Sophelen Research

Consider working in an area where you wish to keep the system running smoothly. A common practice is to continually learn a model f as an error detection system. However, if the domain knowledge is rich enough, like agriculture, geometry would be able to guide us towards stability and optimization. From the Manifold Hypothesis, our parameters live inside some manifold M_t given by some constraint C_t. We can carefully design something similar to C_t and R_t, a reward function. Both C_t and R_t are continually improved along the way.

The manifolds M_t form a fiber bundle

There is another fiber bundle from the tangle bundle of M.

There exist composition of global sections

The second global section tells us that there exists a trajectory that sustains the system, while the composition tells us how to achieve that trajectory. With the reward function, we can slowly nudge the trajectory towards the ridge.

Imagine a person trying to climb an ever-changing mountain, hoping to find the ridge. Each M_t is a rough sketch of the actual map and each R_t is a decent compass that is slowly improving. The global sections tell us that there are sustainable paths on the mountain, and if we follow our compass R_t, we should be able to get close to the ridge of the mountain.