On the stability of POD basis interpolation on Grassmann manifolds for parametric model order reduction

Abstract

Proper Orthogonal Decomposition (POD) basis interpolation on Grassmann manifolds has been successfully applied to problems of parametric model order reduction (pMOR). In this work we address the necessary stability conditions for the interpolation, all defined from strong mathematical background. A first condition concerns the domain of definition of the logarithm map. Second, we show how the stability of interpolation can be lost if certain geometrical requirements are not satisfied by making a concrete elucidation of the local character of linearization. To this effect, we draw special attention to the Grassmannian exponential map and the optimal injectivity condition of this map, related to the cut–locus of Grassmann manifolds. From this, an explicit stability condition is established and can be directly used to determine the loss of injectivity in practical pMOR applications. A third stability condition is formulated when increasing the number p of POD modes, deduced from the principal angles of subspaces of different dimensions p. Definition of this condition leads to an understanding of the non-monotonic oscillatory behavior of the Reduced Order Model (ROM) error-norm with respect to the number of POD modes, and on the contrary, the well-behaved monotonic decrease of the error-norm in the two numerical examples presented herein. We have chosen to perform pMOR in hyperelastic structures using a non-intrusive approach for inserting the interpolated spatial POD ROM basis in a commercial FEM code. The accuracy is assessed by a posteriori error norms defined using the ROM FEM solution and its high-fidelity counterpart simulation. Numerical studies successfully ascertained and highlighted the implication of stability conditions which are general and can be applied to a variety of other linear or nonlinear problems involving parametrized ROMs generation based on POD basis interpolation on Grassmann manifolds.

Appendix: Riemannian geometry of Grassmann manifolds

The purpose of this section is to recall the main results about Grassmann manifolds, as well as new ones about the cut–locus and injectivity condition for the exponential map. As far as we know, the normal coordinates are classically defined using the exponential map restricted on an open disk deduced from the injectivity radius [3, 25]. In fact, it will be possible to go beyond such an injectivity radius, using an open set deduced from the cut–locus of the Grassmann manifold, all this being detailed in Sect. A.4.

Note that some results recalled here are classical, either given in their matrix forms [3, 18, 19, 21, 22, 42], or given in a more abstract one [25, 43], but it was necessary to write them back for our proofs to be clearly established. All details about general differential Riemannian geometry can be found in [23, 32, 44].

From now on, let us consider two integers p, n such that $p\le n$ and take ${\mathcal {G}}(p,n)$ to be the Grassmann manifold of p dimensional subspaces of ${\mathbb {R}}^n$. A first way to obtain a point ${\mathbf {m}}\in {\mathcal {G}}(p,n)$ is to consider a basis ${\mathbf {y}}_1,\dots ,{\mathbf {y}}_p$ of the associated subspace

$$\begin{aligned} {\mathbf {m}}=\text {Vect}({\mathbf {y}}_1,\dots ,{\mathbf {y}}_p). \end{aligned}$$

Without loss of generality, we can assume the case of orthonormal basis, so ${\mathbf {m}}$ can be represented by a matrix

$$\begin{aligned} {\mathbf {Y}}:=[{\mathbf {y}}_1,\dotsc ,{\mathbf {y}}_p]\in \mathrm {Mat}_{n,p}({\mathbb {R}}),\quad {\mathbf {Y}}^T{\mathbf {Y}}={\mathbf {I}}_p. \end{aligned}$$

Such matrix ${\mathbf {Y}}$ is not unique, as any matrix in the set

$$\begin{aligned} \left\{ {\mathbf {Y}}{\mathbf {P}},\quad {\mathbf {P}}\in \mathrm {O}(p)\right\} ,\quad \mathrm {O}(p):=\left\{ {\mathbf {P}}\in \mathrm {Mat}_{p,p}({\mathbb {R}}),\quad {\mathbf {P}}^T{\mathbf {P}}={\mathbf {I}}_p\right\} , \end{aligned}$$

can represent the same point ${\mathbf {m}}$.

From this, the Grassmann manifold ${\mathcal {G}}(p,n)$ is obtained as a quotient space [44, Chapter 21] of the (compact) space of p ordered orthonormal vectors of ${\mathbb {R}}^n$. More specifically [45, Appendix C.2], first define the compact Stiefel manifold ${\mathcal {S}}t^{c}(p,n)$ to be the set of p orthonormal vectors $\{{\mathbf {y}}_1,\dotsc ,{\mathbf {y}}_p\}$ of ${\mathbb {R}}^n$. Taking any basis of ${\mathbb {R}}^n$, such a set can be represented by a rank p matrix

$$\begin{aligned} {\mathbf {Y}}:=[{\mathbf {y}}_1,\dotsc ,{\mathbf {y}}_p]\in \mathrm {Mat}_{n,p}({\mathbb {R}}),\quad {\mathbf {Y}}^T{\mathbf {Y}}={\mathbf {I}}_p. \end{aligned}$$

This led to define a fiber bundle [46, 47], which is also a submersion [44]:

$$\begin{aligned}&\pi \, : \, {\mathbf {Y}}\in {\mathcal {S}}t^{c}(p,n)\mapsto \pi ({\mathbf {Y}})={\mathbf {m}}\nonumber \\&\quad :=\{ {\mathbf {Y}}{\mathbf {P}},\quad {\mathbf {P}}\in \mathrm {O}(p)\} \in {\mathcal {G}}(p,n) \end{aligned}$$

(16)

Informally speaking, it means that any point ${\mathbf {m}}$ of the Grassmann manifold ${\mathcal {G}}(p,n)$ can be represented by any point ${\mathbf {Y}}$ of the fiber $\pi ^{-1}({\mathbf {m}})$ (Fig. 2).

1.1 The Grassmann manifold and its Riemannian metric

From the submersion $\pi$ given by (16), the Grassmann manifold ${\mathcal {G}}(p,n)$ can inherit the geometry of the Stiefel manifold ${\mathcal {S}}t^{c}(p,n)$ and its Riemannian structure [23].

