Let a smooth function be given as well as a grid on of knots , , where denotes the knot distance or grid size. A standard approximation of is obtained by evaluating at the knots and then computing the cubic spline interpolant of the resulting data , , for instance with Hermite boundary conditions , . It is a classic result that this approximation is of quartic order, , proved for instance by de Boor  (analogous convergence results hold for spline interpolation of different order).
The well-known minimum curvature property of cubic splines states that cubic spline interpolants uniquely minimize the accumulated squared acceleration among all functions interpolating , , which allows to generalize the notion of a cubic spline interpolant to functions or curves with values in a smooth Riemannian manifold .
Definition 1 (Riemannian cubic spline interpolation).
Let be a continuous curve and set for and . A corresponding cubic spline interpolation of is defined as
Above, denotes the intrinsic second derivative, the covariant derivative of along the curve , and represents the Riemannian norm on the tangent space bundle . A natural question is whether the approximation properties of cubic splines transfer from the setting of real-valued functions to the manifold-valued generalization. Our main result is that they do.
Theorem 2 (Quartic convergence of Riemannian cubic spline interpolation).
Let be complete with bounded Riemann curvature and its derivatives. Let be four times differentiable. For small enough (depending on and )
the cubic spline interpolation exists and is unique,
and it satisfies
where the constant only depends on , , and .
Above, denotes the Riemannian distance and the -norm of the function for . Note that the conditions for existence and uniqueness can be a little relaxed, see proposition 9. One could envision two strategies to arrive at this approximation result. One strategy would be to reduce the problem to the real-valued case by reformulating the Riemannian cubic spline interpolation as an interpolation problem in a linear space (using charts or a tangent space to the manifold) which can be viewed as a perturbation of Euclidean or real-valued cubic spline interpolation. The complementary strategy consists in the generalization of all ingredients to the Euclidean analysis to the Riemannian setting, using a mostly coordinate-free intrinsic Riemannian formulation. We will follow the latter strategy (but also discuss the former along the way). In particular, the Euclidean proof in  makes use of the fact that the second derivative of the cubic spline interpolant is the -best approximation to the the second derivative of , and for our proof in the nonlinear manifold setting we will quantify the deviation from a Riemannian generalization of that property. This will result in an -bound for the second derivative
, which is crucial for the convergence estimate.
For the sake of completeness, along the way we will deal with linear spline interpolation since it is a widespread alternative for applications in Riemannian manifolds and since its convergence properties follow without additional effort.
Definition 3 (Riemannian linear spline interpolation).
Let be a continuous curve and set for and . A corresponding linear spline interpolation of is defined as
Theorem 4 (Quadratic convergence of Riemannian linear spline interpolation).
Let be complete with bounded Riemann curvature and its derivatives. Let be twice differentiable. For small enough (depending on and )
the linear spline interpolation exists and is unique,
and it satisfies
where the constant only depends on , , and .
Higher order spline interpolation of odd degree could in principle be tackled in a similar way as cubic spline interpolation, but of course requires more technical estimates.
In section 2 we will summarize the required basic Riemannian notions and our notation after which we prove the well-posedness of (linear and) cubic spline interpolation in section 3 and derive the corresponding optimality conditions. We then present two convergence proofs for the classical Euclidean setting in section 4 before adapting these to the Riemannian setting in section 5.
2 Required Riemannian notions and preliminaries
Throughout we will assume to be a smooth, complete, connected, finite-dimensional Riemannian manifold with Riemannian metric
such that the Riemann curvature tensor and all its derivatives are bounded. The tangent space at a pointwill be denoted , and for we will write
where the explicit dependence on the base point is suppressed for better readability. The induced Riemannian distance is denoted by , the Levi-Civita connection on is denoted by , and the Riemann curvature tensor is denoted by . For an absolutely continuous curve we will denote the parameterization variable by and its velocity by
(time derivatives of functions into a vector space will also be denoted with powers ofor dots). A lift of the curve is a mapping from into the tangent bundle such that for all ; the vector space of all liftings is abbreviated . For a vector field along the curve and we will write
where with a slight misuse of notation the covariant derivative of along shall be
with being the parallel transport along from to (start and end point of the parallel transport along a curve will usually be clear from the context). We will also write
for . A family of curves will be parameterized by the variable , and analogously to before, for a family of vector fields along the curves we introduce the notation
where is parallel transport along the curve . Below we collect a few classical Riemannian calculus rules to be used in the sequel.
Lemma 5 (Differentiation rules).
Parallel transport along a curve commutes with differentiation along the curve in the following sense. Let be differentiable and denote by the parallel transport along , then for any differentiable we have
Let be a differentiable family of differentiable curves, then .
Let be a differentiable family of differentiable curves and differentiable, then .
For all we have
Since the Levi-Civita connection is torsion-free, we have for the Lie bracket of vector fields. Now for any smooth function we have
where the last term is zero as shown in the previous point. ∎
For a Lebesgue measurable function on some measurable domain we denote its -norm for by
If the domain is not clear from the context, we will write instead. Similarly, for a function we introduce its Hölder norm of exponent by
where is parallel transport to along .
