Tensor models play an increasingly prominent role in many fields, notably in machine learning. In several applications of such models, such as community detection, topic modeling and Gaussian mixture learning, one must estimate a low-rank signal from a noisy tensor. Hence, understanding the fundamental limits and the attainable performance of estimators of that signal inevitably calls for the study of random tensors. Substantial progress has been achieved on this subject thanks to recent efforts, under the assumption that the tensor dimensions grow large. Yet, some of the most significant among these results--in particular, a precise characterization of the abrupt phase transition (in terms of signal-to-noise ratio) that governs the performance of the maximum likelihood (ML) estimator of a symmetric rank-one model with Gaussian noise--were derived on the basis of statistical physics ideas, which are not easily accessible to non-experts. In this work, we develop a sharply distinct approach, relying instead on standard but powerful tools brought by years of advances in random matrix theory. The key idea is to study the spectra of random matrices arising from contractions of a given random tensor. We show how this gives access to spectral properties of the random tensor itself. In the specific case of a symmetric rank-one model with Gaussian noise, our technique yields a hitherto unknown characterization of the local maximum of the ML problem that is global above the phase transition threshold. This characterization is in terms of a fixed-point equation satisfied by a formula that had only been previously obtained via statistical physics methods. Moreover, our analysis sheds light on certain properties of the landscape of the ML problem in the large-dimensional setting. Our approach is versatile and can be extended to other models, such as asymmetric, non-Gaussian and higher-order ones.
翻译:在许多领域,尤其是机器学习领域,电锯模型发挥着日益突出的作用。在诸如社区探测、主题模型和高斯混合学习等若干应用中,必须从一个吵闹的电压中估计低声信号。因此,了解该信号估计者的基本限度和可达到的性能必然要求随机电压研究。由于最近的努力,在假定电压值大幅增长的假设下,在这个问题上已经取得了显著进展。然而,在这些结果中,一些最显著的直径直径直径直径直径直径直的模型,特别是精确描述突然的阶段过渡(信号至噪音平面平面平面平面平面平面平面平面平面比)的特征,以调节一个最大可能性(ML)的信号信号信号信号信号。因此,根据统计物理思想,高斯的测距平面平面平面模型的性能,非专家难以轻易地了解。在这项工作中,我们开发了一种截然不同的方法,而不是依靠多年的随机矩阵理论带来的标准但有力的工具。关键的想法是研究一个随机矩阵模型的光谱矩阵模型,而这是从甚高距平面平面平面平面平面平面平面平面平面平面平面分析本身,我们的一个例子。我们把一个特定的直压直路路路段平面平面平面平面平面平立立了。