论文标题
关于隐藏的马尔可夫模型和张量分解的实现
On the Realization of Hidden Markov Models and Tensor Decomposition
论文作者
论文摘要
隐藏的马尔可夫模型（HMM）的最低实现问题是具有有限字母的固定离散时间过程的基本问题。在文献中显示，张量分解方法为隐藏的马尔可夫模型提供了最小数量的状态。但是，当观察结果是状态的确定性函数时，张量分解方法并不能解决最小的HMM实现问题，这是一个重要的HMM类别，而不是被通用参数捕获。在本文中，我们表明，当可及可及的子空间不是整个空间或空空间不是零空间时，可以分解从过程概率构建的三阶张量所需的排名一张量的数量。实际上，张量的等级不大于有效子空间的维度或广义Hankel矩阵的等级。
The minimum realization problem of hidden Markov models (HMM's) is a fundamental question of stationary discrete-time processes with a finite alphabet. It was shown in the literature that tensor decomposition methods give the hidden Markov model with the minimum number of states generically. However, the tensor decomposition approach does not solve the minimum HMM realization problem when the observation is a deterministic function of the state, which is an important class of HMM's not captured by a generic argument. In this paper, we show that the reduction of the number of rank-one tensors necessary to decompose the third-order tensor constructed from the probabilities of the process is possible when the reachable subspace is not the whole space or the null space is not the zero space. In fact, the rank of the tensor is not greater than the dimension of the effective subspace or the rank of the generalized Hankel matrix.
