Approximating the Derivative of Manifold-valued Functions
AApproximating the Derivative ofManifold-valued Functions
Ralf Hielscher ∗ Laura Lippert † We consider the approximation of manifold-valued functions by embed-ding the manifold into a higher dimensional space, applying a vector-valuedapproximation operator and projecting the resulting vector back to the man-ifold. It is well known that the approximation error for manifold-valuedfunctions is close to the approximation error for vector-valued functions.This is not true anymore if we consider the derivatives of such functions.In our paper we give pre-asymptotic error bounds for the approximation ofthe derivative of manifold-valued function. In particular, we provide explicitconstants that depend on the reach of the embedded manifold.
Keywords: nonlinear approximation, manifold-valued functions, embeddedmanifolds
1. Introduction
Approximating functions f : X → M with values in some Riemannian manifold M hasattracted lots of interest during the last years. The central challenge is that with M not being linear, the function spaces over X with values in M are not linear as welland hence, all the well established linear approximation methods do not have a straightforward generalization to manifold-valued functions.One successful approach to manifold-valued approximation is to consider the problemlocally. Either one maps the function values locally to some linear approximation spaceor one uses local averaging based on the geodesic distance. These approaches allowto generalize subdivision schemes [32, 33, 9, 10, 29], moving least squares [12], quasi-interpolation [11] or splines [30] to the manifold-valued setting.A different approach is to embed the manifold M into some higher dimensional linearspace R d by a map E : M → R d . Note, that according to Nash’s embedding theorem [21] ∗ Faculty of Mathematics, Chemnitz University of Technology, D-09107 Chemnitz, Germany.E-mail: [email protected] † Faculty of Mathematics, Chemnitz University of Technology, D-09107 Chemnitz, Germany.E-mail: [email protected] a r X i v : . [ m a t h . NA ] F e b uch a mapping always exists and can be guaranteed to be locally isometric provided thedimension d is at least m (3 m + 11) /
2, where m denotes the dimension of M . Embeddingbased approximation methods can be summarized as followsi) Transfer f ∈ C ( X, M ) via ˜ f = E ◦ f into the linear function space C ( f, R d ).ii) Use a linear approximation operator I R d : C ( f, R d ) → C ( f, R d ) to find an approx-imant I R d ˜ f in the embedding.iii) Project the resulting R d -valued function back to the manifold I M f = P M ◦ ( I R d ˜ f )using some projection operator P M : R d → M .Because of its generality and simplicity this approach has already been widely inves-tigated [11, 7] and applied [20, 27]. In particular it has been shown in [7] that theapproximation order of the embedding based approximation operator I M f is the sameas the approximation order of its linear counterpart I R d . It is important to note, thatthe projection operator P M is in general only defined in some neighborhood U ⊃ M of the manifold. Hence, the pre-asymptotic behavior of the approximation operator I M strongly depends on the size of this neighborhood which is directly related to theso-called reach of the embedded manifold.The aim of this paper is to analyze the pre-asymptotic behavior of the approximationoperator I M with respect to the reach of the embedded manifold. While for the error I M f − f the reach only controls the required linear approximation error I R d ˜ f − ˜ f thatallows for a meaningful approximation I M f , the situation is completely different for theerror of the derivatives ( I M f ) (cid:48) − f (cid:48) .Our paper is organized as follows. In section 2 we will first show some general dif-ferential geometric properties of submanifolds of R d . Most importantly, we identify inLemma 2.1 the projection operator P M with an orthogonal projection in the normalbundle over the manifold M . This is only possible within some tubular neighborhoodof the manifold M that is controlled by the reach of M . The relationship between thereach of the manifold M and its curvature or second fundamental form is addressedin section 2.2. In Theorem 2.7 we make use of these relationships to describe the dif-ferential of the projection operator P M in terms of the second fundamental form. Insection 2.4 we end up with the main results of this chapter, that is we show in Theo-rem 2.10 that the derivative dP M ( x ) of the projection operator at some point x ∈ R d satisfies a Lipschitz-condition with respect to x . As our Lipschitz bound is with respectto the Euclidean distance in the embedding it is more sharp then the bound reported in[3] that relies on the geodesic distance.Section 3 is dedicated to manifold-valued approximation. We show that the approachof embedding a manifold M in R d , using a linear approximation operator and projectingback on the manifold inherits the approximation order of the linear approximation.Our main result is stated in Theorem 3.2 and gives a pre-asymptotic bound for theapproximation error of the first derivative that relies exclusively on the reach of theembedded manifold. This result is illustrated in Theorem 3.4 for a specific approximationoperator, the Fourier partial sum operator.2n the final section 4 we consider two real world examples for approximating mani-fold valued data. The first example deals with functions from the two-sphere into thetwo-dimensional projective space that describe the dependency between the propaga-tion direction and the polarization directions of seismic waves. The second example isfrom crystallographic texture analysis where the local alignment of the atom lattice isdescribed by a map with values in the quotient SO (3) /S of the rotation group SO (3)modulo some finite symmetry group S . The derivative of this map has important con-nections microscopic and macroscopic material properties.
2. Submanifolds
In this section we will consider smooth compact Riemannian submanifolds M of R d .We will show some differential geometric properties of submanifolds as well as someestimations for the projection P M and the differential of this projection. We will usethese results for estimating some approximation errors in section 3. Throughout our work we denote by
M ⊂ R d a smooth compact Riemannian submanifoldand by P M : R d → M the projection operator onto M defined as the solution of theminimization problem P M ( x ) = argmin m ∈M (cid:107) x − m (cid:107) . (2.1)In general, this minimization problem does not posses an unique solution for every x ∈ R d . However, if we restrict the domain of the definition of P M to some openneighborhood U ⊂ R d of M uniqueness can be granted.In order to find such a neighborhood U we define on the normal bundle N M = { ( m , v ) ∈ R d × R d : m ∈ M , v ∈ N m M} of M the smooth map E : N M → R d , E ( m , v ) = m + v , that maps every normal space N m M to an affine linear subspace through m ∈ R d .Since we assumed M to be compact and smooth, there exist a maximum constant τ > E restricted to the open subset V = { ( m , v ) ∈ N M : (cid:107) v (cid:107) < τ } of the normal bundle is injective, cf. [17, 6.24]. Setting U = E ( V ) defines the so-called tubular neighborhood of M and the restriction E : V → U becomes a diffeomorphism.The constant τ is commonly called reach and its inverse 1 /τ is the condition number ofthe manifold. The reach τ is affected by two factors: the curvature of the manifold andthe width of the narrowest bottleneck-like structure of M , which quantifies how far M
3s from being self-intersecting. An estimate on the relationship between the reach andthe curvature of the manifold M will be given in Lemma 2.4.Using the mapping E we may now give an explicit definition of the projection operator P M . Lemma 2.1.