First, the Stiefel manifold ${\mathcal {S}}t^{c}(p,n)\subset \mathrm {Mat}_{n,p}({\mathbb {R}})$, is naturally endowed with an inner product given by

$$\begin{aligned} \langle {\mathbf {Z}}_1,\mathbf {Z_2}\rangle :={{\,\mathrm{tr}\,}}({\mathbf {Z}}_1^{T}\mathbf {Z_2}),\quad {\mathbf {Z}}_1,{\mathbf {Z}}_2\in \mathrm {Mat}_{n,p}({\mathbb {R}}). \end{aligned}$$

Now, we need to attach, to each ${\mathbf {m}}\in {\mathcal {G}}(p,n)$ a tangent space $T_{{\mathbf {m}}}{\mathcal {G}}(p,n)$, which is a vector space isomorphic to ${\mathbb {R}}^{p\times (n-p)}$, equipped with a scalar product (depending smoothly on ${\mathbf {m}}$), so that ${\mathcal {G}}(p,n)$ becomes a Riemannian manifold.

In fact, there is no canonical way to get a representation of a velocity vector $v\in T_{{\mathbf {m}}}{\mathcal {G}}(p,n)$, as it depends on the choice of a matrix ${\mathbf {Y}}\in {\mathcal {S}}t^{c}(p,n)$ defining ${\mathbf {m}}$ (see Fig. 2): for any ${\mathbf {Y}}\in \pi ^{-1}({\mathbf {m}})$, we define indeed its associated horizontal space by:

$$\begin{aligned} \text {Hor}_{{\mathbf {Y}}}:=\{ {\mathbf {Z}}\in \mathrm {Mat}_{n,p}({\mathbb {R}}),\quad {\mathbf {Z}}^{T}{\mathbf {Y}}={\mathbf {0}} \}. \end{aligned}$$

(17)

Finally:

• 1. The tangent space $T_{{\mathbf {m}}}{\mathcal {G}}(p,n)$ is isomorphic to any $\text {Hor}_{{\mathbf {Y}}}$ with ${\mathbf {Y}}$ such that $\pi ({\mathbf {Y}})={\mathbf {m}}$. An isomorphism is given by

  $$\begin{aligned} \text {d}\pi _{{\mathbf {Y}}\mid \text {Hor}_{{\mathbf {Y}}}}\, : \, \text {Hor}_{{\mathbf {Y}}}\longmapsto T_{{\mathbf {m}}}{\mathcal {G}}(p,n). \end{aligned}$$

• 2. For any $v\in T_{{\mathbf {m}}}{\mathcal {G}}(p,n)$, the unique ${\mathbf {Z}}\in \text {Hor}_{{\mathbf {Y}}}$ such that

  $$\begin{aligned} \text {d}\pi _{{\mathbf {Y}}}\cdot {\mathbf {Z}}=v \end{aligned}$$

  (18)

  is called a horizontal lift of v.

• 3. For any ${\mathbf {P}}\in \mathrm {O}(p)$, then ${\mathbf {Z}}{\mathbf {P}}$ is another horizontal lift of v (but belonging to the vector space $\text {Hor}_{{\mathbf {Y}}{\mathbf {P}}}$) and

  $$\begin{aligned} \text {d}\pi _{{\mathbf {Y}}{\mathbf {P}}}\cdot ({\mathbf {Z}}{\mathbf {P}})=v. \end{aligned}$$

The Riemannian metric on the Grassmannian ${\mathcal {G}}(p,n)$ is then defined by

$$\begin{aligned} \langle v_1,v_2\rangle _{{\mathbf {m}}}:=\langle {\mathbf {Z}}_1,{\mathbf {Z}}_2\rangle _{{\mathbf {Y}}},\quad \end{aligned}$$

with $\pi ({\mathbf {Y}})={\mathbf {m}}$ and ${\mathbf {Z}}_{1}$ (resp. ${\mathbf {Z}}_{2}$) a horizontal lift of $v_1$ (resp. $v_2$) in $\text {Hor}_{{\mathbf {Y}}}$.

For the proofs of the following subsections, an interesting geometric approach, due to Zhou [48], is given by:

Lemma A.1

Let ${\mathbf {m}}\in {\mathcal {G}}(p,n)$ and $v\in T_{{\mathbf {m}}}{\mathcal {G}}(p,n)$, with $2p\le n$. Then, there exists an orthonormal basis ${\mathbf {y}}_1,\dots ,{\mathbf {y}}_n$ of ${\mathbb {R}}^n$ such that

$$\begin{aligned} {\mathbf {Y}}= & {} [{\mathbf {y}}_1,\dots ,{\mathbf {y}}_p]\in \pi ^{-1}({\mathbf {m}}),\quad \\ {\mathbf {Z}}= & {} [\theta _1{\mathbf {y}}_{p+1},\cdots ,\theta _{p}{\mathbf {y}}_{2p}]\in \text {Hor }_{{\mathbf {Y}}},\\ \theta _{1}\ge & {} \cdots \ge \theta _{p}\ge 0. \end{aligned}$$

Proof

Let us consider any ${\mathbf {Y}}\in \pi ^{-1}({\mathbf {m}})$ and a horizontal lift ${\mathbf {Z}}$ of v such that ${\mathbf {Z}}^{T}{\mathbf {Y}}={\mathbf {0}}$. We define a thin singular value decomposition of ${\mathbf {Z}}$, so we can find orthonormal vectors ${\mathbf {u}}_1,\dots ,{\mathbf {u}}_p$ in ${\mathbb {R}}^n$ and ${\mathbf {v}}_1,\dots ,{\mathbf {v}}_p$ in ${\mathbb {R}}^p$ such that

$$\begin{aligned} {\mathbf {Z}}=\sum \theta _i {\mathbf {u}}_i {\mathbf {v}}_i^{T},\quad \theta _{1}\ge \cdots \ge \theta _p\ge 0. \end{aligned}$$

