于原型图的测量矩阵集合的数值结果，