An Evolution Strategy for Black-box Optimization on Matrix Manifold

This work concerns the black-box approach towards a class of matrix optimization problems defined on Riemannian manifolds. Due to their non-Euclidean nature, these problems are intractable for traditional evolutionary algorithms. In this work, we transform, according to the manifold structure, the originally manifold-constrained problem into a sequence of unconstrained ones in the tangent spaces, and propose a manifold search direction adaptation evolution strategy by extending the covariance matrix adaptation mechanism from Euclidean space to matrix manifolds. The proposed algorithm maintains, in each tangent space, a multi-variate Gaussian distribution to guide the search and iteratively adapts it in order to increase the likelihood of reproducing high-quality solutions. This work provides the method of building the high-dimensional probability model with an identity matrix and only a few tangent search directions. The method of moving probability models between different tangent spaces is also discussed. The proposed algorithm is computationally efficient as all its genetic operators are performed using linear matrix transformations, independent of global coordinates, additional encoding schemes, or specifications of orthogonal bases. We verify the performance of the proposed algorithm on three matrix manifold optimization problems and the results show that it is superior to or competitive with other state-of-the-artsin black-box settings. © 2020, Science Press. All right reserved.