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A Grassmann Manifold Handbook: Basic Geometry and Computational Aspects
Thomas Bendokat, Ralf Zimmermann, P. -A. Absil
TL;DR
Grassmann manifolds support many computational applications, but their geometric tools are spread across multiple representations and research tracks. This handbook unifies quotient and projector viewpoints in matrix-based formulae, with a modified logarithm algorithm and related geometric results. Its constructions provide computational tools while extending logarithm-map use to cut-locus points and enabling analysis of shortest curves and conjugate points.
Problem
Grassmann geometry is important across applications, but basis-based and projector-based research tracks are relatively disjoint and lack a unified collection of algorithm-ready formulae.
Method
The handbook develops essential Grassmann geometry simultaneously through orthogonal-projector and orthogonal-group quotient perspectives, using matrix-based formulae for computation.
Results
The work presents a modified Riemannian logarithm algorithm with favorable numerical properties and derives results for cut loci, conjugate loci, parallel transport, exponential derivatives, and Jacobi fields.
Takeaways & Limitations
The resulting framework supplies computational tools and extends logarithm-map operation to cut-locus points while supporting analysis of shortest curves and conjugate points.
Takeaways & Limitations
Coordinate parameterizations lack the metric special properties of Riemannian normal coordinates and constrain interpolation to the chosen coordinate patch.
Abstract
from arXiv · showhide
The Grassmann manifold of linear subspaces is important for the mathematical modelling of a multitude of applications, ranging from problems in machine learning, computer vision and image processing to low-rank matrix optimization problems, dynamic low-rank decompositions and model reduction. With this mostly expository work, we aim to provide a collection of the essential facts and formulae on the geometry of the Grassmann manifold in a fashion that is fit for tackling the aforementioned problems with matrix-based algorithms. Moreover, we expose the Grassmann geometry both from the approach of representing subspaces with orthogonal projectors and when viewed as a quotient space of the orthogonal group, where subspaces are identified as equivalence classes of (orthogonal) bases. This bridges the associated research tracks and allows for an easy transition between these two approaches. Original contributions include a modified algorithm for computing the Riemannian logarithm map on the Grassmannian that is advantageous numerically but also allows for a more elementary, yet more complete description of the cut locus and the conjugate points. We also derive a formula for parallel transport along geodesics in the orthogonal projector perspective, formulae for the derivative of the exponential map, as well as a formula for Jacobi fields vanishing at one point.
1 Introduction
The handbook develops matrix-based geometric tools for Grassmann manifolds, connecting quotient, basis, orthonormal-basis, projector, and Lie-group perspectives. It also presents new results for logarithms, cut and conjugate loci, parallel transport, exponential derivatives, and Jacobi fields.
- Grassmann manifolds model fixed-dimensional subspaces and support applications in data analysis, signal processing, optimization, low-rank decompositions, model reduction, and computer vision.
- Perspectives: The handbook treats basis, orthonormal-basis, projector, and Lie-group representations, which are closely related but developed in partly disjoint research literatures.
- Scope: The complex case is transferable by replacing O(n) with U(n) and transposes with conjugate transposes, but is not the main application focus.
- Purpose: It bridges these perspectives while collecting matrix formulae for Riemannian metrics, distances, connections, exponentials, logarithms, and normal coordinates.
- Original contributions: The modified logarithm algorithm has favorable numerical features and can non-uniquely map cut-locus points into a single tangent space.
- Original contributions: The work derives explicit shortest curves, a more complete principal-angle description of the conjugate locus, parallel transport, exponential-map derivatives, and Jacobi fields.
2 The Manifold Structure of the Grassmann Manifold
The Grassmannian is represented both as fixed-dimensional subspaces or orthogonal projectors and as quotient spaces of orthogonal matrices. These representations induce smooth manifold structures and identify tangent spaces through horizontal directions.
- Embedded structure: The Grassmann manifold consists of p-dimensional subspaces of R^n and can be identified with symmetric rank-p orthogonal projectors.
- Representations: The projector representation is linked to orthonormal bases through P = UU^T, while orthogonal matrices and Stiefel matrices provide non-unique representatives.
- Embedded structure: The orthogonal-group action makes the Grassmannian the orbit of the canonical projector P0, and compactness of O(n) establishes it as an embedded submanifold of Sym_n.
