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.
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22 March 2024
A Correction to this paper has been published: https://doi.org/10.1007/s10444-024-10115-w
Notes
In these formulae, \([X,Y] = X(Y)-Y(X)\) is the Lie bracket of two vector fields.
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Acknowledgements
This work was initiated when the first author was at UCLouvain for a research visit, hosted by the third author.
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Open access funding provided by University of Southern Denmark. The third author was supported by the Fonds de la Recherche Scientifique – FNRS and the Fonds Wetenschappelijk Onderzoek – Vlaanderen under EOS Project no 30468160.
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Appendices
Basics from Riemannian geometry
For the reader’s convenience, we recap some fundamentals from Riemannian geometry. Concise introductions can be found in [33, Appendices C.3, C.4, C.5], [23] and [3]. For an in-depth treatment, see for example [20, 36, 41].
An n -dimensional differentiable manifold \(\mathcal {M}\) is a topological space \(\mathcal {M}\) such that for every point \(p\in \mathcal {M}\), there exists a so-called coordinate chart \(x:\mathcal {M}\supset \mathcal {D}_p \rightarrow \mathbb {R}^n\) that bijectively maps an open neighborhood \(\mathcal {D}_p\subset \mathcal {M}\) of a location p to an open neighborhood \(D_{x(p)}\subset \mathbb {R}^n\) around \(x(p)\in \mathbb {R}^n\) with the additional property that the coordinate change
of two such charts \(x,\tilde{x}\) is a diffeomorphism, where their domains of definition overlap, see [23, Fig. 18.2, p. 496]. This enables to transfer the most essential tools from calculus to manifolds. An n-dimensional submanifold of \(\mathbb {R}^{n+d}\) is a subset \(\mathcal {M}\subset \mathbb {R}^{n+d}\) that can be locally smoothly straightened, i.e., satisfies the local n-slice condition [40, Thm. 5.8].
Theorem 5.4
([23, Prop. 18.7, p. 500]) Let \(h: \mathbb {R}^{n+d}\supset \Omega \rightarrow \mathbb {R}^{d}\) be differentiable and \(c_0\in \mathbb {R}^d\) be defined such that the differential \(Dh_p\in \mathbb {R}^{d\times (n+d)}\) has maximum possible rank d at every point \(p\in \Omega \) with \(h(p) = c_0\). Then, the preimage
is an n-dimensional submanifold of \(\mathbb {R}^{n+d}\).
This theorem establishes the Stiefel manifold \(\textrm{St}(n,p) = \left\{ U\in \mathbb {R}^{n\times p}\ \big | \ U^TU=I\right\} \) as an embedded submanifold of \(\mathbb {R}^{n\times p}\), since \(\textrm{St}(n,p) = F^{-1}(I)\) for \(F:U\mapsto U^TU\).
Tangent spaces
The tangent space of a submanifold \(\mathcal {M}\) at a point \(p\in \mathcal {M}\), in symbols \(T_p\mathcal {M}\), is the space of velocity vectors of differentiable curves \(c:t \mapsto c(t)\) passing through p, i.e.,
The tangent space is a vector space of the same dimension n as the manifold \(\mathcal {M}\).
Geodesics and the Riemannian distance function
Riemannian metrics measure the lengths and angles between tangent vectors. Eventually, this allows to measure the lengths of curves on a manifold and the Riemannian distance between two manifold locations.
A Riemannian metric on \(\mathcal {M}\) is a family \((g_p(\cdot ,\cdot ))_{p\in \mathcal {M}}\) of inner products \(g_p(\cdot ,\cdot ): T_p\mathcal {M}\times T_p\mathcal {M}\rightarrow \mathbb {R}\) that is smooth in variations of the base point p, or more precisely, a smooth covariant 2-tensor field, c.f. [41, Chapter 2]. The length of a tangent vector \(v\in T_p\mathcal {M}\) is \(\Vert v\Vert _p := \sqrt{g_p(v,v)}\). The length of a curve \(c:[a,b] \rightarrow \mathcal {M}\) is defined as
A curve is said to be parameterized by the arc length, if \(L(c|_{[a,t]}) = t-a\) for all \(t\in [a,b]\). Obviously, unit-speed curves with \(\Vert \dot{c}(t)\Vert _{c(t)}\equiv 1\) are parameterized by the arc length. Constant-speed curves with \(\Vert \dot{c}(t)\Vert _{c(t)}\equiv \nu _0\) are parameterized proportional to the arc length. The Riemannian distance between two points \(p,q\in \mathcal {M}\) with respect to a given metric is
where, by convention, \(\inf \{\emptyset \} =\infty \). A shortest path between \(p,q\in \mathcal {M}\) is a curve c that connects p and q such that \(L(c) = {{\,\textrm{dist}\,}}_{\mathcal {M}}(p,q)\). Candidates for shortest curves between points are called geodesics and are characterized by a differential equation: A differentiable curve \(c:[a,b]\rightarrow \mathcal {M}\) is a geodesic (w.r.t. to a given Riemannian metric), if the covariant derivative of its velocity vector field vanishes, i.e.,
Intuitively, the covariant derivative can be thought of as the standard derivative (if it exists) followed by a point-wise projection onto the tangent space. In general, a covariant derivative, also known as a linear connection, is a bilinear mapping \((X,Y) \mapsto \nabla _XY\) that maps two vector fields X, Y to a third vector field \(\nabla _XY\) in such a way that it can be interpreted as the directional derivative of Y in the direction of X, [41, §4, §5]. Of importance is the Riemannian connection or Levi-Civita connection that is compatible with a Riemannian metric [3, Thm 5.3.1], [41, Thm 5.10]. It is determined uniquely by the Koszul formula
and is used to define the Riemannian curvature tensorFootnote 1
A Riemannian manifold is flat if and only if it is locally isometric to the Euclidean space, which holds if and only if the Riemannian curvature tensor vanishes identically [41, Thm. 7.10].
