Optimization without Retraction on the Random Generalized Stiefel Manifold
This addresses computational bottlenecks in machine learning problems with generalized orthogonality constraints, offering a more efficient solution for practitioners, though it is incremental as it builds on existing Riemannian optimization frameworks.
The paper tackles optimization on the generalized Stiefel manifold, which arises in applications like canonical correlation analysis and independent component analysis, by proposing a stochastic iterative method that uses random estimates of the covariance matrix instead of requiring the full matrix, achieving lower per-iteration cost and matching convergence rates of existing methods.
Optimization over the set of matrices $X$ that satisfy $X^\top B X = I_p$, referred to as the generalized Stiefel manifold, appears in many applications involving sampled covariance matrices such as the canonical correlation analysis (CCA), independent component analysis (ICA), and the generalized eigenvalue problem (GEVP). Solving these problems is typically done by iterative methods that require a fully formed $B$. We propose a cheap stochastic iterative method that solves the optimization problem while having access only to random estimates of $B$. Our method does not enforce the constraint in every iteration; instead, it produces iterations that converge to critical points on the generalized Stiefel manifold defined in expectation. The method has lower per-iteration cost, requires only matrix multiplications, and has the same convergence rates as its Riemannian optimization counterparts that require the full matrix $B$. Experiments demonstrate its effectiveness in various machine learning applications involving generalized orthogonality constraints, including CCA, ICA, and the GEVP.