- Topology: The Grassmannian is connected and path-connected, so any two points can be joined by a path within the manifold.
- Quotient structure: The quotient structure identifies orthogonal matrices modulo the stabilizer O(p) × O(n−p), with πOG = πSG ◦ πOS mapping representatives to subspaces.
- Tangent spaces: A quotient projection splits tangent spaces into vertical and horizontal parts, and Grassmann tangent spaces are identified with horizontal spaces at representatives.
3 Riemannian Structure
The section develops the Grassmannian’s Riemannian structure in projector and quotient representations, giving matrix formulas for its metric, connection, exponential, gradients, and transport-related computations.
- Metric: The canonical quotient metric coincides with one half times the Euclidean metric on tangent projectors and is independent of the chosen lift.The same metric is obtained from orthogonal-group or Stiefel lifts because trace invariance removes dependence on the representative.
- Riemannian connection: The Riemannian connection is obtained by projecting the ambient Levi-Civita connection onto the Grassmannian tangent space and has an equivalent horizontal-lift formulation.The construction is compatible with the metric and torsion-free properties of a Riemannian connection.
- Tangent and horizontal projections: For the projector representation, tangent projection maps a symmetric matrix S to (I_n−P)SP+PS(I_n−P).The corresponding horizontal Stiefel projection is (I_n−UU^T)Z.
- Gradient: Gradients are defined as metric duals of differentials and can be computed by projecting suitable Euclidean gradients or lifting functions to the Stiefel manifold.The lifted gradient has no vertical component, enabling a horizontal representation.
- Exponential map and parallel transport: The exponential map sends tangent vectors to geodesic endpoints, while the section also derives formulas for its derivative and introduces parallel-transport constructions.The derivative supports applications including Jacobi fields vanishing at a point and Hermite manifold interpolation; the paper adds projector-perspective parallel transport.
- Differentiating the exponential: A QR-derivative approach removes the mutually distinct singular-values assumption and improves numerical stability near clustered singular values, while retaining the non-zero singular-values assumption.Matrices close to rank-deficient can still produce instabilities.
- Parallel transport: The projector perspective completes prior ONB treatments by providing an explicit formula for parallel transport along Grassmannian geodesics.Earlier work supplied an ONB formula along geodesics and a differential equation along general curves.
4 Symmetry and Curvature
The section constructs the Grassmannian as a symmetric space and uses that structure to establish completeness properties and explicit curvature formulas and bounds.
- Symmetric space structure: The Grassmannian is shown by explicit construction to possess a metric symmetry at every point, establishing its symmetric-space structure.The constructed isometries fix their reference points and reverse tangent vectors there.
- Completeness and Hopf–Rinow properties: The symmetric-space structure implies geodesic completeness, so every Grassmannian geodesic is defined on the whole real line.Hopf–Rinow consequences include completeness, compactness of closed bounded sets, and minimizing geodesics between any two points.
- Completeness and Hopf–Rinow properties: The exponential map is globally defined and surjective, and any two Grassmannian points can be joined by a minimizing geodesic segment.These properties are presented as consequences equivalent to the relevant completeness statements.
- Sectional curvature: The sectional curvature depends only on the plane spanned by two tangent vectors and admits explicit formulas in projector and horizontal-lift coordinates.The formulas can be expressed through the matrices B_i and their p × p products.
- Sectional curvature: For n > 2, Gr(n,1) and Gr(n,n−1) have constant sectional curvature 1, while strictly positive curvature throughout occurs only in these cases.Gr(2,1) is one-dimensional, so its sectional curvature is not defined.
- Sectional curvature: For min(p, n−p) ≥ 2, sectional curvature is nonnegative and bounded above by the stated sharp inequality; the lower bound is attained when the tangent matrices commute.The upper bound is also attained by specified matrix configurations.
5 Cut Locus and Riemannian Logarithm
The Grassmannian’s cut locus marks where minimizing geodesics cease to be unique, while the extended logarithm algorithm computes valid tangent representatives even at cut points.
- 5.2 Riemannian Logarithm: The Riemannian logarithm is the smallest-norm tangent vector whose exponential reaches F; at cut points, multiple such vectors may exist.This non-uniqueness enables tangent-space mappings for points otherwise excluded from the ordinary logarithm’s domain.