Let u ∈ U and let π : N M → M , ( m , n ) (cid:55)→ m be the canonical projectionoperator. Then P M ( u ) = π ◦ E − ( u ) is the unique solution of the minimization problem (2.1) . Proof.
Let u ∈ U and P M ( u ) = m ∈ M . We show that u − m ∈ N m M . We assumethe opposite and decompose u − m in one part in N m M and a part t in T m M . Thenthere is a curve γ ( s ) in M with γ (0) = m and ˙ γ (0) = t . If we go along this curve,we obtain for sufficient small (cid:15) > u − γ ( (cid:15) ) < u − γ (0) = u − m . That is acontradiction to the definition of P M . Since the projection P M should be unique, wehave to show that π ◦ E − is also unique. For that reason we assume that for u ∈ U thereholds π ◦ E − = m ∈ M and π ◦ E − = m (cid:48) ∈ M . This would imply u = m + v = m (cid:48) + v (cid:48) with v ∈ N m M and v ∈ N m (cid:48) M . That is a contradiction to the uniqueness of E − inthe tubular neighborhood U .Let us illustrate this by a simple example. Example 2.2.
Let the manifold M be the ( d − R d . These manifolds can be described by S d − = (cid:8) x ∈ R d : (cid:107) x (cid:107) = 1 (cid:9) . The projection P S d − easily reads P S d − : R d \{ } → S d − , P S d − ( x ) = x (cid:107) x (cid:107) . This map is well-defined and smooth.
For any point m ∈ M ⊂ R d on the manifold we can decompose R d as the direct sum R d = T m M ⊕ N m M of the tangential space T m M and the normal space N m M . Letus denote by P T : R d → T m M and P N : R d → N m M the corresponding orthogonalprojections. Then the canonical connection ∇ on R d defines a connection ∇ M on M by ∇ M X = P T ∇ X ( P T Y ) + P N ∇ X ( P N Y ) (2.2)where X : M → T M is a tangential and Y : M → R d a general vector field on M .If Y is a tangential vector field as well, the first summand P T ∇ X ( P T Y ) = P T ∇ X Y in (2.2) is just the Levi-Cevita-connection on M , whereas its orthogonal complementII( X , Y ) = P N ( ∇ X Y )4s the second fundamental form on M .We call a vector field Y : M → R d parallel along a curve γ if ∇ M ˙ γ Y = 0. For a geodesic γ with γ (0) = m , ˙ γ (0) = t ∈ T m M and an arbitrary vector y ∈ T m M ⊕ N m M = R d we shall use the abbreviation ∇ t y = ∇ t Y (0)where Y is the parallel transport of the vector y along the curve γ .For a fixed point m ∈ M and a normal direction n ∈ N m M we define the operator B n : T m M → T m M on the tangent space by (cid:104) B n x , y (cid:105) = (cid:104) n , ∇ x y (cid:105) , x , y ∈ T m M . (2.3)We may also express B n x as the tangential part of the covariant derivative of n indirection x . Lemma 2.3.
Let n ∈ N m M be a normal and x ∈ T m M a tangential vector. Then B n x = − P T ∇ x n . Proof.
Let γ be a geodesics in M with γ (0) = m and ˙ γ (0) = x and let N be the paralleltransport of n along γ . Let furthermore, Y be an arbitrary tangent vector field parallelalong γ . Then we have0 = dd s (cid:104) N ( s ) , Y ( s ) (cid:105)| s =0 = (cid:104)∇ x N (0) , Y (0) (cid:105) + (cid:104) N (0) , ∇ x Y (0) (cid:105) . This yields (cid:104) B n x , y (cid:105) = (cid:104) n , ∇ x y (cid:105) = −(cid:104)∇ x n , y (cid:105) . Since the vector field Y was arbitrary, this yields the assertion.The operator B n describes the extrinsic curvature of the manifold in the point m andthe normal direction n . Its norm is bounded by the condition number τ of M . Moreprecisely the following result is shown in [22, Proposition 6.1]. Lemma 2.4.
Let τ be the reach of M , m ∈ M be an arbitrary point on the manifoldand n ∈ N m M be a normal vector. Then the operator B n defined in (2.3) is symmetricand bounded by τ , i.e., we have for tangential vectors x , y ∈ T m M the inequality (cid:104) B n x , y (cid:105) ≤ τ (cid:107) n (cid:107) (cid:107) x (cid:107) (cid:107) y (cid:107) . (2.4)The next lemma bounds the covariant derivative of parallel vector fields by the con-dition number τ of the manifold. Lemma 2.5.
