1 Pole ladder in an affine connection spaces
This section establishes a Taylor expansion of the approximation provided by one step of pole ladder. We first detail the pole ladder algorithm. We then turn to an equivalent on affine manifolds of the BCH formula for Lie group before establishing the approximation order of pole ladder.
1.1 Pole ladder algorithm
Pole ladder is a modification of Schild’s ladder for the parallel transport along geodesic curves which is based on the observation that this geodesic can be taken as one of the diagonals of the geodesic parallelogram. The general procedure is described in Fig.2: the vector (rescaled if needed to be small enough) is identified to the geodesic segment for . One compute the mid-point between and and then extend twice the geodesic from to to obtain the point . The vector is the pole ladder approximation of the parallel transport of at . An alternative is to first compute and then extend twice the geodesic from to to obtain the point . The vector is another approximation of the parallel transport of at .
1.2 Algorithms for one step of the ladder
Pole ladder assumes that the successive points of the geodesic curve are in a sufficiently small convex normal neighborhood so that this mid-point is unique. In order to analyze one single step of the ladder along a geodesic segment, we simplify the notations in the sequel according to figure 3. The original pole ladder procedure is summarized in Algorithm 1.
Instead of encoding a geodesic by its initial point and initial tangent vector , we can also encode a geodesic segment by its end-points . We can also rephrase the doubling of the geodesics with a mid-point formulation. This leads to an alternative version of pole ladder using mid-point geodesic symmetries (Algorithm 2). Although the two versions of pole ladder are theoretically completely equivalent, it was experimentally observed that this second version was numerically much more stable when applied to diffeomorphisms acting on images [JDM18]. This is most probably be due to the more symmetric formulation that minimizes numerical errors.
Notice that the mid-point of a geodesic segment is well defined even in affine connection spaces: this is the point of coordinate in arc-length parametrization. This is in particular an exponential barycenter: . In a Riemannian manifold, it would also be a minimizer of the square distance to the two points, necessarily a Fréchet mean if we assumed that all our points belong to a sufficiently small convex neighborhood.
1.3 A BCH-type formula on affine connection spaces
In order to analyze the pole ladder approximation of parallel transport in a local coordinate system, we need to compute the Taylor expansion of the parallel transport along a geodesic segment and of the composition of two Riemannian exponentials. Low order Taylor expansion of the Riemannian metric and of geodesic equations are traditional in Riemannian geometry since Gauss to establish the existence of normal coordinate system, Gauss lemma or the infinitesimal change of the volume of a geodesic ball due to the curvature. This is also what is used for the proof of [KMN00] for Schild’s ladder. However, establishing expansions above the order 3 is much more difficult, and it turns out that we need even more for the pole ladder. One solution was provided by [Bre97, Bre09] with the symbolic computer algebra system Cadabra. In this setting, the starting point is the (pseudo)-Riemannian metric, and it is not so easy to prove that the formulas continue to hold for affine connection spaces. More recently, a simpler algebraic formulation of higher order covariant derivatives in affine connection spaces [Gav06] led Gavrilov to obtain a combined Taylor expansion of the parallel transport and the composition of exponentials [Gav07]. After recalling this result in our notations, we adapt this formula to the analysis of pole ladder below.
We consider a small convex normal neighborhood of a point in an affine connection space . In a Riemannian manifold, we can chose a sufficiently small regular geodesic ball (see [Kar77, Ken90]). This means that any two points of the neighborhood can be can be joined by a unique geodesic segment and we can define the parallel transport of a tangent vector along this geodesic. The midpoint of each geodesic segment is also well defined in this neighborhood. The double exponential is defined as the mapping :
For sufficiently small vectors , one can define the log of this expression (see Figure 4 for notations), which is a formal analog of the Baker-Campbell-Hausdorff (BCH) formula for Lie groups.
Let be an affine connection space with vanishing torsion and Riemannian curvature tensor . In a sufficiently small neighborhood of , the log of the double exponential:
has the following series expansion:
If the connection is not symmetric, then the torsion appears as a second order term and in all higher order terms.
In the above equations, denotes homogeneous terms of degree 5 and higher in and (the norm of can be taken arbitrarily), and the curvature tensor value is taken at . The full series expansion with torsion up to order 5 is provided in [Gav06].
