We study the problem of learning mixtures of low-rank models, i.e. reconstructing multiple low-rank matrices from unlabelled linear measurements of each. This problem enriches two widely studied settings -- low-rank matrix sensing and mixed linear regression -- by bringing latent variables (i.e. unknown labels) and structural priors (i.e. low-rank structures) into consideration. To cope with the non-convexity issues arising from unlabelled heterogeneous data and low-complexity structure, we develop a three-stage meta-algorithm that is guaranteed to recover the unknown matrices with near-optimal sample and computational complexities under Gaussian designs. In addition, the proposed algorithm is provably stable against random noise. We complement the theoretical studies with empirical evidence that confirms the efficacy of our algorithm.
翻译:我们研究学习低级模型的混合问题,即从无标签线性测量中重建多种低级矩阵。这个问题丰富了两个经过广泛研究的设置 -- -- 低级矩阵感测和混合线性回归 -- -- 通过将潜在变量(即未知标签)和结构前科(即低级结构)纳入考虑而丰富了两个内容。为了处理由无标签的混杂数据和低复杂度结构产生的非混凝土问题,我们开发了一个三阶段元等级,保证在Gaussian设计下用接近最佳的样本和计算复杂性来恢复未知的矩阵。此外,拟议的算法与随机噪声相比是相当稳定的。我们用证实我们算法有效性的经验证据来补充理论研究。