https://arxiv.org/pdf/2305.19693.pdf
Our findings challenge the current dominant conception that the generative process of diffusion models is essentially comprised of a single denoising phase
Symmetry breaking is observed in physical phenomena like electromagnetism. It happens when we consider a stable state of equally influenced points, each direction to that point has equal favoring, but one direction is chosen that breaks the symmetry.
The symmetry does not come from laws of physics but instead from the implicitness of the dataset (just like data manifolds)
In beginning, particle is in symmetric because its dynamics reflect around a symmetric fixed point. However, after a critical time, the fixed point becomes unstable
Second derivatives are interesting because they describe how rapid the function changes at a point. If positive, it is convex. We have hit a minimum. This means it is stable.
Eigenvalues of a hessian matrix tell us how much variance in a direction
When we take second derivative of the potential function $u(x,t)$, we can determine the stability at a point. Then if we set it to 0 and solve for the $\theta_c$, it will show us when it changes from positive to negative, or shows us an inflection point. The concavity changes.
Since the early dynamics is approximately linear and meanreverting, the basic idea is to initialize the samplers just before the onset of the instability. This avoids a wasteful utilization of denoising steps in the early phase while also ensuring that the critical window of instability has been appropriately sampled.