Yogesh Darmwal | Searcharxiv

arXiv2026

Rank-one Riemannian Subspace Descent for Nonlinear Matrix Equations

We propose a rank-one Riemannian subspace descent algorithm for computing symmetric positive definite (SPD) solutions to nonlinear matrix equations arising in control theory, dynamic programming, and stochastic filtering. For solution matrices of size $n\times n$, standard approaches for dense matrix equations typically incur $\mathcal{O}(n^3)$ cost per-iteration, while the efficient $\mathcal{O}(n^2)$ methods either rely on sparsity or low-rank solutions, or have iteration counts that scale poorly. The proposed method entails updating along the dominant eigen-component of a transformed Riemannian gradient, identified using at most $\mathcal{O}(\log(n))$ power iterations. The update structure also enables exact step-size selection in many cases at minimal additional cost. For objectives defined as compositions of standard matrix operations, each iteration can be implemented using only matrix--vector products, yielding $\mathcal{O}(n^2)$ arithmetic cost. We prove an $\mathcal{O}(n)$ iteration bound under standard smoothness assumptions, with improved bounds under geodesic strong convexity. Numerical experiments on large-scale CARE, DARE, and other nonlinear matrix equations show that the proposed algorithm solves instances (up to $n=10{,}000$ in our tests) for which the compared solvers, including MATLAB's \texttt{icare}, structure-preserving doubling, and subspace-descent baselines fail to return a solution. These results demonstrate that rank-one manifold updates provide a practical approach for high-dimensional and dense SPD-constrained matrix equations. MATLAB code implementation is publicly available on GitHub : \href{https://github.com/yogeshd-iitk/nonlinear_matrix_equation_R1RSD}{\textcolor{blue}{https://github.com/yogeshd-iitk/nonlinear\_matrix \_equation\_R1RSD}}

math.NA↗

arXiv2023

Low-complexity subspace-descent over symmetric positive definite manifold

This work puts forth low-complexity Riemannian subspace descent algorithms for the minimization of functions over the symmetric positive definite (SPD) manifold. Different from the existing Riemannian gradient descent variants, the proposed approach utilizes carefully chosen subspaces that allow the update to be written as a product of the Cholesky factor of the iterate and a sparse matrix. The resulting updates avoid the costly matrix operations like matrix exponentiation and dense matrix multiplication, which are generally required in almost all other Riemannian optimization algorithms on SPD manifold. We further identify a broad class of functions, arising in diverse applications, such as kernel matrix learning, covariance estimation of Gaussian distributions, maximum likelihood parameter estimation of elliptically contoured distributions, and parameter estimation in Gaussian mixture model problems, over which the Riemannian gradients can be calculated efficiently. The proposed uni-directional and multi-directional Riemannian subspace descent variants incur per-iteration complexities of $O(n)$ and $O(n^2)$ respectively, as compared to the $O(n^3)$ or higher complexity incurred by all existing Riemannian gradient descent variants. The superior runtime and low per-iteration complexity of the proposed algorithms is also demonstrated via numerical tests on large-scale covariance estimation and matrix square root problems. MATLAB code implementation is publicly available on GitHub : https://github.com/yogeshd-iitk/subspace_descent_over_SPD_manifold

stat.ML↗