Riemannian adaptive stochastic gradient algorithms on matrix manifolds
This work addresses optimization problems in machine learning where data lie on matrix manifolds, offering a novel approach that preserves subspace structure, though it is incremental relative to existing Euclidean adaptive methods.
The paper tackles the challenge of developing adaptive stochastic gradient algorithms for optimization on Riemannian matrix manifolds, such as the Grassmann manifold, by adapting row and column subspaces of gradients, achieving a provable convergence rate of O(log(T)/√T).
Adaptive stochastic gradient algorithms in the Euclidean space have attracted much attention lately. Such explorations on Riemannian manifolds, on the other hand, are relatively new, limited, and challenging. This is because of the intrinsic non-linear structure of the underlying manifold and the absence of a canonical coordinate system. In machine learning applications, however, most manifolds of interest are represented as matrices with notions of row and column subspaces. In addition, the implicit manifold-related constraints may also lie on such subspaces. For example, the Grassmann manifold is the set of column subspaces. To this end, such a rich structure should not be lost by transforming matrices to just a stack of vectors while developing optimization algorithms on manifolds. We propose novel stochastic gradient algorithms for problems on Riemannian matrix manifolds by adapting the row and column subspaces of gradients. Our algorithms are provably convergent and they achieve the convergence rate of order $\mathcal{O}(\log (T)/\sqrt{T})$, where $T$ is the number of iterations. Our experiments illustrate the efficacy of the proposed algorithms on several applications.