We consider the problem of jointly modeling and clustering populations of tensors by introducing a high-dimensional tensor mixture model with heterogeneous covariances. To effectively tackle the high dimensionality of tensor objects, we employ plausible dimension reduction assumptions that exploit the intrinsic structures of tensors such as low-rankness in the mean and separability in the covariance. In estimation, we develop an efficient high-dimensional expectation-conditional-maximization (HECM) algorithm that breaks the intractable optimization in the M-step into a sequence of much simpler conditional optimization problems, each of which is convex, admits regularization and has closed-form updating formulas. Our theoretical analysis is challenged by both the non-convexity in the EM-type estimation and having access to only the solutions of conditional maximizations in the M-step, leading to the notion of dual non-convexity. We demonstrate that the proposed HECM algorithm, with an appropriate initialization, converges geometrically to a neighborhood that is within statistical precision of the true parameter. The efficacy of our proposed method is demonstrated through comparative numerical experiments and an application to a medical study, where our proposal achieves an improved clustering accuracy over existing benchmarking methods.
翻译:我们考虑采用高维多元混合模型,采用多种共差的高维多元混合模型,共同建模和组组群变色体群的问题。为了有效处理高维度的强度物体,我们采用合理维度的减少假设,利用高温体的内在结构,如中位低和共差分等;估计,我们开发了高效的高维预期-条件最大化算法,将M级的棘手优化打破为一系列简单得多的有条件优化问题,其中每个问题都是 convex,接受正规化,并有封闭式更新公式。我们的理论分析受到EM型估算中非兼容性的挑战,而且只能获得M级中有条件最大化的解决办法,从而导致双重非共振度概念。我们证明,拟议的HEMM算法经过适当初始化后,从几何角度将真正参数在统计精确度范围内的邻里相融合。我们拟议方法的功效是通过比较数字实验和对医学基准研究的应用来显示的,从而实现现有改进后的基准方法。