From the condition ${\mathbf {Z}}^T{\mathbf {Y}}={\mathbf {0}}$ we thus deduce that ${\mathbf {y}}_1,\dots ,{\mathbf {y}}_p,{\mathbf {u}}_1,\dots ,{\mathbf {u}}_p$ is a family of orthonormal vectors. Taking now the matrix ${\mathbf {P}}:=[{\mathbf {v}}_1,\dots ,{\mathbf {v}}_p]\in \mathrm {O}(p)$ and ${\mathbf {Y}}':={\mathbf {Y}}{\mathbf {P}}\in \pi ^{-1}({\mathbf {m}})$, we obtain

$$\begin{aligned} {\mathbf {Z}}':=[\theta _1{\mathbf {y}}_{p+1},\cdots ,\theta _{p}{\mathbf {y}}_{2p}]\in \text {Hor}_{{\mathbf {Y}}{\mathbf {P}}},\quad {\mathbf {y}}_{p+i}:={\mathbf {u}}_i, \end{aligned}$$

so we can conclude. $\square$

Remark A.2

In the case when $2p>n$, that is $p>n-p$, then we can only write a horizontal lift as

$$\begin{aligned} {\mathbf {Z}}\!=\![\theta _{1}{\mathbf {y}}_{p+1},\cdots ,\theta _{n-p}{\mathbf {y}}_{n},\underbrace{{\mathbf {0}},\dots ,{\mathbf {0}}}_{2p-n \text { times }}],\, \theta _{1}\ge \cdots \ge \theta _{n-p}\!\ge \! 0. \end{aligned}$$

1.2 Geodesics and distance on Grassmann manifolds

The Grassmann manifold ${\mathcal {G}}(p,n)$ being equipped with a Riemannian metric, it is possible to define the length of any curve $c:[0;1]\rightarrow {\mathcal {G}}(p,n)$:

$$\begin{aligned} L(c)=\int _{0}^{1} \langle {\dot{c}}(t),{\dot{c}}(t)\rangle _{c(t)}\text {d}t \end{aligned}$$

(19)

and so the associated Riemannian distance

$$\begin{aligned} d_{r}({\mathbf {m}},{\mathbf {m}}'):=\text {inf}\{L(c),\quad c(0)={\mathbf {m}},c(1)={\mathbf {m}}'\}. \end{aligned}$$

(20)

To obtain an explicit computation of such a distance, one can use the geodesics obtained from the Riemannian metric and its associated Levi-Civita connection [23, 44] (see also [46, III.6]). First recall that for Grassmann manifold, geodesics are obtained explicitly [22, 43]:

Theorem A.3

Let ${\mathbf {m}}\in {\mathcal {G}}(p,n)$ and $v\in T_{{\mathbf {m}}}{\mathcal {G}}(p,n)$ with horizontal lift given by ${\mathbf {Z}}\in \text {Hor}_{{\mathbf {Y}}}$, where $\pi ({\mathbf {Y}})={\mathbf {m}}$ and ${\mathbf {Y}}^{T}{\mathbf {Y}}={\mathbf {I}}_p$. Let ${\mathbf {Z}}={\mathbf {U}}\varvec{\varTheta }{\mathbf {V}}^{T}$ be a thin singular value decomposition of ${\mathbf {Z}}$. Then

$$\begin{aligned} \alpha _{v} : \, t\in {\mathbb {R}}\mapsto \pi \left( {\mathbf {Y}}{\mathbf {V}}\cos (t\varvec{\varTheta })+{\mathbf {U}}\sin (t\varvec{\varTheta })\right) \in {\mathcal {G}}(p,n) \end{aligned}$$

(21)

is the unique maximal geodesic such that $\alpha _{v}(0)={\mathbf {m}}$ and $\dot{\alpha _{v}}(0)=v$, maximality meaning here that such curve is defined on all ${\mathbb {R}}$.

Remark A.4

There is another approach proposed in [48] which produces a more intrinsic formula for the geodesics. Indeed, let us consider $2p\le n$ and take back the result from Lemma A.1. Then one horizontal lift of v can writes

$$\begin{aligned} {\mathbf {Z}}=[\theta _1{\mathbf {y}}_{p+1},\cdots ,\theta _{p}{\mathbf {y}}_{2p}],\quad \theta _{1}\ge \cdots \ge \theta _{p}\ge 0. \end{aligned}$$

where ${\mathbf {Y}}=[{\mathbf {y}}_1,\dots ,{\mathbf {y}}_p]\in \pi ^{-1}({\mathbf {m}})$ and ${\mathbf {y}}_1,\dots ,{\mathbf {y}}_{2p}$ is an orthonormal family. The unique geodesic obtained from velocity vector v is then defined by $\pi ({\mathbf {Y}}(t))$, with

$$\begin{aligned} {\mathbf {Y}}(t)&=\left[ \cos (\theta _{1}t){\mathbf {y}}_1+\sin (\theta _{1}t){\mathbf {y}}_{p+1},\dots ,\cos (\theta _{p}t){\mathbf {y}}_p\right. \\&\quad \left. +\sin (\theta _{p}t){\mathbf {y}}_{2p}\right] . \end{aligned}$$

We observe that the norm of the velocity vector is given by

$$\begin{aligned} \Vert v\Vert =\sqrt{\sum \theta _{i}^2}. \end{aligned}$$

In fact, all matrices given by (21) are lying in ${\mathcal {S}}t^c(p,n)$:

