Probabilistic Schubert calculus

A.1 Proof of Theorem 3.2

Since the density of A is invariant under the orthogonal group O⁢(n), we can assume that B is spanned by the first l standard vectors, i.e.,

where 1 is the ℓ×ℓ identity matrix, A1 is an ℓ×k matrix and A2 is an (n-ℓ)×k matrix. (Again, abusing notation, we denote by A and B also matrices whose columns span the corresponding spaces.) If we sample A with i.i.d. normal Gaussians, the corresponding probability distribution for the span of its columns is O⁢(n) invariant, and consequently it coincides with the uniform distribution. In order to compute the principal angles between A and B using (3.1), we need to orthonormalize the columns of A. Defining

we see that the span of the columns of A and A^ is the same, and the columns of A^ are orthonormal. The cosines of the principal angles between A and B are the singular values 1≥σ1≥⋯≥σk≥0 of the matrix

The σ1,…,σk coincide with the square roots of the eigenvalues 1≥u1≥⋯≥uk≥0 of the positive semidefinite matrix

which are the same as the eigenvalues of N=(A1T⁢A1+A2T⁢A2)-1⁢A1T⁢A1 (eigenvalues are invariant under cyclic permutations). Consider the Cholesky decomposition

Then the eigenvalues of N equal the eigenvalues of

We use now some facts about the multivariate Beta distribution (see [37, Section 3.3]). By its definition, the matrix U has a Betak⁢(12⁢l,12⁢(n-l)) distribution, and [37, Theorem 3.3.4] states that the joint density of the eigenvalues of U is given by

Recall now that ui=(cos⁡θi)2 for i=1,…,k, which implies the change of variable

and thus (A.1) becomes the stated density in (3.2).

A.2 Proof of Lemma 4.3

We begin with a general reasoning. Assume that A∈e⁢(B). The unit normal vectors of e⁢(B) at A, up to a sign, are uniquely determined by A. Lemma 4.1 provides an explicit description for them as follows. Let a1,a2,…,ak and a1,b2,…,bn-k be the orthonormal bases given by Lemma 3.1 for A and B, respectively; note that dim⁡(A∩B)=1 since A∈e⁢(B). In particular, 〈ai,bi〉=cos⁡θi for i≤k, where θ1≤…≤θk are the principal angles between A and B. Let ℝ⁢f=(A+B)⊥ with ∥f∥=1. According to Lemma 4.1, the unit vector ν:=f∧a2∧⋯∧ak spans the normal space of e⁢(B) at A. (We use here the representation of elements of G⁢(k,n) and its tangent spaces by vectors in Λk⁢ℝn; cf. Section 2.1.)

Fix now B∈G⁢(n-k,n) and recall that the functions θ1,θ2:G⁢(k,n)→ℝ give the smallest and second smallest principal angle, respectively, between A∈G⁢(k,n) and B. Let 0<ε≤δ<π2 and put

We first prove that 𝒯⊥⁢(e⁢(B)δ,ε)⊆T.

For A∈e⁢(B) we consider the curve NA⁢(t):=(a1⁢cos⁡t+f⁢sin⁡t)∧a2∧⋯∧ak for t∈ℝ. We note that NA⁢(t) arises from A by a rotation with the angle t in the (oriented) plane spanned by a1 and f, fixing the vectors in the orthogonal complement spanned by a2,…,ak. We have NA⁢(0)=A and N˙A⁢(0)=ν. It is well known (see, e.g., [13]) that NA⁢(t) is the geodesic through A with speed vector ν, that is, NA⁢(t)=expA⁡(t⁢ν) and NA⁢(0)=A. From the definition of the ε-tube, we therefore have

Since both a1 and f are orthogonal to b2,…,bn-k, the principal angles between NA⁢(t) and B are |t|,θ2,…,θk, where 〈aj,bj〉=cos⁡θj. By our assumption A∈e⁢(B)δ, we have δ≤θ2≤…≤θk. Hence we see that |t| is the smallest principle angle between NA⁢(t) and B if |t|≤δ. We have thus verified that 𝒯⊥⁢(e⁢(B)δ,ε)⊆T.

For the other inclusion, let C∈T; assume that C∉e⁢(B)δ, otherwise we clearly have C∈𝒯⊥⁢(e⁢(B)δ,ε). Let (c1,…,ck) and (b1,…,bn-k) be orthonormal bases of C and B, respectively, as provided by Lemma 3.1. So we have 〈bj,cj〉=cos⁡θj⁢(C) for all j, where θ1⁢(C)≤ε and δ≤θ2⁢(C)≤…≤θk⁢(C) by the assumption C∈T. In particular, b1 is orthogonal to c2,…,ck and b1,c1 are linearly independent. We define A∈G⁢(k,n) as the space spanned by b1,c2,…,ck. By construction, θ1⁢(A)=0 and θj⁢(A)=θj⁢(C) for j≥2. Hence, A∈e⁢(B)δ. Let NA⁢(t)∈G⁢(k,n) be defined as above as the space resulting from A by a rotation with the angle t in the oriented plane spanned by b1 and c1. By construction, we have C=NA⁢(τ), where τ=θ1⁢(C)≤ε. Therefore, we indeed have C∈𝒯⊥⁢(e⁢(B)δ,ε) by (A.2).