Lie groups and orbits
A Lie group is a smooth manifold that is also a group with smooth multiplication and inversion. A matrix Lie group G is a subgroup of the general linear group \(GL(n,\mathbb {C})\) that is closed in \(GL(n,\mathbb {C})\) (but not necessarily in the ambient space \(\mathbb {C}^{n\times n}\)). Basic examples include \(GL(n,\mathbb {R})\) and the orthogonal group \(\textrm{O}(n)\). Any matrix Lie group G is automatically an embedded submanifold of \(\mathbb {C}^{n\times n}\) [29, Corollary 3.45]. The tangent space \(T_IG\) of G at the identity \(I\in G\) has a special role. When endowed with the bracket operator or matrix commutator \([V,W] = VW-WV\) for \(V,W \in T_IG\), the tangent space becomes an algebra, called the Lie algebra associated with the Lie group G, see [29, §3]. As such, it is denoted by \(\mathfrak {g} = T_IG\). For any \(A\in G\), the function “left-multiplication with A” is a diffeomorphism \(L_A:G\rightarrow G,\ L_A(B) = AB\); its differential at a point \(B\in G\) is the isomorphism \(\textrm{d}(L_A)_B:T_BG\rightarrow T_{L_A(B)}G,\ \textrm{d}(L_A)_B(V) = AV\). Using this observation at \(B=I\) shows that the tangent space at an arbitrary location \(A\in G\) is given by the translates (by left-multiplication) of the tangent space at the identity [26, §5.6, p. 160],
A smooth left action of a Lie group G on a manifold M is a smooth map \(\phi :G \times M \rightarrow M\) fulfilling \(\phi (g_1,\phi (g_2,p))=\phi (g_1g_2,p)\) and \(\phi (e,p)=p\) for all \(g_1,g_2 \in G\) and all \(p \in M\), where \(e \in G\) denotes the identity element. One often writes \(\phi (g,p) = g \cdot p\). For each \(p \in M\), the orbit of p is defined as
and the stabilizer of p is defined as
For a detailed introduction see for example [40, Chapters 7 & 21]. We need the following well known result, see for example [33, Section 2.1], where the quotient manifold \(G/G_p\) refers to the set \(\{g G_p \mid g \in G\}\) endowed with the unique manifold structure that turns the quotient map \(g\mapsto g G_p\) into a submersion.
Proposition 2.1
Let G be a compact Lie group acting smoothly on a manifold M. Then for any \(p \in M\), the orbit \(G \cdot p\) is an embedded submanifold of M that is diffeomorphic to the quotient manifold \(G/G_p\).
Proof
The continuous action of a compact Lie group is always proper, [40, Corollary 21.6]. Therefore [5, Proposition 3.41] shows the claim. \(\square \)
B Matrix analysis necessities
Throughout, we consider the matrix space \(\mathbb {R}^{m\times n}\) as a Euclidean vector space with the standard metric
Unless noted otherwise, the singular value decomposition (SVD) of a matrix \(X\in \mathbb {R}^{m\times n}\) is understood to be the compact SVD
The SVD is not unique.