Lemma A.5

Let ${\mathbf {m}}\in {\mathcal {G}}(p,n)$ and $v\in T_{{\mathbf {m}}}{\mathcal {G}}(p,n)$. Take ${\mathbf {Y}}\in \pi ^{-1}({\mathbf {m}})$ and ${\mathbf {Z}}\in \text {Hor}_{{\mathbf {Y}}}$ like in statement of Theorem A.3. Then for any $t\in {\mathbb {R}}$ we have

$$\begin{aligned} {\mathbf {Y}}(t):= & {} {\mathbf {Y}}{\mathbf {V}}\cos (t\varvec{\varTheta })+{\mathbf {U}}\sin (t\varvec{\varTheta })\\\in & {} {\mathcal {S}}t^c(p,n) \text {, meaning that } {\mathbf {Y}}(t)^{T}{\mathbf {Y}}(t)={\mathbf {I}}_p. \end{aligned}$$

Proof

By direct computation we have:

$$\begin{aligned} {\mathbf {Y}}(t)^{T}{\mathbf {Y}}(t)&=\cos ^2(t\varvec{\varTheta })+\sin ^{2}(t\varvec{\varTheta })+{\mathbf {X}}+{\mathbf {X}}^{T},\\ {\mathbf {X}}&:=\sin (t\varvec{\varTheta }){\mathbf {U}}^{T}{\mathbf {Y}}{\mathbf {V}}\cos (t\varvec{\varTheta }) \\&={\mathbf {I}}_p+{\mathbf {X}}+{\mathbf {X}}^{T}. \end{aligned}$$

As we have ${\mathbf {Z}}={\mathbf {U}}\varvec{\varTheta }{\mathbf {V}}^{T}$ and ${\mathbf {Z}}^{T}{\mathbf {Y}}=0$ we deduce that

$$\begin{aligned} {\mathbf {V}}\varvec{\varTheta }{\mathbf {U}}^{T}{\mathbf {Y}}=0,\quad {\mathbf {V}}\in \mathrm {O}(p)\implies \varvec{\varTheta }{\mathbf {U}}^{T}{\mathbf {Y}}=0. \end{aligned}$$

and thus $\sin (t\varvec{\varTheta }){\mathbf {U}}^{T}{\mathbf {Y}}=0$ for all t, which concludes the proof. $\square$

As a consequence of Hopf-Rinow Theorem [23, Theorem 2.103], any two points of the Grassmann manifold can be joined by a length minimizing geodesic. An explicit expression of such a geodesic is given also in [25]:

Theorem A.6

Let ${\mathbf {m}},{\mathbf {m}}'\in {\mathcal {G}}(p,n)$ be any two points on the Grassmann manifold ${\mathcal {G}}(p,n)$. Then, for $2p\le n$:

• (1) There exists an orthonormal family ${\mathbf {y}}_1,\dots ,{\mathbf {y}}_n$ of $~{\mathbb {R}}^n$ such that

  $$\begin{aligned} {\mathbf {Y}}'&=[\cos (\theta _1){\mathbf {y}}_1+\sin (\theta _1){\mathbf {y}}_{p+1},\dots ,\cos (\theta _p){\mathbf {y}}_p\\&\quad +\sin (\theta _p){\mathbf {y}}_{2p}]\in \pi ^{-1}({\mathbf {m}}'),\\ {\mathbf {Y}}&=[{\mathbf {y}}_1,\dots ,{\mathbf {y}}_p]\in \pi ^{-1}({\mathbf {m}}), \end{aligned}$$

  with $\theta _i\in \left[ 0,\pi /2\right]$ are the Jordan’s principal angles between ${\mathbf {Y}}$ and ${\mathbf {Y}}'$, meaning that $\theta _i=\arccos (\sigma _{p-i+1})$, where $0\le \sigma _p\le \dots \le \sigma _1$ are the singular values of ${\mathbf {Y}}^{T}{\mathbf {Y}}'$.

• (2) A length minimizing geodesic from ${\mathbf {m}}$ to ${\mathbf {m}}'$ is given by $t\in [0,1] \mapsto \pi ({\mathbf {Y}}(t))$ with

  $$\begin{aligned} {\mathbf {Y}}(t):= & {} [\cos (t\theta _1){\mathbf {y}}_1+\sin (t\theta _1){\mathbf {y}}_{p+1},\dots ,\cos (t\theta _p){\mathbf {y}}_p\\&+\sin (t\theta _p){\mathbf {y}}_{2p}]. \end{aligned}$$

  Furthermore, such length minimizing geodesic is unique if and only if $\theta _{1}<\pi /2$.

In the case $2p>n$, the same result holds using

$$\begin{aligned} {\mathbf {Y}}'= & {} [\cos (\theta _1){\mathbf {y}}_1+\sin (\theta _1){\mathbf {y}}_{p+1},\dots ,\cos (\theta _{n-p}){\mathbf {y}}_{n-p}\\&+\sin (\theta _{n-p}){\mathbf {y}}_{n-p}, {\mathbf {y}}_{n-p+1},\dots ,{\mathbf {y}}_p]\in \pi ^{-1}({\mathbf {m}}). \end{aligned}$$

Proof

Take any ${\mathbf {Y}}\in \pi ^{-1}({\mathbf {m}})$ and ${\mathbf {Y}}'\in \pi ^{-1}({\mathbf {m}}')$. Let now consider a reordered SVD of the square matrix ${\mathbf {Y}}^{T}{\mathbf {Y}}'$:

$$\begin{aligned} {\mathbf {Y}}^{T}{\mathbf {Y}}'={\mathbf {U}}\varvec{\varSigma }{\mathbf {V}}^{T},\quad \varvec{\varSigma }=\begin{pmatrix} \sigma _p &{} \dots &{} 0 \\ \vdots &{} \ddots &{} \vdots \\ 0 &{} \dots &{} \sigma _{1} \end{pmatrix},\quad {\mathbf {U}},{\mathbf {V}}\in \mathrm {O}(p), \end{aligned}$$

