Geometry Rescaling

One of the major issues with the stability of the timemachine is whether or not your Hessians are PSD. A non-PSD Hessian induces a transition matrix with a spectral radius , causing a divergent trajectory.

Recall Brownian dynamics:

Assuming a zero mixed partial, we can differentiate with respect to

G and H are the Gradients and Hessians, respectively. Since is rarely PSD in Cartesian coordinates, how can we fix this?

By playing dirty: consider an alternative dynamical equation where we shrink the geometry by a small amount:

Which has derivative:

Which is equivalent to adding a diagonal to the Hessian. So long as is bigger than the sum of each row/column, our Hessian is now PSD.

The downside to all of us? No one really knows what statistical ensemble this belongs to anymore.