- 5.1 Cut Locus: A point lies in the cut locus of P exactly when at least one principal angle between P and that point equals π/2.Equivalently, the target subspace contains a direction orthogonal to every direction in P.
- 5.1 Cut Locus: Geodesics with largest horizontal-lift singular value below, equal to, or above π/2 are respectively uniquely minimizing, non-uniquely minimizing, or non-minimizing.The cut time is determined by the first occurrence of cos(tσ1)=0.
- 5.2 Riemannian Logarithm: Algorithm 5.3 computes the unique logarithm outside the cut locus and one valid cut-locus logarithm, while retaining Stiefel representatives without matrix inversion.The algorithm is designed to behave reliably near the cut locus, where the standard unprojected method develops increasing error around τ≈10^-3.
- 5.2 Riemannian Logarithm: The projector-perspective logarithm is explicit but uses n×n matrices, whereas lifting to the Stiefel manifold reduces computational complexity.The alternative QR-derivative approach removes the distinct-singular-values assumption but retains a non-zero-singular-values assumption.
6 Local Parameterizations of the Grassmann Manifold
The section constructs local coordinate charts for the Grassmannian using orthogonal-projector representations, offering alternatives to Riemannian exponential and logarithm mappings. These charts support Euclidean computation but are restricted to open subsets and lack the metric properties of normal coordinates.
- Projector-based charts: The Grassmannian has dimension (n − p)p, and local parameterizations can be constructed from open subsets of R^(n−p)×p.The construction uses the orthogonal-projector representation P = UU^T.
- Projector-based charts: A chart around a reference projector is obtained by mapping matrix coordinates B to projectors with an invertible p × p upper diagonal block A.The inverse chart maps such a projector to BA^−1, establishing a bijection onto the parameterization image.
- Projector-based charts: The local parameterization can be moved around the Grassmannian by conjugating the reference projector with an orthogonal matrix Q.This produces local parameterizations φ_Q(B) = Qφ(B)Q^T around other points.
- Computational use: Coordinate charts can replace Riemannian exponential and logarithm mappings for interpolation and optimization using standard Euclidean tools.These procedures avoid evaluating matrix exponentials and logarithms when data and objectives remain inside the chart domain.
- Computational limitations: Charts lack the metric special properties of Riemannian normal coordinates and constrain computations to an open subset, which may be unnatural for some data sets.Switching charts can also cause information gathered by an optimizer to lose relevance, although chart-based methods can still succeed.
7 Jacobi Fields and Conjugate Points
The section characterizes Jacobi fields and the Grassmannian’s conjugate locus through geodesic variations and principal angles. Repeated principal angles, and certain zero-angle cases, determine conjugacy, while cut and conjugate loci only partially coincide.
- Jacobi fields: Jacobi fields are vector fields along geodesics satisfying the Jacobi equation and representing variations through nearby geodesics.They are tangent vectors along the geodesic, not necessarily point-to-point offset vectors in the ambient Euclidean space.
- Jacobi fields: The derivative of the Grassmann exponential map yields explicit Jacobi fields with prescribed initial conditions J(0) = 0 and D_tJ(0) = Δ2.The construction uses compact SVDs of horizontal lifts and a corresponding horizontal-lift formula.
- Conjugate locus: The conjugate locus consists of points with at least two identical principal angles, or at least one zero principal angle when p < n/2.For p > n/2, the zero-angle condition requires at least 2p − n + 1 zero principal angles.
- Conjugate locus: Repeated principal angles generate families of geodesics and nontrivial Jacobi fields, establishing conjugacy even when the relevant geodesic is not minimizing.The proof exploits SVD indeterminacy and smooth variations among geodesics sharing endpoints.
- Cut versus conjugate loci: The cut locus and conjugate locus are neither nested, although conjugate points along minimizing geodesics are cut points when multiple principal angles equal π/2.Points with only one principal angle equal to π/2 are in the cut locus but not the conjugate locus.
- Relation to prior results: The real Grassmannian’s conjugate-locus description is more complete than the earlier treatment, which covered only repeated principal angles equal to π/2.The complex case additionally includes points with one principal angle equal to π/2.