with singular values $0\le \sigma _p\le \dots \le \sigma _{1}$. Define

$$\begin{aligned} {\widehat{{\mathbf {Y}}}}:={\mathbf {Y}}{\mathbf {U}}\in \pi ^{-1}({\mathbf {m}}),\quad {\widehat{{\mathbf {Y}}}}':={\mathbf {Y}}'{\mathbf {V}}\in \pi ^{-1}({\mathbf {m}}') \end{aligned}$$

and write

$$\begin{aligned} {\widehat{{\mathbf {Y}}}}= & {} [{\mathbf {y}}_1,\dots ,{\mathbf {y}}_p]\in \mathrm {Mat}_{n,p}({\mathbb {R}}),\quad \\ {\widehat{{\mathbf {Y}}}}'= & {} [{\mathbf {x}}_1,\dots ,{\mathbf {x}}_p]\in \mathrm {Mat}_{n,p}({\mathbb {R}}) \end{aligned}$$

so we can deduce from ${\widehat{{\mathbf {Y}}}}^{T}{\widehat{{\mathbf {Y}}}}'=\varvec{\varSigma }$ the inner products

$$\begin{aligned} \langle {\mathbf {y}}_i,{\mathbf {x}}_j\rangle =\sigma _{p-i+1}\delta _{ij},\quad \sigma _{p-i+1}\in [0,1]. \end{aligned}$$

Using a direct induction on i, we obtain a family of orthonormal vectors ${\mathbf {y}}_{p+1},\dots ,{\mathbf {y}}_{2p}$ such that

$$\begin{aligned} {\mathbf {x}}_i= & {} \cos (\theta _i){\mathbf {y}}_i+\sin (\theta _{i}){\mathbf {y}}_{p+i},\quad \\ \theta _i:= & {} \arccos (\sigma _{p-i+1}),\quad \langle {\mathbf {y}}_i,{\mathbf {y}}_{p+j}\rangle =0 \end{aligned}$$

which conclude the proof of (1).

Now, from Remark A.4, any other geodesic from ${\mathbf {m}}$ to ${\mathbf {m}}'$ reads $t\mapsto \pi ({\mathbf {Y}}(t))$ with

$$\begin{aligned} {\mathbf {Y}}(t)= & {} \left[ \cos (\alpha _{1}t){\mathbf {y}}_1+\sin (\alpha _{1}t){\mathbf {y}}_{p+1},\dots ,\cos (\alpha _{p}t){\mathbf {y}}_p\right. \\&\left. +\sin (\alpha _{p}t){\mathbf {y}}_{2p}\right] ,\quad \pi ({\mathbf {Y}}(1))={\mathbf {m}}' \end{aligned}$$

so that $\cos (\alpha _{i})=\cos (\theta _i)$ and $\alpha _{i}=\theta _i+k_i\pi$, with $k_i\in {\mathbb {Z}}$. We deduce that the length of this geodesic is given by

$$\begin{aligned} \left( \sum _{i=1}^{p} (\theta _i+k_i\pi )^2\right) ^{1/2}. \end{aligned}$$

As $(\theta +k\pi )^2\ge \theta ^2$ for all $k\in {\mathbb {Z}}$ and $\theta \in [0,\pi /2]$, we deduce length minimization for $k_i=0$. Non unicity can only occur if and only if there is non-zero $k_i\in {\mathbb {Z}}$ such that $\theta _{i}+k_i\pi =-\theta _i$, so that

$$\begin{aligned} k_i=\frac{-2\theta _{i}}{\pi }\in {\mathbb {Z}}-\{0\} \end{aligned}$$

which translate into $\theta _i=\theta _{i-1}=\dots =\theta _1=\pi /2$, which conclude the proof. $\square$

As a consequence of Theorem A.6, for any two points ${\mathbf {m}}$ and ${\mathbf {m}}'$ of ${\mathcal {G}}(p,n)$ the Riemannian distance is given by

$$\begin{aligned} d_{r}({\mathbf {m}},{\mathbf {m}}')=\bigg (\sum _{i=1}^{p} \theta ^2_i \bigg )^{1/2} \end{aligned}$$

(22)

with $\theta _{i}$ the Jordan’s principal angles as defined in the statement of the theorem. Finally, the diameter of ${\mathcal {G}}(p,n)$ (the maximum distance between two points) is given by

$$\begin{aligned} \text {diam}=\sqrt{r}\frac{\pi }{2},\quad r=\min (p,n-p). \end{aligned}$$

(23)

1.3 Exponential and logarithm map on Grassmann manifolds

By exploiting geodesics of a Riemannian manifold, it is possible to establish local maps using normal coordinates [23] defined from the exponential map.

In the case of Grassmann manifolds, the exponential map is obtained from the exact formulation of the geodesics (see Theorem A.3).

Definition A.7

(Exponential map) For any point ${\mathbf {m}}\in {\mathcal {G}}(p,n)$, let consider the tangent plane $T_{{\mathbf {m}}}{\mathcal {G}}(p,n)\simeq {\mathbb {R}}^d$, with $d=p(n-p)$ the dimension of ${\mathcal {G}}(p,n)$. Then the exponential map is defined by

$$\begin{aligned}&{{\,\mathrm{Exp}\,}}_{{\mathbf {m}}}\,:\, v\in T_{{\mathbf {m}}}{\mathcal {G}}(p,n) \mapsto \pi \left( {\mathbf {Y}}{\mathbf {V}}\cos \varvec{\varTheta }\right. \nonumber \\&\left. +{\mathbf {U}}\sin \varvec{\varTheta }\right) \in {\mathcal {G}}(p,n) \end{aligned}$$

(24)

where ${\mathbf {Y}}\in \pi ^{-1}({\mathbf {m}})$ and ${\mathbf {Z}}={\mathbf {U}}\varvec{\varTheta }{\mathbf {V}}^{T}$ is a thin SVD of a horizontal lift ${\mathbf {Z}}\in \text {Hor}_{{\mathbf {Y}}}$ of v.

