Nonconvex optimization on data manifold by accelerated curvature transport*

In this paper, we explore a new approach to optimization of cost or utility functions defined over a surface, a manifold, or its simplicial decomposition. In the era of Big Data, heterogeneous signal samples sometimes embed with less distortion in a lower dimensional space if the embedding space is a manifold rather than the traditional Euclidean space. If a utility function is defined over the data and if there is a need to identify significant events defined by extreme values of the utility function, we are faced with the problem of identifying the extreme minima/maxima points of the cost/utility function defined over the manifold or its triangulation. The fundamental idea developed here is to observe that at the extreme points the graph of the utility function has extreme curvature. Accordingly, the celebrated Ricci/Yamabe flow for uniformization of the curvature of the graph will show significant "curvature transport" in the vicinity of the extreme values, hence allowing their rapid identification, obviating the classical sorting. The novel theoretical contribution is to accelerate the process by compounding the Laplace operator.