with $\widetilde{{\boldsymbol{\varOmega}}}$, notably comparing with that of <span style="font-variant:small-caps;">Theta-Update</span>. In terms of initialization, which is important since we are optimizing a non-convex problem, we propose to use the method by @huang2016nips to obtain an initialization for ${\boldsymbol{M}}$; for ${\boldsymbol{\varTheta}}$, it is best if we start with a feasible point (so that the Newton’s iterates will remain feasible), and a simple choice is scaling the matrix ${\boldsymbol{I}}+{\boldsymbol{\mathit{1}}}\!{\boldsymbol{\mathit{1}}}^{{\!\top\!}}$ to sum up to one. Finally, we show that this algorithm converges to a stationary point of Problem , with proof relegated to the supplementary material based on [@razaviyayn2013unified].

Assume <span style="font-variant:small-caps;">ThetaUpdate</span> solves Problem  with respect to ${\boldsymbol{\varTheta}}$ exactly, then Algorithm \[alg:hmm\_id\] converges to a stationary point of Problem .

![The total variation difference between the ground truth and estimated transition probability (top) and emission probability (bottom). The total variation difference of the emission probabilities is calculated as $\frac{1}{2K}\|{\boldsymbol{M}}_\natural-{\boldsymbol{M}}_\star\|_1$, since each column of the matrices indicates a (conditional) probability, and the total variation difference is equal to one half of the $L_1$-norm; and similarly for that of the transition probabilities after rescaling the rows of ${\boldsymbol{\varOmega}}_\natural$ and ${\boldsymbol{\varOmega}}_\star$ to sum up to one. The result is averaged over 10 random problem instances.[]{data-label="fig:sim_exp1"}](sim_exp1){width="\linewidth"}

Validation on 