Constrained dynamics for searching saddle points on general Riemannian manifolds

TL;DR

This work develops constrained saddle dynamics on general Riemannian manifolds by formulating over the Grassmann bundle $\mathrm{Gr}_k(\mathrm{T}\mathcal{M})$ and incorporating the second fundamental form $\mathrm{II}_{\boldsymbol{x}}$ to account for tangent-space variation. It proves global well-definedness, local linear stability of index-$k$ constrained saddle points, and the first local convergence results for discretized manifold saddle-search algorithms, while removing strict nondegeneracy requirements through intrinsic quotient geometry. The methodology is validated numerically on linear eigenvalue problems and electronic excited-state calculations, highlighting the necessity of nonredundant parametrizations and the benefits of Grassmann over Stiefel formulations. These results broaden the applicability of saddle-point searches to manifolds with quotient structures, including electronic structure contexts, and provide foundational theory for manifold-constrained optimization in nonconvex energy landscapes.

Abstract

Finding constrained saddle points on Riemannian manifolds is significant for analyzing energy landscapes arising in physics and chemistry. Existing works have been limited to special manifolds that admit global regular level-set representations, excluding applications such as electronic excited-state calculations. In this paper, we develop a constrained saddle dynamics applicable to smooth functions on general Riemannian manifolds. Our dynamics is formulated compactly over the Grassmann bundle of the tangent bundle. By analyzing the Grassmann bundle geometry, we achieve universality via incorporating the second fundamental form, which captures variations of tangent spaces along the trajectory. We rigorously establish the local linear stability of the dynamics and the local linear convergence of the resulting algorithms. Remarkably, our analysis provides the first convergence guarantees for discretized saddle-search algorithms in manifold settings. Moreover, by respecting the intrinsic quotient structure, we remove unnecessary nondegeneracy assumptions on the eigenvalues of the Riemannian Hessian that are present in existing works. We also point out that locating saddle points can be more ill-conditioning than finding local minimizers, and requires using nonredundant parametrizations. Finally, numerical experiments on linear eigenvalue problems and electronic excited-state calculations showcase the effectiveness of the proposed algorithms and corroborate the established local theory.

Figures (6)

• Figure 1: Convergence curves of Algorithm \ref{['alg:discretized deterministic constrained saddle dynamics lifted']} of Stiefel (orange lines) and Grassmann (blue lines) versions on the linear eigenvalue problem ($n=64$, $p=8$, and $\xi=1.01$). Left: position (solid lines) and direction (dashed lines) relative residuals vs iteration. Middle: function value vs iteration. Right: estimated lowest eigenvalue of Riemannian Hessian vs iteration.
• Figure 2: Average iteration number (left) and estimated condition number (right) vs $(n,p)$ on the linear eigenvalue problem class.
• Figure 3: Average iteration number (left) and estimated condition number (right) vs $(n,p)$ on the linear eigenvalue problem class.
• Figure 4: Average iteration number vs $(\eta_P,\eta_\Gamma)$ on the linear eigenvalue problem ($n=10$, $p=2$, and $\xi=1.01$). Left: perturbation level of $10^{-3}$. Right: perturbation level of $10^{-1}$.
• Figure 5: An overview of the constrained SPs of RHF found across the bond length interval on the H$_2$ molecule. The blue solid and dashed lines represent the energies of FCI states belonging to the irreps ${\rm A_g}$ and ${\rm A_u}$, respectively. The red dots, deepblue squares, yellow stars, and purple diamonds stand for the energies of the RHF SPs of indices 0, 1, 2, and 3, respectively.

Theorems & Definitions (28)

• Lemma 1: Tangent space to $\mathrm{St}_k(\mathrm{T}\mathcal{M})$
• proof
• Lemma 2: Tangent space to $\mathrm{Gr}_k(\mathrm{T}\mathcal{M})$
• Lemma 3: Decomposition of the tangent space to $\mathrm{St}_k(\mathrm{T}\mathcal{M})$
• proof
• Definition 1: Sasaki-type metric for $\mathrm{Gr}_k(\mathrm{T}\mathcal{M})$
• Lemma 4: Retraction over $\mathrm{Gr}_k(\mathrm{T}\mathcal{M})$
• proof
• Example 1: Second fundamental form in special cases
• Theorem 1: Global well-definedness of the dynamics \ref{['eqn:deterministic constrained saddle dynamics']}