Let Y be a parallel vector field along a geodesic γ in M . Then its covariantderivative in R d is bounded by (cid:107)∇ ˙ γ Y (cid:107) ≤ τ (cid:107) Y (cid:107) (cid:107) ˙ γ (cid:107) . roof. Let Y = T + N be the decomposition of Y into a tangent vector field T and anormal vector field N . Since Y is parallel along γ we have0 = ∇ M ˙ γ Y = P T ∇ ˙ γ T + P N ∇ ˙ γ N and, hence, ∇ ˙ γ Y = P N ∇ ˙ γ T + P T ∇ ˙ γ N . Let n = P N ∇ ˙ γ ( s ) T ( s ). Then we obtain by Lemma 2.4 (cid:107) n (cid:107) = (cid:10) n , ∇ ˙ γ ( s ) T ( s ) (cid:11) = (cid:104) B n ˙ γ ( s ) , T ( s ) (cid:105) ≤ τ (cid:107) ˙ γ (cid:107) (cid:107) n (cid:107) (cid:107) T ( s ) (cid:107) . For the tangential part we have by Lemma 2.3 (cid:107) P T ∇ ˙ γ ( s ) N ( s ) (cid:107) = (cid:107) B N ( s ) ˙ γ ( s ) (cid:107) ≤ τ (cid:107) ˙ γ (cid:107) (cid:107) N ( s ) (cid:107) . The assertion follows now from Parsevals inequality.
The differential d P M ( m ) : R d → T m M of the projection P M : R d → M is especiallyeasy to compute at points m ∈ M on the manifold. In this case it is simply the linearprojection P T m M : R d → T m M onto the tangential space attached to m , i.e.d P M ( m ) = P T m M . (2.5)We can verify this by observing that for normal vectors n ∈ N m M we haved P M ( m ) n = lim h → P M ( m + h n ) − P M ( m ) h = lim h → m − m h = , while for tangent vectors t ∈ T m M we obtaind P M ( m ) t = lim h → P M ( m + h t ) − P M ( m ) h = lim h → m + exp( h t ) − m h = lim h → exp( h t ) h = t , where exp denotes the exponential map to the manifold.The differential d P M ( m + v ), v ∈ N m M at a point not in the manifold is a little bitmore tricky. We start by observing that the tangential T ( m , v ) N M ⊂ R d of the normalbundle at a point ( m , v ) ∈ N M is T ( m , v ) N M = { ( , n ) | n ∈ N m M} ⊕ { ( t , ∇ t v ) | t ∈ T m M} = { ( t , u ) | t ∈ T m M , P T u = ∇ t v } . The following lemma describes the differential d P M ( m + v ).6 emma 2.6. Let m ∈ M be an arbitrary point on the manifold M and v ∈ N m M bea normal vector with (cid:107) v (cid:107) < τ , i.e. m + v is in the tubular neighborhood of M . Thenthe derivative d P M ( m + v ) satisfies for every tangent direction t ∈ T m M , (d P M ( m + v )) ( t + ∇ t v ) = t . while it vanishes for any normal direction n ∈ N m M , i.e. d P M ( m + v ) n = . Proof.
According to Lemma 2.1 we have P M = π ◦ E − , where π : N M → M , ( m , v ) (cid:55)→ m is the projection operator. Its differential at the point ( m , v ) ∈ N M is the projectiond π ( m , v ) : T ( m , v ) N M → T m M , ( t , u ) (cid:55)→ t . The differential of the mapping E : N M → R d in a point ( m , v ) ∈ N M is given byd E ( m , v ) : T ( m , v ) N M → R d , ( t , u ) (cid:55)→ t + u . Since m + v is within the tubular neighborhood of M , E is invertible in some neighbor-hood of m + v . Then d E ( m , v ) is invertible as well and we have for any normal vector n ∈ N m M d E − ( m + v ) n = ( , n )and for any tangent vector t ∈ T m M d E − ( m + v ) ( t + ∇ t v ) = ( t , ∇ t v ) . Together with the chain rule this implies the assertion.The image of d P M ( m + v ) is contained in the tangential space T m M , especiallyd P M ( m + v ) is the projection P T m M up to a factor matrix. We will write this lin-ear operator d P M ( m + v ) in another way, to see the difference to the linear operatord P M ( m ). Theorem 2.7.
Let m ∈ M be a point on the manifold, let v ∈ N m M be a normalvector with (cid:107) v (cid:107) < τ and let B v : T m M → T m M be the symmetric operator defined in (2.3) , extended to B v : R d → R d by B v n = for all normal vectors n ∈ N m M . Thenthe derivative of the projection operator P M satisfies d P M ( m + v ) = P T m M ( I − B v ) − = d P M ( m ) − B v ( I + B v ) − , where I : R d → R d is the identity.Proof. Using Lemma 2.6 we obtain for all tangential vectors t ∈ T m M , P T m M t = t = (d P M ( m + v )) ( t + ∇ t v )= (d P M ( m + v )) ( t − B v t ) = (d P M ( m + v )) ( I − B v ) t . n ∈ N m M , = P T m M n = (d P M ( m + v )) ( I − B v ) n . Consequently, we have P T m M = d P M ( m + v ) ( I − B v ) . By our assumption and (2.4) we have (cid:107) B v (cid:107) ≤ τ (cid:107) v (cid:107) < I − B v is invertible. This yields the first part of the assertion. For the second part we use (2.5)and compute d P M ( m + v ) = P T m M ( I − B v ) − = P T m M (cid:0) I + B v ( I − B v ) − (cid:1) = d P M ( m ) + P T m M B v ( I − B v ) − = d P M ( m ) + B v ( I − B v ) − , where the last equality follows from the fact that the image of B n is in the tangentspace T m M , so the projection on T m M is unnecessary.We consider again the manifold from example 2.2. Example 2.8.
