Questions about PbD

1.Based on your explain, and since after LBG every point could only be assigned to a unique cluster, I think there would not exit a point that both stands on the end of a cluster and the beginning of the other. Do you actually mean like this that in the following fig ,both point 3 and 4 could be a key-point?

2.Since LBG has made a segment to the trajectory, why do we still need a HMM step to make another segment?And why do we need DTW to cluster the key-points after HMM?(Because I think the result of HMM could be a kind of clustering.Since all the demonstrations are assigned by the same state sequence, then the $\kappa_{j,m}^{HMM}$κj,mHMM​ connected to the same hidden state could be mapped to the same time index$\tau_j$τj​)

3.Can I make a understand about HMM step as follows:We do not need to configure the explicit position and pose of hidden states. What really matters is the number of hidden states, so we need the LBG step to give an initialization of total number of the hidden state.If my understand is not proper,then how we determine the spatial position of hidden state?

4.What is the meaning of the subscript of $s$s(e.g.$s_1,s_2\dots$s1​,s2​…)?(If $s_1$s1​ could jump directly to $s_3$s3​,then why not just view $s_3$s3​ as $s_2$s2​?hhh)Is it modified by the $\kappa_{j}^{LBG}$κjLBG​ of $X_\delta$Xδ​?Is the following fig right?

form of transition matrix:$\left[ \begin{matrix} a_{11} & a_{12} & a_{13} &0&0\\ 0 & a_{22} & a_{23} &a_{24} &0\\ 0&0 & a_{33} & a_{34} &a_{35}\\ \vdots&\vdots&\vdots&\ddots&\cdots\\ 0&0&0&0&a_{55} \end{matrix}\right]$⎣⎢⎢⎢⎢⎢⎡​a11​00⋮0​a12​a22​0⋮0​a13​a23​a33​⋮0​0a24​a34​⋱0​00a35​⋯a55​​⎦⎥⎥⎥⎥⎥⎤​
5.How could we get the $K_m^{HMM}$KmHMM​using hidden state sequence$\{s_i\}${si​}?