Say that I have a discretized differential path vector of $n$ discrete points defined in $\mathbb R^k$:
$${\bf d} = \left[\begin{array}{cccc}dx_{11}&dx_{12}&\cdots&dx_{1k}\\dx_{21}&dx_{22}&\cdots&dx_{2k}\\\vdots&\vdots&\ddots&\vdots\\dx_{n1}&dx_{n2}&\cdots&dx_{nk}\end{array}\right]$$
Now we can convince ourselves that the matrix representing a discretized integral and differential operator will be $${\bf S} = \left[\begin{array}{cccccc}1&0&0&0&0&0\\1&1&0&0&0&0\\1&1&1&0&0&0\\1&1&1&1&0&0\\ 1&1&1&1&1&0\\1&1&1&1&1&1\end{array}\right]\hspace{1cm}{\bf D = S}^{-1} = \left[\begin{array}{rrrrrr}1&0&0&0&0&0\\-1&1&0&0&0&0\\ 0&-1&1&0&0&0\\0&0&-1&1&0&0\\0&0&0&-1&1&0\\0&0&0&0&-1&1\end{array}\right]$$
That is, the operator that "sums up the snake" in this sense:
$$\bf Sd = \left[\begin{array}{cccc}x_{11}&x_{12}&\cdots&x_{1k}\\x_{21}&x_{22}&\cdots&x_{2k}\\\vdots&\vdots&\ddots&\vdots\\x_{n1}&x_{n2}&\cdots&x_{nk}\end{array}\right]\hspace{1cm} {\bf D(Sd)} = {\bf d}$$
I.e. we integrate $\bf d$ to get the coordinates of each discretized point along the snake and we differentiate $\bf Sd$ to get the differentials at each point - like a "current direction" vector.
We may want to decide some initial point $\bf x_0 \neq \bf 0$ but there is nothing preventing us from baking it into the $\bf d$ matrix in a first row for as long as we remember to modify the operators accordingly.
Now to my question, can we combine this curve representation with a joint grid optimization in an exclusively linear least squares program. So that we can put constraints on all $dx_{ij}, x_{ij}$ and also simultaneously optimize some scalar or vector valued function on some grid with it's own constraints and costs.
Own work I have not managed to do this without taking the intermediate steps of some functional parametrization, for example polynomial regression or updating the cost matrix for the grid for each new step in the solver ( which may be numerically sound, but essentially breaks linearity of the problem ).
Here is an animation that tries to explain the relationship between $dx_{ij}$ and $x_{ij}$ in case it is too complicated.