For M = S d − ⊂ R d any normal vector v ∈ N m S d − has the repre-sentation v = v m . Let { t i } d − i =1 ⊂ T m S d − be an orthonormal basis of T m S d − . Then ∇ t i m = t j and hence B v = − v d − (cid:88) i =1 t i t (cid:62) i . By Theorem 2.7 and the orthonormality of { m } ∩ { t i } d − i =1 of we obtain for v > − x = m + v m ,d P S d − ( x ) = 11 + v d − (cid:88) i =1 t i t (cid:62) i = 1 (cid:107) x (cid:107) (cid:32) I d × d − x (cid:107) x (cid:107) (cid:18) x (cid:107) x (cid:107) (cid:19) (cid:62) (cid:33) . In this section we are interested in the change of the derivative d P M ( m ) of the projectionoperator for small deviations of m . We shall show that for two points m and z on M and v ∈ N m M with (cid:107) v (cid:107) < τ we can bound the difference (cid:107) d P M ( m + v ) − d P M ( z ) (cid:107) by a multiple of the euclidean distance (cid:107) m + v − z (cid:107) .As usual we start with the case that both points are on the manifold. According to[3, Lemma 6] the difference of the differentials is then bounded by (cid:107) d P M ( m ) − d P M ( z ) (cid:107) ≤ τ d ( m , z ) , where d ( m , z ) denotes the geodesic distance between the points m , z ∈ M . If theEuclidean distance between the two points is bounded by (cid:107) m − z (cid:107) ≤ τ we have by83, Lemma 3] and the fact that arcsin( x ) ≤ π x for 0 ≤ x ≤
1, the following estimatebetween geodesic distance and Euclidean distance in the embedding d ( m , z ) ≤ π (cid:107) m − z (cid:107) , (2.6)which leads to the local estimate (cid:107) d P M ( m ) − d P M ( z ) (cid:107) ≤ π τ (cid:107) m − z (cid:107) . In the following Theorem we prove a sharper and global bound for this difference.
Theorem 2.9.
For all m , z ∈ M the difference between the projection operators P T m M and P T z M onto the respective tangential spaces is bounded by (cid:107) P T m M − P T z M (cid:107) ≤ τ (cid:107) m − z (cid:107) . Proof.
First of all we note that for (cid:107) m − z (cid:107) ≥ τ the assertion is immediately satisfiedsince (cid:107) P T m M − P T z M (cid:107) ≤ m , z ∈ M . We may therefore assume (cid:107) m − z (cid:107) < τ for the rest of the proof.In order to estimate the difference between the two projection operators we considera geodesic γ with γ (0) = m , γ ( t ) = z and (cid:107) ˙ γ (cid:107) = 1. Furthermore, we consider anorthonormal basis { t i } Di =1 in T m M and an orthonormal basis { n j } d − Dj =1 in N m M . Theparallel transport of these basis vectors along γ defines a rotation R ∈ SO( d ) that mapsthe tangent space T m M onto the tangent space T z M . Using the rotation R we mayrewrite the difference between the projection operators as P T m M − P T z M = P T m M − R P T m M R T . By Lemma A.1 in the appendix we obtain (cid:107) P T m M − P T z M (cid:107) = (cid:107) P T m M R − R P T z M (cid:107) ≤ (cid:107) I − R (cid:107) (2.7)and hence, it suffices to bound for any normalized x ∈ R d (cid:107) ( I − R ) x (cid:107) = 2 − (cid:104) x , Rx (cid:105) . (2.8)By definition Rx is the result of the parallel transport of x along the curve γ in γ ( t ) = z . Let us denote by X ( s ) the parallel transport of x along γ for all times s ∈ [0 , t ]. Viewing s (cid:55)→ X ( s ) as a curve on S d − with velocity bounded according toLemma 2.5 by (cid:107) ˙ X ( s ) (cid:107) = (cid:107)∇ ˙ γ ( s ) X ( s ) (cid:107) ≤ τ , we conclude that ∠ ( X ( η ) , X ( ξ )) ≤ τ | η − ξ | , η, ξ ∈ [0 , t ] . (2.9)Since γ is a geodesic we can set in (2.9), X = ˙ γ . As | η − ξ | ≤ t and t is the geodesicdistance between z and m we can use (2.6) and our assumption (cid:107) m − z (cid:107) < τ tobound the right hand side of (2.9) by ∠ ( ˙ γ ( ξ ) , ˙ γ ( η )) ≤ τ | η − ξ | ≤ tτ ≤ π τ (cid:107) z − m (cid:107) ≤ π. ∠ ( ˙ γ ( ξ ) , ˙ γ ( η )) > cos ξ − ητ , ξ, η ∈ [0 , t ] . (2.10)Considering again the general vector field X we use (2.9) and (2.10) to bound (2.8) by2 − (cid:104) X (0) , X ( t ) (cid:105) = 2 − ∠ ( X (0) , X ( t ))) ≤ − tτ = 1 τ (cid:90) t (cid:90) t cos ξ − ητ d η d ξ ≤ τ (cid:90) t (cid:90) t cos ∠ ( ˙ γ ( ξ ) , ˙ γ ( η )) d η d ξ = 1 τ (cid:90) t (cid:90) t (cid:104) ˙ γ ( ξ ) , ˙ γ ( η ) (cid:105) d η d ξ = 1 τ (cid:107) m − z (cid:107) . In combination with (2.7) and (2.8) this proves (cid:107) P T m M − P T z M (cid:107) ≤ τ (cid:107) m − z (cid:107) . Using the example of the unit circle it can be easily verified that our new bound issharp.So far we bounded the variation of the projection operator for points on the manifold.For the general case that only one point is on the manifold we have the following result.
Theorem 2.10.
Let m , z ∈ M and v ∈ N m M with (cid:107) v (cid:107) < τ . Then (cid:107) d P M ( m + v ) − d P M ( z ) (cid:107) ≤ τ (cid:107) m − z (cid:107) + 1 τ − (cid:107) v (cid:107) (cid:107) v (cid:107) ≤ (cid:18) τ + 1 τ − (cid:107) v (cid:107) (cid:19) (cid:107) m + v − z (cid:107) . Proof.