8 Conclusion
The conclusion presents the handbook as a matrix-based reference for Grassmannian geometry and computation. It connects quotient, Stiefel, and projector viewpoints while introducing a logarithm algorithm that supports more explicit analyses of cut and conjugate points.
- Scope and tools: The handbook collects explicit formulas and algorithms for local coordinates, exponential and logarithm mappings, connections, parallel transport, and sectional curvature.These tools are intended for Riemannian computations on the Grassmann manifold.
- Applications: The collected concepts can serve as building blocks for optimization, interpolation, averaging, clustering, and other data-processing problems on Grassmannians.The conclusion frames these uses as tools for theoretical analysis and computational applications.
- Geometric viewpoints: The Grassmannian is treated both as a quotient of the orthogonal and Stiefel manifolds and as fixed-rank orthogonal projectors, with connections between the viewpoints exposed.The resulting formulas are purely matrix-based and designed for direct algorithmic use.
- Original contributions: The original Grassmann logarithm approach extends its operational domain, improves numerical properties, and enables detailed analyses of shortest curves to cut points and conjugate points.The resulting findings are described as more explicit and complete than previous research results.
A Basics from Riemannian Geometry
This section introduces the manifold, metric, distance, geodesics, curvature, tangent spaces, and Lie-group quotient foundations used in Riemannian analysis.
- Manifolds: An n-dimensional manifold is locally described by coordinate charts mapping neighborhoods smoothly and bijectively to open subsets of R^n.Compatible coordinate changes are diffeomorphisms, enabling calculus on manifolds.
- Tangent Spaces: A tangent space consists of velocity vectors of differentiable curves through a point and has the same dimension as the manifold.
- Riemannian Metrics and Distance: A Riemannian metric assigns smoothly varying inner products to tangent spaces, thereby defining tangent-vector lengths, curve lengths, and distances.The distance is the infimum of lengths over piecewise-smooth connecting curves.
- Geodesics and Connections: Geodesics are curves whose velocity has vanishing covariant derivative, while the Levi-Civita connection provides the relevant directional-derivative structure.
- Curvature: The Riemannian connection defines curvature through covariant derivatives, and a manifold is flat exactly when its curvature tensor vanishes identically.
- Lie Groups and Quotients: Matrix Lie groups are smooth matrix subgroups whose tangent space at the identity forms a Lie algebra, with other tangent spaces obtained by left translation.Smooth group actions yield orbits that, for compact groups, are embedded submanifolds diffeomorphic to quotient manifolds.
B Matrix Analysis Necessities
This section collects matrix-space, SVD, QR, and matrix-function facts needed for computational work with Grassmannian formulas.
- Matrix Space: The matrix space R^m×n is treated as a Euclidean vector space with its standard metric.
- Singular Value Decomposition: The compact SVD is used by default, but SVD factors are nonunique because orthogonal transformations may rotate equal-singular-value and null-space components.
- Differentiating the SVD: When singular values are distinct and nonzero, the singular values and singular vectors vary differentiably along a sufficiently small differentiable matrix curve.
- Differentiating the QR Decomposition: Differentiating a QR factorization uses Y=QR, orthogonality of Q, and the triangular structure of R to solve for the factor derivatives.The algorithm first computes X=Q^T dot Q as a skew-symmetric matrix, then obtains dot R and dot Q.
- Differentiating the QR Decomposition: Algorithm B.3 takes T, dot T, and a compact QR factorization as input and returns dot Q and dot R through triangular projection and matrix products.
- Matrix Functions: The principal matrix logarithm is well-defined for matrices with no eigenvalues on the negative real axis.
C Computational Complexity
The handbook reports FLOP counts for commonly used formulas, emphasizing Stiefel representatives in the regime n ≫ p.
- FLOP Counts: The reported FLOP counts depend on the specific implementation of the SVD and other operations.
- FLOP Counting Convention: Scalar evaluations such as sin(·), cos(·), and √· are counted as one flop for simplicity.
- FLOP Counts: Table C.1 counts floating point operations for commonly used formulas implemented with Stiefel representatives under the assumption n ≫ p.
Funding and/or Conflicts of Interest/Competing Interests
The paper acknowledges external funding for one author and reports no conflicts of interest.
- Funding: The third author was supported by FNRS and FWO Vlaanderen through EOS Project no 30468160.
- Conflicts of Interest: The authors declare that they have no conflict of interest.