1.4 Taylor expansion of pole ladder
We now have the tools to study the second version of pole ladder (algorithm 2). We consider that all points belong to a sufficiently small convex normal coordinate system centered at with the conventions of Figure 5. The geodesic starting at with tangent vector is a straight line in this chart going from point at time -1 to point at time 1. We denote by the parallel transport of the vector along the geodesic segment . This is equivalent to . We denote by the coordinates of in the normal coordinate system at : with . Using the previous BCH formula, we have:
Notice that the geodesic from to deviates from the straight line in the normal coordinate system at chart because of curvature.
We have an equivalent construction for the symmetric part of the structure: the symmetric of with respect to is . Taking the log at , we get the opposite of the pole ladder transport . In order to compare this vector to the exact parallel transport in a symmetric way, we now parallel transport it along the geodesic segment to get , that has to be compared with . Rephrasing this combination of transformations in the reverse way, we get that is the solution of with . Using Gavrilov’s formula, we can write the following polynomial expansion of in the variables and :
However, what we want to obtain is the polynomial expansion of with respect to the variables and . Such a polynomial can be written:
where each term is a multi-linear map in the indexed variables. Plugging this expression into Gavrilov’s formula above, we can identify each term iteratively for increasing orders: the first order is trivially and the second order terms have to vanish because there are none in the BCH-type formula. Introducing the third order into Gavrilov’s formula, we get:
Simplifying the expression thanks to the skew symmetries, we obtain:
The forth order is obtained by:
which gives after simplification:
Finally, we can substituting the value of from Equation (3) into this expression. This leads to:
Theorem 2 (Pole ladder is a third order scheme for parallel transport).
In a sufficiently small convex normal neighborhood of a point in an affine connection space with symmetric connection, the error on one step of pole ladder to transport the vector along a geodesic segment of tangent vector (all quantities being parallel translated at the mid-point, see Figure 5) is:
Instead of performing first a symmetry at and then at , an alternative version of pole ladder is to first perform a symmetry at and then at . One could think of averaging the two versions of the pole ladder to get a more accurate transport. The error on this alternative version is:
Unfortunately, we see that averaging does not kill all the third order terms.
2 Pole ladder in symmetric spaces
Theorem 2 shows that the main source of error in the pole ladder procedure is not the curvature itself but its covariant derivative. Since a vanishing covariant derivative is the characteristic of a locally symmetric space, this suggests that the error could vanish also in higher order terms. Thus, we investigate in this section local and global symmetric spaces with affine or Riemannian structure. The description of these spaces closely follows the definitions of [Pos01][Chapter 4]. The interested read should also refer to [Hel62]. We show that pole ladder is actually locally exact in one step, and even almost surely globally exact in Riemannian symmetric manifolds. The key feature is that the differential of the symmetry is the negative of the parallel transport, so that the parallel transport along a geodesic segment can be realized using the composition of two symmetries, which is called a transvection. Even if this result was not really known in the geometric computing community, the construction was already pictured for instance in [Tro94, Fig.5, p.168]).
2.1 Locally symmetric affine connection spaces
An affine connection space is said to be locally symmetric if the connection is torsion free and if the the curvature tensor is covariantly constant (). This second condition is equivalent to the preservation of the curvature tensor by the parallel transport along any path. When the manifold is endowed with a (pseudo-) Riemannian metric, one says that it is a locally symmetric (pseudo-) Riemannian space if the covariant derivative of its curvature tensor with respect to the Levi-Civita connection vanishes identically.
Locally symmetric affine connection spaces can be characterized by the geodesic symmetry which maps any point of a normal neighborhood of to the reverse point on the geodesic . The differential of this mapping is clearly a central symmetry on (a symmetric neighborhood of 0 in) the tangent space at : , and can thus be seen as a midpoint of the geodesic segment using the exponential barycenter definition since .
Using this geodesic symmetry, pole ladder can be rewritten as the composition of a symmetry at followed by a symmetry at (or alternatively as a symmetry at followed by a symmetry at ). Such a composition of two symmetries preserves the orientation and is called a transvection (or simply a translation). Moreover, its differential realizes the parallel transport:
Theorem 3 (Pole ladder is locally exact in locally symmetric spaces).
In a locally symmetric affine connection space with geodesic symmetry at point , the pole ladder transport along the geodesic segment of the tangent vector (identified to the geodesic segment ) is the tangent vector (identified to the geodesic segment ). Alternatively, one can start by the symmetry at to compute . For a sufficiently small time , both procedure realize exactly the parallel transport.