A.3 Proof of Lemma 4.5

More generally, we consider a semialgebraic set Y⊆ℝ⁢Pn-1 of dimension d≤n-k-1. Consider the semialgebraic set

together with the projections on the two factors: π1:C⁢(Y)→G⁢(k,n) and π2:B⁢(Y)→Y. As Z⁢(Y)=π1⁢(C⁢(Y)), the set Z⁢(Y) is semialgebraic. Note that C⁢(Y) is compact if Y compact, and C⁢(Y) is connected if Y is connected (since π1 is continuous). In order to determine the dimension of C⁢(Y), we note that the fiber π2-1⁢(y) over y∈Y is isomorphic to G⁢(k-1,n-1). As a consequence, we get

The fibers π1-1⁢(A) over A∈Z⁢(Y) consist of exactly one point, except for the A lying in the exceptional set

Note that ℙ⁢(A)∩Y consists of one point only, for all A∈Z⁢(Y)∖Z(2)⁢(Y).

In order to show that dim⁡Z(2)⁢(Y)<dim⁡Z⁢(Y), we consider the semialgebraic set

with the corresponding projections π3:C(2)⁢(Y)→G⁢(k,n) and π4:C(2)⁢(Y)→Y×Y. Note that Z(2)⁢(Y)=π3⁢(C(2)⁢(Y)). For (y1,y2)∈Y×Y such that y1≠y2, we have

since the fibers of π4 are isomorphic to G⁢(k-2,n-2). Therefore,

and we see that dim⁡(C⁢(Y)∖C(2)⁢(Y))=dim⁡C⁢(Y). For the projection f:C(2)⁢(Y)→C⁢(Y) defined by f⁢(A,y1,y2):=(A,y1), we have π1-1⁢(Z(2)⁢(Y))=f⁢(C(2)⁢(Y)), hence

Moreover, the projection C⁢(Y)∖C(2)⁢(Y)→Z⁢(Y)∖Z(2)⁢(Y), (A,y)↦y is bijective, hence

Using dim⁡Z⁢(Y)≤dim⁡C⁢(Y), we see that

and

In the special case where dim⁡X=n-k-1, we conclude that Z⁢(X) is a hypersurface.

We consider now the following set of “bad” A∈Z⁢(X):

We claim that this semialgebraic set has dimension strictly less than Z⁢(X). This follows for Z(2)⁢(X) from (A.6), for Z⁢(Sing⁡X) from (A.5) applied to Y=Sing⁡(X), for π1(Sing(C(X)) from Proposition 2.2 and (A.5), and finally for Sing⁡(Z⁢(X)) by Proposition 2.2.

Thus generic points A∈Z⁢(X) are not in S⁢(X) and hence satisfy the following:

1. The intersection ℙ⁢(A)∩X consists of one point only (let us denote this point by p),

2. The point p is a smooth point of X,

3. The point A is a regular point of Z⁢(X),

4. The point (A,p) is a regular point of C⁢(X).

It remains to prove that for every A∈Z⁢(X)∖S⁢(X) we have (4.3). To this end, let us take (A,p)∈C⁢(X) with A∉S⁢(X). We work in local coordinates (w,y)∈ℝN×ℝn-1 on a neighborhood U of (A,p) in G⁢(k,n)×ℝ⁢Pn-1, where N=k⁢(n-k). For simplicity we center the coordinates on the origin, so that (0,0) are the coordinates of (A,p). In this coordinates, the set C⁢(X)∩U can be described as

where F⁢(w,x)=0 represents the reduced local equations describing the condition x∈ℙ⁢(W) and G⁢(x)=0 the reduced local equations giving the condition x∈X. Since (A,p) is a regular point of C⁢(X), the tangent space of C⁢(X) at (A,p) is described by

Note that p is a smooth point of X, hence (D0⁢G)⁢x˙=0 is the equation for the tangent space to X at p. On the other hand, since Z⁢(X)=π1⁢(C⁢(X)), we have

Let now B⊆ℝn be the linear space corresponding to Tp⁢X and let us write the equations for C⁢(B) in the same coordinates as above (by construction we have (A,p)∈C⁢(B)):

Note that the same equation F⁢(w,x)=0 as in (A.7) appears here (recall that this is the equation describing x∈ℙ⁢(W)), but now G=0 is replaced with its linearization at zero. In particular,

which coincides with (A.8). Since Ω⁢(B)=π1⁢(C⁢(B)), this finally implies

which finishes the proof.

A.4 Proof of Proposition 5.12

We extend the function f to ℝk×m∖{0} by setting

denoting it by the same symbol. Similarly, we extend g by setting

We assume now that X∈ℝk×m has i.i.d. standard Gaussian entries. Then we can write

where pSVD denotes the joint density of the ordered singular values of X. This density can be derived as follows. The joint density of the ordered eigenvalues λ1>⋯>λk>0 of the Wishart distributed matrix X⁢XT is known to be [37, Corollary 3.2.19]

where

From this, using λi⁢(X⁢XT)=σi⁢(X)2 and the change of variable d⁢λi=2⁢σi⁢d⁢σi, we obtain

As a consequence, we obtain

where in the second line we have switched to polar coordinates σ=r⁢θ with θ∈Sk-1 and r≥0. Note that the power of the r-variable arises as

Using

we obtain

It is immediate to verify that

which completes the proof.

A.5 Generalized Poincaré’s formula in homogeneous spaces

The purpose of this subsection is to prove the kinematic formula in homogeneous spaces for multiple intersections and to derive Theorem 3.19. The proofs are similar to [26], to which we refer the reader for more details.