Proposition 3.1
(Ambiguity of the Singular Value Decomposition)[35, Theorem 3.1.1’] Let \(X \in \mathbb {R}^{m \times n}\) have a (full) SVD \(X=U \Sigma V^T\) with singular values in descending order and \({{\,\textrm{rank}\,}}(X)=r\). Let \(\sigma _1>\dots>\sigma _k>0\) be the distinct nonzero singular values with respective multiplicity \(\mu _1,\dots ,\mu _k\). Then \(X=\tilde{U}\Sigma \tilde{V}^T\) is another SVD if and only if \(\tilde{U}=U{{\,\textrm{diag}\,}}(D_1,\dots ,D_k,W_1)\) and \(\tilde{V}=V{{\,\textrm{diag}\,}}(D_1,\dots ,D_k,W_2)\), with \(D_i \in \textrm{O}(\mu _i)\), \(W_1 \in \textrm{O}(m-r)\), and \(W_2 \in \textrm{O}(n-r)\) arbitrary.
Differentiating the singular value decomposition
Let \(p\le n\in \mathbb {N}\) and suppose that \(t \mapsto Y(t)\in \mathbb {R}^{n\times p}\) is a differentiable matrix curve around \(t_0\in \mathbb {R}\). If the singular values of \(Y(t_0)\) are mutually distinct and non-zero, then the singular values and both the left and the right singular vectors depend differentiable on \(t \in [t_0 -\delta t, t_0+\delta t]\) for \(\delta t\) small enough.
Let \( t \mapsto Y(t) = U( t)\Sigma ( t) V( t)^T \in \mathbb {R}^{n\times p}\), where \(U(t)\in \textrm{St}(n,p)\), \(V( t)\in O(p)\) and \(\Sigma ( t)\in \mathbb {R}^{p\times p}\) diagonal and positive definite. Let \(u_j\) and \(v_j\), \(j=1,\ldots ,p\) denote the columns of \(U( t_0)\) and \(V( t_0)\), respectively. For brevity, write \(Y = Y(t_0), \dot{Y} = \frac{\textrm{d}}{\textrm{d}t}\big \vert _{t=t_0}Y(t)\), likewise for the other matrices that feature in the SVD. The derivatives of the matrix factors of the SVD can be calculated with Algorithm 2. A proof can for example be found in [19, 30].
Differentiating the QR-decomposition
Let \(t\mapsto Y(t)\in \mathbb {R}^{n\times r}\) be a differentiable matrix function with Taylor expansion \(Y(t_0+ h) = Y(t_0) + h \dot{Y}(t_0) + \mathcal {O}(h^2)\). Following [59, Proposition 2.2], the QR-decomposition is characterized via the following set of matrix equations.
In the latter, \(P_L = \small \begin{pmatrix} 0 &{} \cdots &{}\cdots &{} 0\\ 1 &{} \ddots &{} &{} \vdots \\ \vdots &{} \ddots &{}\ddots &{}\vdots \\ 1 &{} \cdots &{}1 &{} 0 \end{pmatrix}\) and ‘\(\odot \)’ is the element-wise matrix product so that \(P_L\odot R\) selects the strictly lower triangle of the square matrix R. For brevity, we write \(Y= Y(t_0),\ \dot{Y} = \frac{\textrm{d}}{\textrm{d}t}\big \vert _{t=t_0}Y(t)\), likewise for Q(t), R(t). By the product rule
According to [59, Proposition 2.2], the derivatives \(\dot{Q}, \dot{R}\) can be obtained from Algorithm 3. The trick is to compute \(X = Q^T\dot{Q}\) first and then use this to compute \(\dot{Q} = QQ^T\dot{Q} + (I_n-QQ^T)\dot{Q}\) by exploiting that \(Q^T\dot{Q}\) is skew-symmetric and that \(\dot{R}R^{-1}\) is upper triangular.
Matrix exponential and the principal matrix logarithm
The matrix exponential and the principal matrix logarithm are defined by
The latter is well-defined for matrices that have no eigenvalues on \(\mathbb {R}^-\).
C Computational complexity
For the benefit of the reader, we include Table 1 of the floating point operation (FLOP) counts of some of the most commonly used formulas in this handbook. Note that the FLOP count of the SVD and other operations depends on the specific implementation. Furthermore, we counted \(\sin (\cdot )\), \(\cos (\cdot )\), \(\sqrt{\cdot }\) etc. for scalars as one flop for simplicity.
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Bendokat, T., Zimmermann, R. & Absil, PA. A Grassmann manifold handbook: basic geometry and computational aspects. Adv Comput Math 50, 6 (2024). https://doi.org/10.1007/s10444-023-10090-8
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DOI: https://doi.org/10.1007/s10444-023-10090-8
Keywords
- Grassmann manifold
- Stiefel manifold
- Orthogonal group
- Riemannian exponential
- Geodesic
- Riemannian logarithm
- Cut locus
- Conjugate locus
- Curvature
- Parallel transport
- Quotient manifold
- Horizontal lift
- Subspace
- Singular value decomposition