I am trying to define the dynamics of a system to follow a specific trajectory in the state space indicated by a point cloud/previous measurements/observations.
Let us consider a simple example: In the following figure, I want my trajectory to start from one green circle and end at the second green circle and follow the trajectory indicated by the red points.
The system will follow some baseline simple dynamics (indicated here by the orange line) that ensures that the second point will be reached, and at every timestep I'd like to add some penalty cost that will drive the system towards the lower dimensional manifold indicated by the red points.
So I would like to be able to set some cost that maximizes the projection of the trajectory on the said empirical manifold.
My problem is that most manifold learning methods, identify a lower dimensional manifold, but they only return lower dimensional co-ordinates of the point cloud to a lower dimensional space. Here, I would need either the geodesic of the learnt manifold so I can constrain my trajectory to follow the geodesic, or at least some mapping that would transform a random point of the ambient space to the lower dimensional space.
Has anyone seen any method or publication that performs what I describe here? or does anyone has any indication of what to read and how to proceed onward?
OK, here is one possible path: Let $R_1,...,R_N$ denote the points in the cloud, $A$ the aimed point and $r$ the current point then you may calculate a 'local' center of gravity $C$ of the cloud in the vicinity of $r$ by taking the average of $R_i$ with respect to a probability vector which could be chosen e.g. as (with $\sigma>0$): $$ const \times \exp ( - \sum_i (R_i-r)^2 / \sigma^2 )$$
You may chose a vector field like: $v(r) = A-r + \lambda (C-r)$ and solve $\dot{r}=v(r)$. I made a test for 1000 points randomly distributed on the annulus $0.95<x^2+y^2 < 1.05$. I chose $\lambda=25$ and $\sigma^2=0.05$ and it worked out pretty well.
The first term $A-r$ just directs towards the aimed for point. The other term $\lambda (C-r)$ drifts towards the 'local' center of gravity of the constraint cloud and $\sigma$ is the spread on which compute this local center.
Choosing $\lambda$ large gives a strong constraint but gives a 'bumpy' ride and sometimes block the trajectory at some isolated point in the cloud. Choosing $\sigma$ small is good if you have a very dense point set in the cloud. There is some trade off between these aspects. Also this simple approach may get stuck in some local minimum distance to A (notably if your curve for the cloud is not convex).