Using Theorem 2.7 and Theorem 2.9 we find (cid:107) d P M ( m + v ) − d P M ( z ) (cid:107) ≤ (cid:107) d P M ( m + v ) − d P M ( m ) (cid:107) + (cid:107) P T m M − P T z M (cid:107) ≤ (cid:107) B v ( I + B v ) − (cid:107) + τ (cid:107) m − z (cid:107) . From Lemma 2.4 we know that (cid:107) B v (cid:107) ≤ (cid:107) v (cid:107) τ <
1. This allows us to bound the secondterm by (cid:107) B v ( I + B v ) − (cid:107) ≤ (cid:107) B v (cid:107) − (cid:107) B v (cid:107) ≤ (cid:107) v (cid:107) τ − (cid:107) v (cid:107) , which implies the first inequality of the theorem (cid:107) d P M ( m + v ) − d P M ( z ) (cid:107) ≤ τ (cid:107) m − z (cid:107) + 1 τ − (cid:107) v (cid:107) (cid:107) v (cid:107) . (2.11)Since (cid:107) v (cid:107) < τ the point m + v is within the tubular neighborhood of M and, hence (cid:107) v (cid:107) ≤ (cid:107) m + v − z (cid:107) . Together with the triangle inequality this gives us (cid:107) m − z (cid:107) ≤ (cid:107) m + v − z (cid:107) . Including these two inequalities into (2.11) we obtain the assertion.We observe that the constants in Theorem 2.10 become large if either the reach ofthe manifold becomes small or the point m + v is close to the boundary of the tubularneighborhood of M . 10 . Manifold-valued Approximation In this section we generalize arbitrary approximation operators for vector valued func-tions to approximation operators for manifold-valued functions. To this end we considerfor an arbitrary domain D the space C ( D, R d ) of continuous functions f : D → R d together with the norm (cid:107) f (cid:107) L ∞ ( D ) ,p = sup x ∈ D (cid:107) f ( x ) (cid:107) p . For a generic approximation operator I R d : C ( D, R d ) → C ( D, R d ) and an embeddedmanifold M ⊂ R d with reach τ and projection operator P M : U → M , U = { x ∈ R d | min m ∈M (cid:107) x − m (cid:107) < τ } , we define the approximation operator I M : ˜ C ( D, M ) → C ( D, M ) for manifold-valuedfunctions as I M f = P M ◦ I R d f. It is important to note that I M is not defined for all continuous functions f : D → M ,but only for those for which I R d is within the reach of the manifold M , i.e., for thefunctions in ˜ C ( D, M ) = { f ∈ C ( D, M ) | (cid:107) I R d f − f (cid:107) < τ } . It is straight forward to see that operator I M has the same order of approximation as I R d , c.f., [7]. Theorem 3.1.
Let f ∈ ˜ C ( D, M ) be a continuous M -valued function such that for all x ∈ D , ˜ f ( x ) is contained in the reach of M . We then have (cid:107) I M f ( x ) − f ( x ) (cid:107) ≤ (cid:107) I R d f ( x ) − f ( x ) (cid:107) . Proof.
Since f has function values on M , it follows from the definition of P M in equa-tion (2.1) for all x ∈ D that (cid:107) I M f ( x ) − I R d f ( x ) (cid:107) ≤ (cid:107) f ( x ) − I R d f ( x ) (cid:107) . Because of the triangle inequality and the definition of I M we have (cid:107) I M f ( x ) − f ( x ) (cid:107) ≤ (cid:107) I M f ( x ) − I R d f ( x ) (cid:107) + (cid:107) I R d f ( x ) − f ( x ) (cid:107) , x ∈ D. As we will see later, considering the error of the differential, things become morecomplicated.
In this section we are interested in the approximation error (cid:107) d I M f − d f (cid:107) betweenthe differential of the manifold-valued approximation d I M f and the original differential11 f . To this end we assume from now on that both, f : D → R d and the vector-valuedapproximation ˜ f = I R d f , are differentiable.While the error bound for I M f is independent of the geometry of the manifold M ,we will see that this is not true for the differential d I M f of the manifold-valued approx-imation. Moreover, it is not sufficient to ensure that ˜ f is contained in the reach of M ,but instead, it must be bounded away from the reach by some positive constant. Theorem 3.2.
Let τ be the reach of the manifold M , ε < τ and f ∈ C ( D, M ) , suchthat ˜ f ( x ) = I R d f ( x ) satisfies for all x ∈ D , (cid:107) f ( x ) − ˜ f ( x ) (cid:107) ≤ ε and, consequently, is contained in the ε -tubular neighborhood of M . Then we have thefollowing upper bound on the approximation error of the differential d I M f , (cid:107) d I M f − d f (cid:107) L ∞ ( D ) , ≤ (cid:107) d ˜ f − d f (cid:107) L ∞ ( D ) , + ε (cid:18) τ + 1 τ − ε (cid:19) (cid:16) (cid:107) d ˜ f − d f (cid:107) L ∞ ( D ) , + (cid:107) d f (cid:107) L ∞ ( D ) , (cid:17) . Proof.
By the chain rule we obtain for all x ∈ D ,d I M f ( x ) = d P M ˜ f ( x ) ◦ d ˜ f ( x )and from P M f = f , d f ( x ) = d P M f ( x ) = d P M f ( x ) ◦ d f ( x ) . Using the expansiond I M f ( x ) − d f ( x ) = d P M ˜ f ( x ) ◦ d ˜ f ( x ) − d P M f ( x ) ◦ d f ( x ) ≤ (d P M ˜ f ( x ) − d P M f ( x )) ◦ d ˜ f ( x ) + d P M f ( x ) ◦ ( d ˜ f ( x ) − d f ( x ))we conclude that (cid:107) d I M f ( x ) − d f ( x ) (cid:107) ≤ (cid:107) d P M ˜ f ( x ) − d P M f ( x ) (cid:107) (cid:107) d ˜ f ( x ) (cid:107) + (cid:107) d P M f ( x ) (cid:107) (cid:107) d ˜ f ( x ) − d f ( x ) (cid:107) . Since f ( x ) ∈ M and (cid:107) f ( x ) − ˜ f ( x ) (cid:107) < ε we have by Theorem 2.10 (cid:107) d P M ˜ f ( x ) − d P M f ( x ) (cid:107) ≤ (cid:18) τ + 1 τ − ε (cid:19) ε. Together with the fact that (cid:107) d P M f ( x ) (cid:107) = 1 we obtain (cid:107) d I M f ( x ) − d f ( x ) (cid:107) ≤ ε (cid:18) τ + 1 τ − ε (cid:19) (cid:107) d ˜ f ( x ) (cid:107) + (cid:107) d ˜ f ( x ) − d f ( x ) (cid:107) . This implies the assertion by triangle inequality.We found that the order of approximation of the differential d I M f of the manifold-valued approximant is the same as the order of the vector-valued approximant, as itwas already reported in [7]. However, the pre-asymptotic behavior depends strongly onreach of the embedding of the manifold M .12 .2. Fourier Interpolation In this section we want to illustrate Theorem 3.2 using Fourier-Interpolation as theapproximation operator I R d . More precisely, we define for a function f ∈ C ( T , R d ) onthe torus T the Fourier partial sum I R d f ( t ) = S n f ( t ) = n (cid:88) k = − n c k ( f ) e πikt with the vector-valued Fourier coefficients c k ( f ) = (cid:90) f ( x ) e − πikx d x. The Fourier-Interpolation satisfies the following well known approximation inequali-ties, cf. [26].