Such a map is only a diffeomorphism locally, meaning that there exists some open set $\mathrm {W}\subset T_{{\mathbf {m}}}{\mathcal {G}}(p,n)$ containing 0 such that $\left( {{\,\mathrm{Exp}\,}}_{{\mathbf {m}}}\right) _{\mid \mathrm {W}}$ is a diffeomorphism, which thus makes it possible to define local coordinates on $\mathrm {W}$. A first way to do so is to consider the injectivity radius and thus the open disk:

$$\begin{aligned} \mathrm {D}_{{\mathbf {m}}}:=\left\{ v\in T_{{\mathbf {m}}}{\mathcal {G}}(p,n),\quad \Vert v\Vert <\pi /2\right\} , \end{aligned}$$

(25)

where $\pi /2$ is the injectivity radius of Grassmann manifolds [25]. We obtain here a local map

$$\begin{aligned} \left( {{\,\mathrm{Exp}\,}}_{{\mathbf {m}}}\right) _{\mid _{\mathrm {D}_{{\mathbf {m}}}}}\, : \, \mathrm {D}_{{\mathbf {m}}} \longrightarrow {{\,\mathrm{Exp}\,}}_{{\mathbf {m}}}\left( \mathrm {D}_{{\mathbf {m}}}\right) . \end{aligned}$$

It turns out that in our case, it is possible to go beyond this injectivity radius. To do so, a logarithm map is directly define at each point of the Grassmann manifold.

First, for any point ${\mathbf {m}}\in {\mathcal {G}}(p,n)$, let us define the open set

$$\begin{aligned} \mathrm {U}_{{\mathbf {m}}}:= & {} \{ {\mathbf {m}}'\in {\mathcal {G}}(p,n),\quad {\mathbf {Y}}^{T}{\mathbf {Y}}' \text { is invertible},\quad \nonumber \\ \pi ({\mathbf {Y}})= & {} {\mathbf {m}},\quad \pi ({\mathbf {Y}}')={\mathbf {m}}\}. \end{aligned}$$

(26)

A more geometric insight of such an open set is given by a lemma directly deduced from Jordan’s principal angles (see Theorem A.6):

Lemma A.8

For any ${\mathbf {m}},{\mathbf {m}}'\in {\mathcal {G}}(p,n)$, take $0\le \theta _p\le \dots \le \theta _1\le \pi /2$ to be their corresponding Jordan’s principal angles. Then ${\mathbf {m}}'\in \mathrm {U}_{{\mathbf {m}}}$ if and only if $\theta _1<\pi /2$.

From now on, let us suppose that $2p\le n$, while the case $2p>n$ is straightforward.

Following Theorem A.6, we can find an orthonormal family ${\mathbf {y}}_1,\dots ,{\mathbf {y}}_n$ of ${\mathbb {R}}^n$ such that

$$\begin{aligned} {\mathbf {Y}}'&=[\cos (\theta _1){\mathbf {y}}_1+\sin (\theta _1){\mathbf {y}}_{p+1},\dots ,\cos (\theta _p){\mathbf {y}}_p\nonumber \\&\quad +\sin (\theta _p){\mathbf {y}}_{2p}]\in \pi ^{-1}({\mathbf {m}}'),\nonumber \\ {\mathbf {Y}}&=[{\mathbf {y}}_1,\dots ,{\mathbf {y}}_p]\in \pi ^{-1}({\mathbf {m}}), \end{aligned}$$

(27)

and then ${\mathbf {Y}}^T{\mathbf {Y}}'=\cos \varvec{\varTheta }$. The classical definition of the logarithm map [3] makes use of a thin SVD of

$$\begin{aligned} {\mathbf {Y}}'\left( {\mathbf {Y}}^{T}{\mathbf {Y}}'\right) ^{-1}-{\mathbf {Y}}=[\tan (\theta _{1}){\mathbf {y}}_{p+1},\dots ,\tan (\theta _p){\mathbf {y}}_{2p}] \end{aligned}$$

(28)

where singular values are well-defined (as a consequence of Lemma A.8). From all this, it is possible to have the following definition, using the $\arctan$ function:

Definition A.9

(Logarithm map in Grassmann manifolds) For any ${\mathbf {m}}\in {\mathcal {G}}(p,n)$, take the open set $\mathrm {U}_{{\mathbf {m}}}$ defined by (26). Then the logarithm map at ${\mathbf {m}}$ is given by

$$\begin{aligned} {{\,\mathrm{Log}\,}}_{{\mathbf {m}}}\,: \, {\mathbf {m}}'\in \mathrm {U}_{{\mathbf {m}}}\mapsto {{\,\mathrm{Log}\,}}_{{\mathbf {m}}}({\mathbf {m}}')\in T_{{\mathbf {m}}}{\mathcal {G}}(p,n) \end{aligned}$$

where an horizontal lift ${\mathbf {Z}}$ of ${{\,\mathrm{Log}\,}}_{{\mathbf {m}}}({\mathbf {m}}')$ is defined using a thin SVD

$$\begin{aligned} {\mathbf {Y}}'\left( {\mathbf {Y}}^{T}{\mathbf {Y}}'\right) ^{-1} -{\mathbf {Y}}={\mathbf {U}}\varvec{\varSigma }{\mathbf {V}}^{T},\quad {\mathbf {Y}}'\in \pi ^{-1}({\mathbf {m}}'), \end{aligned}$$

so that

$$\begin{aligned} {\mathbf {Z}}:={\mathbf {U}}\arctan (\varvec{\varSigma }) {\mathbf {V}}^{T}. \end{aligned}$$

