A Grassmann Manifold Handbook: Basic Geometry and Computational Aspects

TL;DR

This work delivers a comprehensive, matrix-centric handbook for the Grassmann manifold $Gr(n,p)$, unifying projector, ONB, and quotient viewpoints to enable practical, scalable computations. It introduces a novel Grassmann logarithm algorithm with strong handling of cut loci and conjugate points, and provides explicit formulas for the Riemannian exponential, its derivative, and parallel transport in both projector and Stiefel representations. The authors also derive curvature quantities, symmetry properties, local parameterizations, and Jacobi-field machinery, all with $O(np^2)$-oriented formulas suitable for large-scale applications such as subspace tracking, dynamic low-rank modeling, and model reduction. Collectively, the paper equips practitioners with rigorous, implementable tools for optimization, interpolation, and data processing on the Grassmannian."

Abstract

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.

Figures (4)

• Figure 2.1: Conceptual visualization of the quotient structure of the Grassmann manifold. The double brackets $[[\cdot]]$ denote an equivalence class with respect to $\pi^\mathrm{OG}=\pi^\mathrm{SG} \circ \pi^\mathrm{OS}$, while the single brackets $[\cdot]$ denote an equivalence class with respect $\pi^\mathrm{OS}$ or $\pi^\mathrm{SG}$, depending on the element inside the brackets. The tangent vectors along an equivalence class for a projection are vertical with respect to that projection, while the directions orthogonal to the vertical space are horizontal. Correspondingly, the horizontal lift of a tangent vector $\Delta \in T_P\mathrm{Gr}(n,p)$ to $T_U\mathrm{St}(n,p)$ or $T_Q\mathrm{O}(n)$ is orthogonal to all vertical tangent vectors at that point. With respect to the projection $\pi^\mathrm{SG}$ from the Stiefel to the Grassmann manifold, the green tangent vector $\Delta^\mathsf{hor}_U$ is horizontal and the magenta tangent vector (along the equivalence class) is vertical. On the other hand, the magenta tangent vectors in $\mathrm{O}(n)$ (pointing to the left) are horizontal with respect to $\pi^\mathrm{OS}$ but vertical with respect to $\pi^\mathrm{OG}$.
• Figure 5.1: The manifold of one-dimensional subspaces of $\mathbb{R}^3$, i.e., $\mathrm{Gr}(3,1)$, can be seen as the upper half sphere with half of the equator removed. For points in the cut locus of a point $P \in \mathrm{Gr}(3,1)$ (like $F_3$ in the figure), there is no unique velocity vector in $T_P\mathrm{Gr}(3,1)$ that sends a geodesic from $P$ to the point in question, but instead a set of two starting velocities ($\Delta_{3,+1}$ and $\Delta_{3,-1}$) that can be calculated according to Theorem\ref{['thm:multiple_shortest_geodesics']}. Since the points actually mark one dimensional subspaces through the origin, $F_3$ is identical to its antipode on the equator.
• Figure 5.2: The error of the new log algorithm (blue stars) versus standard log algorithm with horizontal projection (red crosses) by subspace distance over $\tau$. For comparison, the error of the standard log algorithm without projection onto the horizontal space is also displayed (yellow plus). The cut locus is approached as $\tau$ goes to zero. It can be observed that the new log algorithm still produces reliable results close to the cut locus.
• Figure 7.1: The Jacobi field $J$ points from the geodesic $\gamma$ towards close-by geodesics (dotted) and vanishes at $P$. Note that $J(t) \in T_{\gamma(t)}\mathrm{Gr}(3,1)$ is a tangent vector and not actually the offset vector between points on the respective geodesics in $\mathbb{R}^3$. Nevertheless, $J$ is the variation field of a variation of $\gamma$ through geodesics, c.f. Lee2018riemannian.

Theorems & Definitions (35)

• Proposition 1: Tangent vector characterization
• proof
• Remark
• Proposition 2: Riemannian Connection
• proof
• Proposition 3: Grassmann Exponential: Projector Perspective
• proof
• Proposition 4: Grassmann Exponential: ONB Perspective
• proof
• Proposition 5: Derivative of the Grassmann Exponential