Theorem 3.3.
Let r ∈ N and f ∈ C r ( T , R d ) . Then (cid:107) f − S n f (cid:107) L ( T ) , ≤ √ d (2 π ) r √ nn r (cid:107) f ( r ) (cid:107) L ( T ) , , (cid:107) d f − d( S n f ) (cid:107) L ( T ) , ≤ √ d (2 π ) r √ nn r − (cid:107) f ( r ) (cid:107) L ( T ) , . Proof.
The first bound follows from the point-wise estimates in [4, Theorem 4.3], whichinduces the estimate in the L ( T )-norm. The second bound follows from d( S n f ) = S n ( d f ) and the fact that the regularity of d f is one less than the regularity of f . Thefactor √ d comes from the fact that the function f maps in the d-dimensional space.In [26, Theorem 1.39] a similar bound for the L ∞ ( T )-norm can be found. Using theFourier partial sum operator as the approximation operator I R d Theorem 3.2 becomesthe following.
Theorem 3.4.
Let τ be the reach of the manifold M , r ∈ N , f ∈ C r ( T , M ) and n ∈ N such that n > (cid:32) √ d (2 π ) r τ (cid:107) f ( r ) (cid:107) L ( T ) , (cid:33) r − . (3.1) Then we have (cid:107) I M f − f (cid:107) L ( T ) , ≤ √ d (2 π ) r n − r (cid:107) f ( r ) (cid:107) L ∞ ( T ) , , whereas for the differential we have (cid:107) d( I M f ) − d f (cid:107) L ( T ) , ≤ C n − r (cid:107) f ( r ) (cid:107) L ( T ) , + C n − r (cid:107) f ( r ) (cid:107) L ( T ) , . ith the constants C = √ d (2 π ) r ,C = √ d (2 π ) r τ + τ − √ d (2 π ) r n − r (cid:107) f ( r ) (cid:107) L T ) , (cid:18) √ d (2 π ) r n − r + 1 (cid:19) . Proof.
The condition in (3.1) ensures that S n f ( x ) is in the tubular neighborhood of M for every x ∈ T . The first bound follows directly from Theorem 3.1 together withTheorem 3.3. We use the L ( T )-norm instead of the L ∞ ( T )-norm, since the proof ofTheorem 3.2 shows a pointwise estimation. The second bound follows from Theorem 3.2together with Theorem 3.3. The first summand together with the constant C is thefirst term in Theorem 3.2. For the second term we have to choose ε = √ d (2 π ) r n − r (cid:107) f ( r ) (cid:107) L ( T ) , . As the torus T has the length 1 we have (cid:107) d f (cid:107) L ( T ) , ≤ (cid:107) f ( r ) (cid:107) L ( T ) , . Hence, we can use (cid:16) (cid:107) d S n f − d f (cid:107) L ( T ) , + (cid:107) d f (cid:107) L ( T ) , (cid:17) ≤ (cid:32) √ d (2 π ) r n − r (cid:33) (cid:107) f ( r ) (cid:107) L ( T ) , , which finally yields the assertion.Theorem 3.4 states that the approximation order of the manifold-valued Fourier partialsum operator coincides with the approximation order pf the vector-valued operator.The same holds true for the approximation order of the differential. However, the pre-asymptotic second summand, controlled by the constant C , grows to infinity when S n f − f is close to the reach of the manifold M . Thus, one has to assure that n issufficiently large, such that (cid:107) f ( x ) − S n f ( x ) (cid:107) is bounded away from τ .
4. Examples
In this section we apply our findings to two real world examples of manifold-valuedapproximation. Both examples are related to the analysis of crystalline materials. Inthe first example we consider functions that relate propagation directions of waves topolarization directions and in the second example we consider functions that relate pointswithin crystalline specimen to the local orientation of its crystal lattice.
In crystalline materials the propagation velocity and polarization direction of waves isoften isotropic, i.e., it depends on the propagation direction relative to the crystal lattice.This posses an important issue in seismology where one analyzes the distribution ofearthquake waves in order to get a deeper understanding of the core of the earth, cf.1418]. Each earthquake wave decomposes into a p-wave and two perpendicular shear-wavecomponents. The polarization vectors of p-wave components as well as of the two s-wavecomponents depend on the propagation direction of the wave relative to the crystal, cf.[24, 5]. Mathematically, the directional dependency of the polarization directions fromthe propagation direction is modeled as function f : S → R P from the two-sphere S into the two–dimensional projective space R P . Our goal is toapproximate this function from finite measurements y n = f ( x n ) ∈ R P , n = 1 , . . . , N .To this end, we identify the two dimensional projective space R P with the quotient S / ∼ with respect to the equivalence relation x ∼ − x and consider the embedding E : S / ∼ → R × , E ( x ) = xx (cid:62) . The reach of this embedding is τ = √ as we show inthe following lemma. Lemma 4.1.