As a direct consequence of (27) and (28), the horizontal lift ${\mathbf {Z}}$ of $v={{\,\mathrm{Log}\,}}_{{\mathbf {m}}}({\mathbf {m}}')$ encodes the Jordan’s principal angles between ${\mathbf {m}}$ and ${\mathbf {m}}'$, as we can write in the orthonormal basis ${\mathbf {y}}_1,\dots ,{\mathbf {y}}_n$ of ${\mathbb {R}}^n$:

$$\begin{aligned} {\mathbf {Z}}=[\theta _{1}{\mathbf {y}}_{p+1},\dots ,\theta _p{\mathbf {y}}_{2p}]. \end{aligned}$$

From Remark A.4, we deduce that we have ${{\,\mathrm{Exp}\,}}_{{\mathbf {m}}}(v)={\mathbf {m}}'$, leading to:

Lemma A.10

For any ${\mathbf {m}}\in {\mathcal {G}}(p,n)$, the map ${{\,\mathrm{Log}\,}}_{{\mathbf {m}}}$ is a diffeomorphism from $\mathrm {U}_{{\mathbf {m}}}$ onto ${{\,\mathrm{Log}\,}}_{{\mathbf {m}}}\left( \mathrm {U}_{{\mathbf {m}}}\right)$, with inverse map given by the exponential map at ${\mathbf {m}}$:

$$\begin{aligned} {{\,\mathrm{Exp}\,}}_{{\mathbf {m}}}\circ {{\,\mathrm{Log}\,}}_{{\mathbf {m}}}=\text {id} _{\mathrm {U}_{{\mathbf {m}}}}. \end{aligned}$$

As a conclusion of this subsection, we obtain here normal coordinates on all the open set $\mathrm {U}_{{\mathbf {m}}}$, which is in fact an improvement compared to the open set deduced from the injectivity radius disk, thanks to the lemma:

Lemma A.11

For any ${\mathbf {m}}\in {\mathcal {G}}(p,n)$ and n, p such that $\min (p,n-p)\ge 2$, the open set $\mathrm {U}_{{\mathbf {m}}}$ given by (26) strictly contains ${{\,\mathrm{Exp}\,}}_{{\mathbf {m}}}\left( \mathrm {D}_{{\mathbf {m}}}\right)$, with $\mathrm {D}_{{\mathbf {m}}}$ given by (25):

$$\begin{aligned} {{\,\mathrm{Exp}\,}}_{{\mathbf {m}}}\left( \mathrm {D}_{{\mathbf {m}}}\right) \varsubsetneq \mathrm {U}_{{\mathbf {m}}}. \end{aligned}$$

Proof

The inclusion follows from Theorem A.15 as any $v\in \mathrm {D}_{{\mathbf {m}}}$ is such that

$$\begin{aligned} \Vert v\Vert <\frac{\pi }{2}. \end{aligned}$$

To obtain a strict inclusion we follow Remark A.4 in the case $2p\le n$. Let us consider an orthonormal basis ${\mathbf {y}}_1,\dots ,{\mathbf {y}}_n$ and v with horizontal lift given by

$$\begin{aligned} {\mathbf {Z}}=[\theta _{1}{\mathbf {y}}_{p+1},\dotsc ,\theta _{p}{\mathbf {y}}_{2p}]. \end{aligned}$$

Then we can find $\theta _1,\dots ,\theta _p$ such that

$$\begin{aligned} \Vert v\Vert =\left( \sum \theta _{i}^2\right) ^{1/2}\ge \pi /2 \text { and } \theta _{1}<\pi /2 \end{aligned}$$

using for instance

$$\begin{aligned} \theta _{i}:=\alpha <\frac{\pi }{2} \text { with } \frac{\pi }{2} \le \sqrt{p}\alpha . \end{aligned}$$

$\square$

1.4 Cut–locus and exponential map injectivity on Grassmann manifolds

In this final subsection, it is proposed to establish the link between the open set $\mathrm {U}_{{\mathbf {m}}}$ defined by (26) and the cut–locus of Grassmann manifolds. Such a notion of cut–locus is particularly related to the loss of injectivity of the exponential map. As far as we know, such a result about the cut–locus was suggested in [24], but without any clear proof nor statement.

Let us take back here the geodesic $t\in {\mathbb {R}}\mapsto \alpha _{v}(t)$ from (21), with non-zero initial velocity $v\in T_{{\mathbf {m}}}{\mathcal {G}}(p,n)$. Define now

$$\begin{aligned} I_{v}:=\{t\in {\mathbb {R}},\quad (\alpha _v)_{\mid _{[0,t]}} \text { is length minimal}\}=[0,\rho (v)], \end{aligned}$$

where $\rho (v)$ is some bounded real number (see [23, Section 2.C.7]). A first result is given by [23, Theorem 3.77]:

Theorem A.12

Let ${\mathbf {m}}\in {\mathcal {G}}(p,n)$ and

$$\begin{aligned} \mathrm {V}_{{\mathbf {m}}}:=\left\{ v\in T_{{\mathbf {m}}}{\mathcal {G}}(p,n),\quad \rho (v)>1 \right\} \cup \{0\}. \end{aligned}$$

(29)

Then $\mathrm {V}_{{\mathbf {m}}}$ is an open neighborhood of $0\in T_{{\mathbf {m}}}{\mathcal {G}}(p,n)$ and the map

$$\begin{aligned} \left( {{\,\mathrm{Exp}\,}}_{{\mathbf {m}}}\right) _{\mid _{\mathrm {V}_{\mathbf {m}}}} \, : \, \mathrm {V}_{{\mathbf {m}}} \longrightarrow {{\,\mathrm{Exp}\,}}_{{\mathbf {m}}}(\mathrm {V}_{\mathbf {m}}) \end{aligned}$$

is a diffeomorphism.

