The Baum-Welch (B-W) algorithm is the most widely accepted method for inferring hidden Markov models (HMM). However, it is prone to getting stuck in local optima, and can be too slow for many real-time applications. Spectral learning of HMMs (SHMMs), based on the method of moments (MOM) has been proposed in the literature to overcome these obstacles. Despite its promises, asymptotic theory for SHMM has been elusive, and the long-run performance of SHMM can degrade due to unchecked propogation of error. In this paper, we (1) provide an asymptotic distribution for the approximate error of the likelihood estimated by SHMM, and (2) propose a novel algorithm called projected SHMM (PSHMM) that mitigates the problem of error propogation, and (3) develop online learning variantions of both SHMM and PSHMM that accommodate potential nonstationarity. We compare the performance of SHMM with PSHMM and estimation through the B-W algorithm on both simulated data and data from real world applications, and find that PSHMM not only retains the computational advantages of SHMM, but also provides more robust estimation and forecasting.

翻译：Baum-Welch (B-W) 算法是推断隐马尔可夫模型 (HMM) 最广泛接受的方法。然而，该算法易陷入局部最优解，且对于许多实时应用而言速度过慢。基于矩方法 (MOM) 的隐马尔可夫模型谱学习 (SHMM) 已在文献中被提出以克服这些障碍。尽管前景广阔，但SHMM的渐近理论尚不明确，且由于误差的无约束传播，其长期性能可能退化。本文 (1) 给出了SHMM估计似然近似误差的渐近分布；(2) 提出了一种名为投影SHMM (PSHMM) 的新算法，以缓解误差传播问题；(3) 开发了SHMM和PSHMM的在线学习变体以应对潜在的非平稳性。我们通过模拟数据和实际应用数据比较了SHMM、PSHMM与B-W算法估计的性能，发现PSHMM不仅保留了SHMM的计算优势，还提供了更稳健的估计与预测。