Riemannian gradient descent for Hartree-Fock theory

• URL: http://arxiv.org/abs/2603.15870v1
• Date: Mon, 16 Mar 2026 19:58:42 GMT
• Title: Riemannian gradient descent for Hartree-Fock theory
• Abstract summary: We present an optimization framework for Hartree-Fock theory formulated directly in the Sobolev space $H1$.<n>Explicit expressions for Euclidean and Riemannian gradients, tangent-space projections, and retractions are derived.<n>The proposed formulation provides a geometrically consistent and discretization-independent perspective on electronic structure optimization.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We present a Riemannian optimization framework for Hartree-Fock theory formulated directly in the Sobolev space $H^1$. The orthonormality constraints are interpreted geometrically via infinite-dimensional Stiefel and Grassmann manifolds endowed with the embedded $H^1$ metric. Explicit expressions for Euclidean and Riemannian gradients, tangent-space projections, and retractions are derived using resolvent operators, avoiding distributional formulations. The resulting algorithms include Riemannian steepest descent and a preconditioned nonlinear conjugate gradient method equipped with Armijo backtracking and Powell-type restarts. Particular attention is given to physically motivated preconditioning based on inversion of the kinetic energy operator. The framework is naturally compatible with adaptive multiwavelet discretizations, where Coulomb-type convolutions can be evaluated efficiently. Numerical experiments demonstrate robust convergence and competitive performance compared to conventional SCF-DIIS schemes. In addition, for small molecules the gradient descent method converges from random initial guesses. The proposed formulation provides a geometrically consistent and discretization-independent perspective on electronic structure optimization and offers a foundation for further developments in infinite-dimensional Riemannian methods for quantum chemistry.

Related papers

• Fitting Unknown Number of Hyperplanes with Manifold Optimization [57.48093263119306]
  Fitting unknown number of linearplanes to data is a fundamental yet challenging problem in machine learning.<n>Existing approaches often struggle with optimum optima or lack geometric consistency.
  arXiv  Detail & Related papers  (2026-05-27T14:02:20Z)
• BOOOM: Loss-Function-Agnostic Black-Box Optimization over Orthonormal Manifolds for Machine Learning and Statistical Inference [0.0]
  We present a systematic exploration of.<n>the set of $p.<n>orthonormal column-orthonormal escape through a.<n>novel applied to data data.<n>problems.<n>We show a diagonal optimization framework showing a.<n>strong, low-rank decomposition, independent decomposition analysis,.<n>and global convergence in under conditions.
  arXiv  Detail & Related papers  (2026-04-23T19:32:27Z)
• On Higher-Order Geometric Refinements of Classical Covariance Asymptotics: An Approach via Intrinsic and Extrinsic Information Geometry [0.0]
  In curved models, including mixtures, curved exponential families, latent-variable models, and manifold-root parameter spaces, finite-sample behavior can deviate systematically from predictions.<n>We develop a coordinate-invariant, curvature-aware refinement by viewing a regular parametric family as a ((,g)) with Fisher-constrained-Rao metric.<n>We describe the resolved metric, the role of the real log canonical threshold in learning rates and posterior mean-squared error, and a curvature-based covariance expansion on the resolved space that recovers the regular theory as a special case.
  arXiv  Detail & Related papers  (2026-04-14T13:40:13Z)
• Riemannian Langevin Dynamics: Strong Convergence of Geometric Euler-Maruyama Scheme [51.56484100374058]
  Low-dimensional structure in real-world data plays an important role in the success of generative models.<n>We prove convergence theory of numerical schemes for manifold-valued differential equations.
  arXiv  Detail & Related papers  (2026-03-04T01:29:35Z)
• Provable Non-Convex Euclidean Distance Matrix Completion: Geometry, Reconstruction, and Robustness [8.113729514518495]
  The Euclidean Distance Matrix Completion problem arises in a broad range of applications, including sensor network localization, molecular robustness, and manifold learning.<n>In this paper, we propose a low-rank matrix completion task over the space of positive semi-definite Gram matrices.<n>The available distance measurements are encoded as expansion coefficients in a non-orthogonal basis, and optimization over the Gram matrix implicitly enforces geometric consistency through nonnegativity and the triangle inequality.
  arXiv  Detail & Related papers  (2025-07-31T18:40:42Z)
• Euclidean Distance Matrix Completion via Asymmetric Projected Gradient Descent [25.846262685970164]
  This paper proposes and analyzes a gradient-type algorithm based on Burer-Monteiro factorization.<n>It reconstructs the point set configuration from partial Euclidean distance measurements.
  arXiv  Detail & Related papers  (2025-04-28T07:13:23Z)
• Last-Iterate Convergence of Adaptive Riemannian Gradient Descent for Equilibrium Computation [52.73824786627612]
  This paper establishes new convergence results for textitgeodesic strongly monotone games.<n>Our key result shows that RGD attains last-iterate linear convergence in a textitgeometry-agnostic fashion.<n>Overall, this paper presents the first geometry-agnostic last-iterate convergence analysis for games beyond the Euclidean settings.
  arXiv  Detail & Related papers  (2023-06-29T01:20:44Z)
• Flow Matching on General Geometries [43.252817099263744]
  We propose a simple yet powerful framework for training continuous normalizing flows on manifold geometries. We show that it is simulation-free on simple geometries, does not require divergence, and computes its target vector field in closed-form. Our method achieves state-of-the-art performance on many real-world non-Euclidean datasets.
  arXiv  Detail & Related papers  (2023-02-07T18:21:24Z)
• The Dynamics of Riemannian Robbins-Monro Algorithms [101.29301565229265]
  We propose a family of Riemannian algorithms generalizing and extending the seminal approximation framework of Robbins and Monro. Compared to their Euclidean counterparts, Riemannian algorithms are much less understood due to lack of a global linear structure on the manifold. We provide a general template of almost sure convergence results that mirrors and extends the existing theory for Euclidean Robbins-Monro schemes.
  arXiv  Detail & Related papers  (2022-06-14T12:30:11Z)
• First-Order Algorithms for Min-Max Optimization in Geodesic Metric Spaces [93.35384756718868]
  min-max algorithms have been analyzed in the Euclidean setting. We prove that the extraiteient (RCEG) method corrected lastrate convergence at a linear rate.
  arXiv  Detail & Related papers  (2022-06-04T18:53:44Z)
• Optimization on manifolds: A symplectic approach [127.54402681305629]
  We propose a dissipative extension of Dirac's theory of constrained Hamiltonian systems as a general framework for solving optimization problems. Our class of (accelerated) algorithms are not only simple and efficient but also applicable to a broad range of contexts.
  arXiv  Detail & Related papers  (2021-07-23T13:43:34Z)
• A Unifying and Canonical Description of Measure-Preserving Diffusions [60.59592461429012]
  A complete recipe of measure-preserving diffusions in Euclidean space was recently derived unifying several MCMC algorithms into a single framework. We develop a geometric theory that improves and generalises this construction to any manifold.
  arXiv  Detail & Related papers  (2021-05-06T17:36:55Z)
• Curvature-Dependant Global Convergence Rates for Optimization on Manifolds of Bounded Geometry [6.85316573653194]
  We give curvature-dependant convergence rates for weakly convex functions defined on a manifold of 1-bounded geometry. We compute these bounds explicitly for some manifold commonly used in the optimization literature. We present self-contained proofs of fully general bounds on the norm of the differential of the exponential map.
  arXiv  Detail & Related papers  (2020-08-06T08:30:35Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.