The two dimensional projective space R P embedded into the space ofsymmetry × matrices E ( S / ∼ ) ⊂ R × has the reach τ = √ .Proof. Following [1, Thm. 2.2], we can estimate the reach by the following infimum τ = inf x (cid:54) = y ∈ S / ∼ (cid:107)E ( x ) − E ( y ) (cid:107) d ( E ( x ) − E ( y ) , T y S / ∼ ) . (4.1)Since in our setting in both spaces, S / ∼ and R × the metric is invariant with respectto the action of SO(3), it suffices to take the infimum for y = e = (1 , , (cid:62) . We definethe other canonical basis vectors in R as e = (0 , , (cid:62) and e = (0 , , (cid:62) The tangentvector space in e is then given by these tangent vectors: T (1 , , (cid:62) S / ∼ = span (cid:26) √ e e (cid:62) + e e (cid:62) ) , √ e e (cid:62) + e e (cid:62) ) (cid:27) . Hence, we write v = E ( x ) − E ( e ) = xx (cid:62) − e e (cid:62) and therefore we can calculate thereach by τ = inf x (cid:54) = e ∈ S / ∼ (cid:107) v (cid:107) (cid:107) v − (cid:104) v , √ e e (cid:62) (cid:105) − (cid:104) v , √ e e (cid:62) (cid:105)(cid:107) = inf x (cid:54) = e ∈ S / ∼ − x (cid:112) − x − x x − x x = inf x (cid:54) = e ∈ S / ∼ − x √ (cid:112) (1 − x ) = 1 √ , which finishes the proof.The calculation of the reach gives us the constants in Theorem 3.1 and 3.2 for thisspecific manifold. 15igure 1.: Polarization directions of the fastest shear wave in dependency of the propaga-tion direction: theoretical function values (left) with sampling points markedred, approximated values (middle) and approximation error (right).Let g : S → R × , g = E ◦ f be the embedded function. A common method, cf. [19], ofapproximating the spherical function g is by linear combinations of spherical harmonics Y (cid:96),k , (cid:96) = 0 , . . . , L , k = − (cid:96), . . . , (cid:96) up to a fixed bandwidth L , S L g ( x ) = L (cid:88) (cid:96) =0 (cid:96) (cid:88) k = − (cid:96) c (cid:96),k Y (cid:96),k ( x ) , where the coefficients c (cid:96),k ∈ R × , k = − (cid:96), . . . , (cid:96) , (cid:96) = 0 , . . . , L are elements of theembedding space. In our little example we simply assume that the measurement points x n together with some weights ω n form a spherical quadrature rule up to degree 2 L which allows us to determine the Fourier coefficients c (cid:96),k , by c k,(cid:96) = N (cid:88) n =1 ω n E ( y n ) Y (cid:96),k ( x n ) . Fig. 1 displays the theoretical polarization directions of an Olivine crystal in depen-dency of the propagation direction. We clearly observe the two points of singularity atthe hemisphere displayed. In order to approximate this non-smooth function we fixedthe bandwidth L = 4 and used Chebyshev quadrature nodes x , . . . , x ∈ S as sam-pling points, cf. [8]. The approximated function P R P ◦ S g is depicted in Fig. 1 (middle)and shows very good approximation with the original function away from the singularitypoints. This corresponding approximation error on R P is shown in the right figure. The subject of crystallographic texture analysis is the microstructure of polycrystallinematerials. Locally the microstructure is described by the orientation of the atom latticewith respect to some specimen fixed reference frame. More specifically, one describesthe local orientation of the atom lattice by a coset [ R ] S ∈ SO(3) / S of the rotation groupSO(3) modulo the finite subgroup S ⊂
SO(3), called point group. The point group of acrystal consists of all symmetries of its atom lattice and is either one of the cyclic groups16 a) global color key (b) local color key
Figure 2.: The raw EBSD data. Each square corresponds to a single orientation mea-surement at the surface of the specimen. The color is computed by proceduredescribed in [23]. C , C , C , C , C , the dihedral groups D , D , D , D , the tetragonal group T or theoctahedral group O . Assuming a monophase material, i.e., a material consisting only ofa single type of crystals, the variation of the local orientation of the atom lattice at thesurface Ω ⊂ R of the specimen is modeled by the map f : Ω → SO(3) / S . The gradient of the function f , also called lattice curvature tensor κ ( x ), is closelyrelated to elastic and plastic deformations the specimen has been exposed to. Morespecifically, it is related via the Nye equation to the dislocation density tensor α ( x ),[25, 15] that describes how many lattice dislocations are geometrically necessary in orderto preserve the compatibility of the lattice for a given deformation. Hence, estimating f and its derivatives from experimental data is a central problem in material science.Electron back scatter diffraction (EBSD) is an experimental technique [2, 16] fordetermining the local lattice orientations f ( x (cid:96) ) ∈ SO(3) / S at discrete sampling points x i,j ∈ Ω. An example of such EBSD data is the SO(3) / S - valued image displayed inFig. 