Unlabeled sensing is a linear inverse problem with permuted measurements. We propose an alternating minimization (AltMin) algorithm with a suitable initialization for two widely considered permutation models: partially shuffled/$k$-sparse permutations and $r$-local/block diagonal permutations. Key to the performance of the AltMin algorithm is the initialization. For the exact unlabeled sensing problem, assuming either a Gaussian measurement matrix or a sub-Gaussian signal, we bound the initialization error in terms of the number of blocks $s$ and the number of shuffles $k$. Experimental results show that our algorithm is fast, applicable to both permutation models, and robust to choice of measurement matrix. We also test our algorithm on several real datasets for the linked linear regression problem and show superior performance compared to baseline methods.

翻译：无标记感知是一个具有置换测量的线性逆问题。针对两种广泛考虑的置换模型——部分混洗/$k$-稀疏置换与$r$-局部/块对角置换，我们提出了一种配备合适初始化方案的交替最小化算法。该算法性能的关键在于初始化过程。对于精确无标记感知问题，在假设测量矩阵服从高斯分布或信号服从亚高斯分布的前提下，我们以块数$s$与混洗次数$k$为参数对初始化误差进行了界定。实验结果表明，所提算法具有运行速度快、适用于两种置换模型、对测量矩阵选择具有鲁棒性等优势。在线性关联回归问题的多个真实数据集测试中，本算法相较于基线方法亦展现出更优越的性能。