The image of the boundary $\partial \mathrm {V}_{{\mathbf {m}}}$ then define the cut-locus:

Definition A.13

(Cut-locus) For any point ${\mathbf {m}}\in {\mathcal {G}}(p,n)$, the cut-locus of ${\mathbf {m}}$ is given by

$$\begin{aligned} \text {Cut}({\mathbf {m}}):=\left\{ {{\,\mathrm{Exp}\,}}_{{\mathbf {m}}}(\rho (v)v),\quad \Vert v\Vert =1 \right\} . \end{aligned}$$

In the specific case of Grassmann manifolds, there is a way to explicitly obtain the bound $\rho (v)$, while the main ideas are directly taken from [25, Theorem 12.5]:

Lemma A.14

Let ${\mathbf {m}}\in {\mathcal {G}}(p,n)$ and $v\in T_{{\mathbf {m}}}{\mathcal {G}}(p,n)$, with horizontal lift given by some ${\mathbf {Z}}\in \mathrm {Mat}_{n,p}({\mathbb {R}})$. Then we have

$$\begin{aligned} \rho (v)=\frac{\pi }{2\theta _{1}}, \end{aligned}$$

where $\theta _1$ is the maximal singular value of ${\mathbf {Z}}$ and thus, taking back the open set $\mathrm {V}_{{\mathbf {m}}}$ defined by (29) we have

$$\begin{aligned} \mathrm {V}_{{\mathbf {m}}}=\left\{ v\in T_{{\mathbf {m}}}{\mathcal {G}}(p,n),\quad \theta _{1}<\frac{\pi }{2} \right\} \cup \{0\}. \end{aligned}$$

(30)

Proof

From Lemma A.1, we can consider an orthonormal basis ${\mathbf {y}}_1,\dots ,{\mathbf {y}}_n$ of ${\mathbb {R}}^n$ such that ${\mathbf {Y}}\in \pi ^{-1}({\mathbf {m}})$ and a horizontal lift ${\mathbf {Z}}$ of v are given by (for $2p\le n$):

$$\begin{aligned} {\mathbf {Y}}=[{\mathbf {y}}_1,\dots ,{\mathbf {y}}_{p}],\quad {\mathbf {Z}}=[\theta _{1}{\mathbf {y}}_{p+1},\dotsc ,\theta _{p}{\mathbf {y}}_{2p}], \end{aligned}$$

where $0\le \theta _{p}\le \dots \le \theta _{1}$ are the singular values of any horizontal lift of v.

Now, from Theorem A.6 the geodesic $\alpha (t)=\pi ({\mathbf {Y}}(t))$ with

$$\begin{aligned}&{\mathbf {Y}}(t)=\left[ \cos (\theta _{1}t){\mathbf {y}}_1+\sin (\theta _{1}t){\mathbf {y}}_{p+1},\dots ,\cos (\theta _{p}t){\mathbf {y}}_p\right. \\&\left. +\sin (\theta _{p}t){\mathbf {y}}_{2p}\right] \end{aligned}$$

is minimal for all $t\le \pi /(2\theta _1)$, and is not unique anymore for $t=\pi /(2\theta _1)$. From [23, Corollary 2.111], $\alpha$ is no longer minimal on $[0,\pi /(2\theta _1)+\varepsilon ]$ for all $\varepsilon >0$, so we can conclude (the proof being the same for $2p>n$). The last equation (30) is straightforward. $\square$

Our main result is now:

Theorem A.15

For any ${\mathbf {m}}\in {\mathcal {G}}(p,n)$ we have

$$\begin{aligned} {{\,\mathrm{Exp}\,}}_{{\mathbf {m}}}(\mathrm {V}_{\mathbf {m}})=\mathrm {U}_{{\mathbf {m}}} \end{aligned}$$

with $\mathrm {U}_{{\mathbf {m}}}$ and $\mathrm {V}_{{\mathbf {m}}}$ respectively defined by (26) and (29). Furthermore, the cut-locus at ${\mathbf {m}}$ is given by:

$$\begin{aligned}&\text {Cut} ({\mathbf {m}})1=\left\{ {\mathbf {m}}',{\mathbf {Y}}^{T}{\mathbf {Y}}' \text { is singular},\pi ({\mathbf {Y}})\right. \left. ={\mathbf {m}},\pi ({\mathbf {Y}}')={\mathbf {m}}' \right\} . \end{aligned}$$

Proof

Taking back Lemma A.14 recall that

$$\begin{aligned} \mathrm {V}_{{\mathbf {m}}}=\left\{ v\in T_{{\mathbf {m}}}{\mathcal {G}}(p,n),\quad \theta _{1}<\frac{\pi }{2} \right\} \cup \{0\} \end{aligned}$$

where $\theta _1$ is the maximal singular value of any horizontal lift ${\mathbf {Z}}\in \mathrm {Mat}_{n,p}({\mathbb {R}})$ of v. Take now any $v\in \mathrm {V}_{{\mathbf {m}}}$ and define an orthonormal basis ${\mathbf {y}}_1,\dots ,{\mathbf {y}}_n$ of ${\mathbb {R}}^n$ like in Lemma A.1, so that for $2p\le n$

$$\begin{aligned} {{\,\mathrm{Exp}\,}}_{{\mathbf {m}}}(v)= & {} \pi \left( [\cos (\theta _{1}){\mathbf {y}}_1+\sin (\theta _{1}){\mathbf {y}}_{p+1},\dots ,\cos (\theta _{p}){\mathbf {y}}_p\right. \\&\left. +\sin (\theta _{p}){\mathbf {y}}_{2p}] \right) ,\quad \theta _1<\pi /2. \end{aligned}$$

From Lemma A.8 we deduce that ${{\,\mathrm{Exp}\,}}_{{\mathbf {m}}}(v)\in \mathrm {U}_{{\mathbf {m}}}$ and thus ${{\,\mathrm{Exp}\,}}_{{\mathbf {m}}}(\mathrm {V}_{{\mathbf {m}}})\subset \mathrm {U}_{{\mathbf {m}}}$.

The converse is a direct consequence of Theorem A.6 and Lemma A.8, all proof being the same for $2p>n$. Finally, the statement for $\text {Cut}({\mathbf {m}})$ follows in the same way, so we can conclude. $\square$