Foundations of Riemannian Geometry for Riemannian Optimization: A Monograph with Detailed Derivations

• URL: http://arxiv.org/abs/2605.02279v1
• Date: Mon, 04 May 2026 07:11:11 GMT
• Title: Foundations of Riemannian Geometry for Riemannian Optimization: A Monograph with Detailed Derivations
• Abstract summary: This work provides a self-contained and rigorous treatment of the foundations of Riemannian geometry.<n>We develop the key geometric structures -- including tangent and cotangent spaces, tensor calculus, metric tensors, Levi-Civita connections, curvature, and geodesics.<n>We specialize these constructions to important matrix manifold, including the Stiefel, Grassmann, and SPD.
• Score: 0.10152838128195464
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Riemannian geometry provides the fundamental framework for optimization on nonlinear spaces such as matrix manifolds, which arise in machine learning, signal processing, and robotics. While the underlying theory is classical, existing literature often presents results at a high level of abstraction, omitting the detailed coordinate-level derivations required for implementation and algorithm development. This work provides a self-contained and rigorous treatment of the foundations of Riemannian geometry, with a focus on explicit derivations tailored to Riemannian optimization. We systematically develop the key geometric structures -- including tangent and cotangent spaces, tensor calculus, metric tensors, Levi-Civita connections, curvature, and geodesics -- emphasizing step-by-step derivations in coordinates and matrix form. Building on these foundations, we derive the Riemannian gradient, Hessian, exponential map, and retraction in a form suitable for numerical computation. We further specialize these constructions to important matrix manifolds, including the Stiefel, Grassmann, and SPD (Symmetric Positive Definite) manifolds, providing explicit formulas widely used in optimization and geometric machine learning. This monograph develops a unified and implementation-oriented treatment of Riemannian geometry for optimization on manifolds. Its main contribution is the systematic organization and detailed derivation of classical geometric constructions in forms directly usable for algorithm design and numerical implementation. By connecting coordinate-level differential geometry with matrix-manifold formulas, the monograph bridges the gap between abstract theory and practical computation, and provides a reference for researchers and practitioners working in Riemannian optimization and related fields.

Related papers

• Riemannian Deep Learning: Modules, Networks, and Geometries [0.0]
  This thesis develops a unified framework for Riemannian deep learning from three complementary perspectives.<n>It generalizes batch normalization from Euclidean spaces to broad classes of Lie groups and gyrogroups.<n>It also develops neural networks for several important geometric representations, including hyperbolic space, Busemann-based hyperbolic learning, and full-rank correlation matrices.
  arXiv  Detail & Related papers  (2026-07-21T17:17:35Z)
• Riemannian gradient descent for Hartree-Fock theory [0.0]
  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.
  arXiv  Detail & Related papers  (2026-03-16T19:58:42Z)
• Dimensionality Reduction on Riemannian Manifolds in Data Analysis [0.0]
  Experimental results show improved representation quality and classification performance compared to Euclidean counterparts.<n>This study underscores the importance of geometry aware dimensionality reduction in modern machine learning and data science applications.
  arXiv  Detail & Related papers  (2026-02-05T17:46:58Z)
• Score-based Pullback Riemannian Geometry: Extracting the Data Manifold Geometry using Anisotropic Flows [10.649159213723106]
  We propose a framework for data-driven Riemannian geometry that is scalable in both geometry and learning.<n>We show that the proposed framework produces high-quality geodesics passing through the data support.<n>This is the first scalable framework for extracting the complete geometry of the data manifold.
  arXiv  Detail & Related papers  (2024-10-02T18:52:12Z)
• RMLR: Extending Multinomial Logistic Regression into General Geometries [64.16104856124029]
  Our framework only requires minimal geometric properties, thus exhibiting broad applicability. We develop five families of SPD MLRs under five types of power-deformed metrics. On rotation matrices we propose Lie MLR based on the popular bi-invariant metric.
  arXiv  Detail & Related papers  (2024-09-28T18:38:21Z)
• 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)
• A singular Riemannian geometry approach to Deep Neural Networks I. Theoretical foundations [77.86290991564829]
  Deep Neural Networks are widely used for solving complex problems in several scientific areas, such as speech recognition, machine translation, image analysis. We study a particular sequence of maps between manifold, with the last manifold of the sequence equipped with a Riemannian metric. We investigate the theoretical properties of the maps of such sequence, eventually we focus on the case of maps between implementing neural networks of practical interest.
  arXiv  Detail & Related papers  (2021-12-17T11:43:30Z)
• Geometry-aware Bayesian Optimization in Robotics using Riemannian Mat\'ern Kernels [64.62221198500467]
  We show how to implement geometry-aware kernels for Bayesian optimization. This technique can be used for control parameter tuning, parametric policy adaptation, and structure design in robotics.
  arXiv  Detail & Related papers  (2021-11-02T09:47:22Z)
• On Geometric Connections of Embedded and Quotient Geometries in Riemannian Fixed-rank Matrix Optimization [5.876141028192136]
  This paper proposes a general procedure for establishing the geometric landscape connections of a Riemannian optimization problem under the embedded and quotient geometries. We observe an algorithmic connection between two geometries with some specific Riemannian metrics in fixed-rank matrix optimization. Results provide a few new theoretical insights to unanswered questions in the literature.
  arXiv  Detail & Related papers  (2021-10-23T03:13:56Z)
• Automatic differentiation for Riemannian optimization on low-rank matrix and tensor-train manifolds [71.94111815357064]
  In scientific computing and machine learning applications, matrices and more general multidimensional arrays (tensors) can often be approximated with the help of low-rank decompositions. One of the popular tools for finding the low-rank approximations is to use the Riemannian optimization.
  arXiv  Detail & Related papers  (2021-03-27T19:56:00Z)
• Bayesian Quadrature on Riemannian Data Manifolds [79.71142807798284]
  A principled way to model nonlinear geometric structure inherent in data is provided. However, these operations are typically computationally demanding. In particular, we focus on Bayesian quadrature (BQ) to numerically compute integrals over normal laws. We show that by leveraging both prior knowledge and an active exploration scheme, BQ significantly reduces the number of required evaluations.
  arXiv  Detail & Related papers  (2021-02-12T17:38:04Z)

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.