# 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 $p(r) > 1$, causing a divergent trajectory.

Recall Brownian dynamics: $x_t = x_{t-1} - k G(x, \theta) + noise$

Assuming a zero mixed partial, we can differentiate with respect to $\theta$ $x'_t = x'_{t-1} - k H(x_{t-1}) x'_{t-1}$ $x'_t = (I - k H) x'_{t-1}$

G and H are the Gradients and Hessians, respectively. Since $H$ 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: $x_t = (1 - k c) x_{t-1} - k G(x, \theta) + noise$

Which has derivative: $x'_t = (1 - k c) x'_{t-1} - k H(x_{t-1}) x'_{t-1}$ $x'_t = (I - k c I - k H) x'_{t-1}$ $x'_t = (I - k (c I + H) ) x'_{t-1}$

Which is equivalent to adding a diagonal to the Hessian. So long as $c$ 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.