Finally, the Riemannian exponential with base point is denoted by . Letting be the injectivity radius of , the inverse of can be defined on the geodesic ball of radius and is called the Riemannian logarithm . We close by providing some bounds on the Riemannian logarithm and exponential.
Lemma 6 (Bounds on Riemannian logarithm and exponential).
Let be a -manifold, , with injectivity radius , sectional curvature bounded from above by , and bounds on and on . Then the bivariate mapping is in for . If , assume additionally . The operator norms of the first derivatives of satisfy
where denotes parallel transport along the geodesic from to . The covariant second order derivatives, denoted by , satisfy
for a constant depending only on and . Similarly, the bivariate mapping is in , and for small enough, depending only on , , and , , it satisfies
The proof for the derivatives of the logarithm follows from Jacobi field estimates and can be found in [5, Prop. A.1-A.2] (the estimates for the first order derivatives can also be found in ). The estimates for the derivatives of the exponential are then straightforward applications of the inverse function theorem.
3 Well-posedness and Euler–Lagrange equations of Riemannian spline interpolations
All throughout the article, will be a given curve, times differentiable ( for linear and for cubic spline interpolation) in the sense that is continuous and is well-defined and bounded for . Furthermore, will denote the spline interpolation according to definition 1 or definition 3 at points , , where .
Remark 7 (Boundary conditions for Riemannian cubic spline interpolation).
In definition 1 we chose to impose the so-called Hermite boundary conditions . An alternative would be to solely require at all interpolation points. This is known to result in so-called natural boundary conditions, essentially a vanishing acceleration of at and . Unfortunately, though, existence of cubic spline interpolations with natural boundary conditions cannot be guaranteed due to lacking control of the curve velocity (a simple example is provided in ).
We begin with straightforward a priori bounds that follow from the coercivity of the spline interpolation energy, which is nothing else but in the case of linear splines and in case of cubic splines. Boundedness of higher derivatives for the cubic spline interpolation will be proven later.
Proposition 8 (A priori bounds).
Let be an absolutely continuous curve.
If is bounded, then is Hölder continuous with exponent , and for any we have
If is bounded, then is even Hölder continuously differentiable with exponent and satisfies
For any , by Hölder’s inequality we have
so that is indeed Hölder continuous with exponent . Furthermore, this implies
from which the estimate follows.
Obviously, an immediate consequence is that the linear spline interpolation of , if it exists, is Hölder continuous with
and that the cubic spline interpolation, if it exists, is Hölder continuously differentiable with
Proposition 9 (Well-posedness of Riemannian spline interpolation).
Linear and cubic spline interpolations of exist. Furthermore, they are unique if is continuous (for linear spline interpolation) or has finite (for cubic spline interpolation) and is small enough (depending on and ).
Obviously, a linear spline interpolation is obtained by minimizing for each the energy
among all curve segments that satisfy and . Those curve segments are known to be exactly the constant speed-parameterized geodesics connecting and , which always exist on a complete Riemannian manifold and which are unique if the end points are close enough to each other (depending on ). By uniform continuity of the latter condition will be fulfilled for all as soon as is small enough.
As for the well-posedness of the cubic spline interpolation, consider a minimizing sequence , , of curves satisfying the interpolation conditions such that the cubic spline energy converges monotonically to the infimum, . Letting denote parallel transport to along a curve , we have
so that the functions have uniformly bounded -seminorm. Together with it follows by Poincaré’s inequality that the functions even have uniformly bounded -norm so that a subsequence converges weakly in (and thus strongly in ) to some function . Now define
as the solution to the ordinary differential equation
with initial value , where we write for the parallel transport of to along . The solution exists and is unique by the theorem of Picard and Lindelöf, and it satisfies the interpolation conditions by Gronwall’s lemma. Indeed, let us express all quantities in local coordinates and indicate this by a hat. Further, let denote the matrix representation in local coordinates of . Then solves the initial value problem
where is the Christoffel operator at coordinates and the second entry of refers to the matrix defined by . The function is obviously Lipschitz so that the Picard–Lindelöf theorem can be applied (it is straightforward to show that and stay bounded for ). Since satisfies the same initial value problem, only with replaced by , the curves converge uniformly on to by Gronwall’s lemma, as do the corresponding . Consequently, satisfies the interpolation conditions. Finally note that
due to the sequential weak lower semi-continuity of the -norm so that minimizes under the interpolation constraints.
It remains to prove uniqueness of . To this end we will show local convexity of the problem for small enough. Indeed, for small enough the interpolation problem can equivalently be formulated in terms of functions
for , which by lemma 6 implies for small enough. Now introduce
which equals . Due to the above bounds on and we can apply lemma 6 to obtain for small enough. Thus, the cubic spline interpolation problem in terms of the can thus be written as
It is obvious that is convex, so it remains to show convexity of on . Abbreviating for , the Gâteaux derivatives of in a direction with and are
Now by lemma 6,
for a constant depending only on . Now recall that for we have