We consider two points and that are sufficiently close in a symmetric normal neighborhood of their midpoint . The geodesic starting at with tangent vector remains in for small enough. The geodesic symmetry is an affine mapping in that maps geodesics to geodesics [Pos01][Chapter 3]. Thus, it maps to a geodesic starting at with tangent vector . Now, the key property is that the differential of the central symmetry at is is a linear maps from to which realizes exactly the (negative of) the parallel transport along the geodesic segment : . This property is described in [Hel62, Eq.(6), p.164] and in the proof of Proposition 4.3 of [Pos01, p.53-54]. The second symmetry is well defined for a sufficiently small time and reverses the sign. ∎
Because we only assumed so far a locally symmetric affine connection, the size of the normal neighborhood containing and and the maximal time cannot be specified without an auxiliary metric. More traditional symmetric (pseudo)-Riemannian spaces provide additional tools to extend this result.
2.2 Riemannian symmetric spaces
A connected affine connection space is a (globally) symmetric space if there exists at each point a smooth involution for which is an isolated point (i.e. a symmetry) and which is stable by composition: for any two points , . A symmetric space is geodesically complete and is obviously locally symmetric. The converse holds modulo discrete quotients: in essence, for each connected and geodesically complete locally symmetric space, there exists a symmetric covering space, which can be taken to be simply connected. Given a symmetric space, an even number of composition of symmetries generate a translation (or transvection) on the manifold. The Lie group of translations is acting transitively on , and its quotient space by the isotropy subgroup of one point of the manifold can be identified with the symmetric space itself.
Conversely, Cartan theorem shows that a homogeneous space with an involutive automorphism satisfying specific properties detailed below is a symmetric space. Because is an involution, its differential
has eigenvalueswhich decompose the Lie algebra into the direct sum
: the +1-eigenspaceis the Lie algebra of , and the -eigenspace is an -invariant complement to in that can be identified to the tangent space at . These two conditions read in the Lie theoretic characterization and , which means that a symmetric space is a reductive homogeneous space. However, reductive homogeneous spaces are not all symmetric: the key feature is that (the composition of two symmetries on has to be a translation on ).
A natural structure on a symmetric space has to be invariant under its symmetry. This condition uniquely define a symmetric connection on each symmetric space. For instance, a connected Lie group is a symmetric space with the symmetry for two points of the group . The associated canonical connection is the mean (or symmetric) Cartan-Schouten connection which was used for defining bi-invariant means on Lie groups even in the absence of an invariant metric [PA12]. On a more general symmetric space , the canonical connection is induced by the symmetric Cartan-Schouten connection on the Lie group of translations , and geodesics of the symmetric space with this canonical connection are realized by the action of one-parameter subgroups of on a point of the manifold.
When is compact, one can moreover obtain an invariant metric on (simply choose a metric on the Lie algebra and average it over ). We obtain in this case a Riemannian symmetric manifold where the symmetry is now an isometry. The Levi-Civita connection of this invariant Riemannian metric coincides with the canonical connection. The invariant metric need not be uniquely define up to a scalar factor. For instance, that there is a one-parameter family of -invariant metrics on SPD matrices [Pen06].
With an invariant Riemannian metric, we can define the maximal injectivity domain of the exponential map: let be the set of all vectors such that is a minimizing geodesic for for some . This is an open star-shape domain containing limited by the tangential cut locus where geodesics stop to be minimizing. The image of the tangential cut locus by the exponential map is the cut locus , which has null measure with respect to the canonical Riemannian measure. The maximal injectivity domain is mapped diffeomorphically by to the normal neighborhood of in the manifold. The shortest distance from a point to its cut locus is the injection radius . Since a Riemannian symmetric space is homogeneous, this radius does not depend on the point and is equal to its infimum over all the points . These definitions allows us to characterize the exactness of pole ladder transport in symmetric spaces.
Theorem 4 (Pole ladder is almost surely exact in Riemannian symmetric manifolds).
In a Riemannian symmetric manifold, pole ladder is exact when and . This happens almost surely since the cut locus has null measure. More restrictive metric conditions of exactness are and .
The first condition is needed to have a well defined mid-point. The second condition ensures that the geodesic does not meet the cut-locus of for , so that the initial tangent vector of the geodesic from to remains . Thanks to the global symmetry , similar conditions hold for and , so that is the exact parallel transport of along the geodesic segment . ∎
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