Optimizations on unknown low-dimensional structures given by high-dimensional data

Optimization problems on unknown low-dimensional structures given by high-dimensional data belong to the field of optimizations on manifolds. Though recent developments have advanced the theory of optimizations on manifolds considerably, when the unknown low-dimensional manifold is given in the form of a set of data in a high-dimensional space, a practical optimization method has yet to be developed. Here, we propose a neural network approach to these optimization problems. A neural network is used to approximate a neighborhood of a point, which will turn the computation of a next point in the searching process into a local constraint optimization problem. Our method ensures the convergence of the process. The proposed approach applies to optimizations on manifolds embedded into Euclidean spaces. Experimental results show that this approach can effectively solve optimization problems on unknown manifolds. The proposed method provides a useful tool to the field of study low-dimensional structures given by high-dimensional data.