2. It describes the variation of lattice orientation at the surface of an Aluminumalloy. The symmetry group in this case is the octahedral group O . The data is displayedwith respect to two different color keys. In Fig. 2a the colors are assigned globally to thecosets f ( x ) ∈ SO(3) /O as described in [23]. Regions of similar lattice orientation formso-called grains as outlined by the black boundaries. In order to make small changes oflattice orientation visible we computed for each grain an average lattice orientation andcolorized the map according to the deviation of lattice orientation in each pixel withrespect to the grain average lattice orientation in Fig. 2b, cf. [31].Estimating the derivative from such an noisy map of lattice orientations is usuallynot a good idea as it is illustrated in Figure 4a. Reducing the noise be means of localapproximation methods has been discussed in [14, 28]. In order to demonstrate our em-17igure 3.: Approximated EBSD data.bedding based approximation approach we make use of the locally isometric embedding E O : SO(3) /O → R described in [13] and proceed as followsi) Compute an R -valued image u i,j = E O ( f ( x i,j )).ii) Approximate the R -valued image using a cosine series ˜ u : Ω → R computed byrobust, penalized least squares [6].iii) Evaluate the function u at the grid points x i,j to obtain a noise reduced R -valuedimage ˜ u i,j .iv) Compute the projection of ˜ u i,j onto the embedding E O (SO(3) /O ) of the quotientand apply the inverse map E − O to end up with a noise reduced SO(3) /O -valuedimage ˜ f ( x i,j ).The resulting image is depicted in Figure 3. For this embedding E O : SO(3) /O → R we determined the reach numerically using the formula (4.1) as τ = , which ensures ustheoretical bounds from Theorems 3.1 and 3.2.For the computation of the lattice curvature tensor κ we use the skew symmetricmatrices s (1) = −
10 1 0 , s (2) = −
10 0 01 0 0 , s (3) = − , to fix the basis Rs (1) , Rs (2) , Rs (3) in the tangential space T R SO(3) /O at some rotation R ∈ SO(3). With respect to this basis the differential D E ( R ) : T R SO(3) /O → R of the embedding E : SO(3) /O → R can be represented as a full rank 3 × D ˜ u : R → R of the embedded image ˜ u = E ◦ ˜ f : Ω → R at some point x ∈ Ω the matrix product D ˜ u ( x ) = D E ( ˜ f ( x )) Df ( x ).Hence, the lattice curvature tensor ˜ κ of the noise reduced EBSD map evaluates to˜ κ ( x ) = D ˜ f ( x ) = (cid:16) D E ( ˜ f ( x )) D E ( ˜ f ( x )) (cid:62) (cid:17) − D E (˜ uf ( x )) (cid:62) D ˜ u ( x ) . a) κ , of noisy map. (b) κ , of noise reduced map. Figure 4.: First coefficient κ ( x ) of the lattice curvature tensor of the SO (3) /O -valuedmap depicted in Fig. 2 (left) and Fig. 3 (right).The map of the first component ˜ κ , of the lattice curvature tensor obtained from theapproximating function ˜ u is depicted in Fig. 4b. For comparison we plotted in Fig. 4a afinite difference approximation κ ( x i,j ) = log f ( x i,j ) ( f ( x i +1 ,j ))[ x i +1 ,j − x i,j ] , derived from the discrete data f ( x i,j ). Here, we denoted by log R : SO(3) /O → T R SO(3) /O the logarithmic mapping with respect to the base point R ∈ SO(3) /O . As expected,we observe that the lattice curvature tensor ˜ κ derived from the approximated map ˜ u ismuch less noisy. Acknowledgments
The authors would like to thank Prof. Dr. Philipp Reiter for the nice hint for completingTheorem 2.9. The second author acknowledges funding by Deutsche Forschungsgemein-schaft (DFG, German Research Foundation) - Project-ID 416228727 - SFB 1410.
A. Bound for the commutator
To bound the term (cid:107) d P M ( m ) − d P M ( z ) (cid:107) in section 2.4 we need a lemma, which isbased on linear algebra. Lemma A.1.
Let T be a projection matrix and R be a rotation matrix. Then thereholds for the commutator (cid:107) T R − RT (cid:107) ≤ (cid:107) I − R (cid:107) , where again (cid:107)·(cid:107) denotes the spectral norm. roof. Since the spectral norm doesn’t change under change of basis, we choose a matrixrepresentation where the projection matrix has the form T = (cid:18) I
00 0 (cid:19) , where I is the identity matrix of dimension D . Then we also write the rotation matrix R in these blocks: R = (cid:18) R R R R (cid:19) . Simple matrix multiplication yields because of the orthogonality of R ( T R − RT ) (cid:62) ( T R − RT ) = R (cid:62) T R − R (cid:62) T RT − T R (cid:62)
T R + T = (cid:18) I − R (cid:62) R R (cid:62) R (cid:19) = (cid:18) R (cid:62) R R (cid:62) R (cid:19) On the other hand there holds, again with help of the orthogonality of R ,( I − R ) (cid:62) ( I − R ) = 2 I − R − R (cid:62) = (cid:18) I − R − R (cid:62) − R − R (cid:62) − R − R (cid:62) I − R − R (cid:62) (cid:19) = (cid:18) ( I − R (cid:62) )( I − R ) + I − R (cid:62) R − R − R (cid:62) − R − R (cid:62) ( I − R (cid:62) )( I − R ) + I − R (cid:62) R (cid:19) = (cid:18) ( I − R (cid:62) )( I − R ) + R (cid:62) R − R − R (cid:62) − R − R (cid:62) ( I − R (cid:62) )( I − R ) + R (cid:62) R (cid:19) . The spectral norm of a matrix A , i.e., the largest absolute value of the eigenvalues canbe written as (cid:107) A (cid:107) = max (cid:107) x (cid:107) =1 (cid:107) x (cid:62) A (cid:62) Ax (cid:107) . For that reason we choose the vector x as the eigenvector of the matrix ( T R − RT ) (cid:62) ( T R − RT ). Since the eigenvalues and eigenvectors of a block-diagonal matrix are the union ofthe eigenvalues and eigenvectors, i.e., there holds x = (cid:0) x (cid:1) (cid:62) or x = (cid:0) x (cid:1) (cid:62) . Weassume the first case, the other one is analog. Hence, there holds (cid:107) T R − RT (cid:107) = x (cid:62) (cid:18) R (cid:62) R R (cid:62) R (cid:19) x = x (cid:62) R (cid:62) R x . If we look at the norm of the matrix I − R , we get (cid:107) I − R (cid:107) ≥ x (cid:62) ( I − R ) (cid:62) ( I − R ) x = x (cid:62) (cid:0) ( I − R (cid:62) )( I − R ) + R (cid:62) R (cid:1) x ≥ x (cid:62) R (cid:62) R x , since the eigenvalues of ( I − R (cid:62) )( I − R ) are positive. Putting this together and takingthe square root, yields the assertion. 20 eferences [1] E. Aamari, J. Kim, F. Chazal, B. Michel, A. Rinaldo, and L. Wasserman. Estimat-ing the reach of a manifold. Electron. J. Statist. , 13(1):1359–1399, 2019.[2] B. L. Adams, S. I. Wright, and K. Kunze. Orientation imaging: The emergence ofa new microscopy.
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