Nonlinear Markov semigroups and refinement schemes on metric spaces.
This article settles the convergence question for multivariate barycentric subdivision schemes with nonnegative masks on complete metric spaces of nonpositive Alexandrov curvature, also known as Hadamard spaces. We establish a link between these types of refinement algorithms and the theory of Markov chains by characterizing barycentric subdivision schemes as nonlinear Markov semigroups. Exploiting this connection, we subsequently prove that any such scheme converges on arbitrary Hadamard spaces if and only if it converges for real valued input data. Moreover, we generalize a characterization of convergence from the linear theory, and consider approximation qualities of barycentric subdivision schemes. A concluding section addresses the relationship between the convergence properties of a scheme and its so-called characteristic Markov chain.
Key words and phrases:Hadamard space; conditional expectation; Markov chain; barycentric subdivision scheme
2010 Mathematics Subject Classification:53C23, 60J20, 65D17
Introduction and main results
The convergence and smoothness analysis of refinement schemes processing data from manifolds and more generally metric spaces has been a subject of intense research over the last few years. As to convergence, complete spaces of nonpositive curvature have proven most accessible in terms of generalizing well-known facts from the linear theory to the nonlinear setting. An example of such a structure prominent in applications is the space of positive definite symmetric matrices, which represent measurements in diffusion tensor imaging.
While the question whether the smoothness properties of the linear model scheme prevail when passing to the nonlinear setting was successfully addressed in , the corresponding convergence problem remained unsolved. The aim of this article is to fill this gap in the theory.
Relying on a martingale theory for discrete-time stochastic processes with values in negatively curved spaces developed in , we observe that the refinement processes in question actually act on bounded input data as nonlinear Markov semigroups. This fact will substantially facilitate their convergence analysis.
Let us specify the general setup. Given a metric space , a refinement scheme is a map . We call convergent if for all there exists a continuous function such that
Visualizing as a function on the refined grid , convergence to is tantamount to .
Throughout the present paper we are mostly concerned with so-called barycentric refinement schemes associated to nonnegative real-valued -variate sequences of finite support, henceforth referred to as masks, which we require to fulfill the basic sum rule
Barycentric refinement schemes act on data from a complete metric space of nonpositive curvature in the sense of A. D. Alexandrov according to the following rule:
Much is known about the convergence of these type of refinement algorithms in the case . On complete manifolds of nonpositive sectional curvature convergence analysis was initiated in the article . The author in  recently proved general convergence statements for arbitrary Hadamard spaces using the principle of contractivity: A scheme is called contractive with respect to some nonnegative function if and only if there is such that
The function is referred to as a contractivity function for . An important class of contractivity functions is associated to balanced, convex and bounded subsets of :
where denotes the Minkowski functional of . Contractivity functions of this type are called admissible, cf. . The following result is taken from loc. cit.:
A barycentric refinement scheme with nonnegative mask which is contractive with respect to some admissible contractivity function also converges. This implies convergence in case the support of the mask coincides with the set of lattice points within a centered unimodular zonotope or a lattice quad with nonempty interior.
The main result of the present article is a substantial extension of this statement and describes a phenomenon which could be referred to as linear equivalence:
A barycentric refinement scheme converges on arbitrary Hadamard spaces if and only if it converges on the real line.
The proof of this fact, given in Section 2, relies on a stochastic interpretation of the subdivision rule (3). More precisely, for each nonnegative mask satisfying the basic sum rule (2) one finds a so-called characteristic Markov chain with state space and transition matrix in terms of which the iterates of the refinement algorithm acting on may be written as
may be considered a (nonlinear) Markov semigroup. Here refers to the set of maps satisfying the Lipschitz condition
Combining Theorem 1 with other recent developments in the theory of linear subdivision schemes with nonnegative masks and their barycentric counterparts on nonlinear objects, one comes up with a variety of remarkable results:
A barycentric subdivision scheme with nonnegative mask converges under each of the following circumstances:
(i) The support of coincides with the set of grid points inside a balanced zonotope.
(ii) The grid dimension and, if, after a possible index translation, , the integers within the support are relatively prime and . This also constitutes a necessary condition for convergence.
On a smooth Hadamard manifold, a barycentric subdivision scheme with nonnegative mask converges and produces -times differentiable limit functions if and only if the same is true for the corresponding linear scheme.
1. Stochastic preliminaries
This section is devoted to a stochastic interpretation of the subdivision rule (3). More precisely, we view barycentric subdivision as the semigroup acting on associated to the so-called characteristic Markov chain of . This result requires some prerequisites about the notion of conditional expectation of random variables with values in Hadamard spaces. This type of metric space was first investigated by A.D. Alexandrov in his seminal articles  and . In loc. cit., Alexandrov uses the Gauss-Bonnet theorem, i.e. the fact that the deficiency of the angle sum in a geodesic triangle can be expressed by means of the ambient space’s curvature, to generalize the notion of curvature bounds to a purely metric setting. A comprehensive introduction to the nowadays well-established theory of such spaces is . Furthermore we refer to the survey article  on probability measures and their centers of mass on Hadamard spaces. A treatise of the smooth case can be found in , which investigates infinite-dimensional Riemannian manifolds of nonpositive curvature.
Definition 1 (Hadamard spaces).
A complete metric space is called Hadamard- or global NPC-space if and only if for one finds such that the so-called Hadamard inequality holds true for all :
In other words, a Hadamard space is a complete metric space with nonpositive curvature in the sense of A. D. Alexandrov. The Hadamard inequality (6) expresses the fact that geodesic triangles are ’slim’ compared to Euclidean ones with the same edge lengths.
Recall that a continuous curve , , is called geodesic if and only if for all . A well-known property of Hadamard spaces is that they are strongly geodesic: given any two points , one finds a unique geodesic joining them. In particular, there is meaningful notion of convexity. More precisely, a subset of a global NPC space is called convex if and only if for each two , the joining geodesic remains in . Nonpositive curvature turns out to have a major impact on the convexity properties of the metric . As announced above, we are particularly interested in the concept of conditional expectation for random variables with values in Hadamard spaces. Although several approaches appear in the literature, a definition due to K.-T. Sturm, see , serves our purposes best. Sturm’s definition relies on a convex projection property enjoyed by NPC spaces:
Suppose is a global NPC space, and a closed and convex subset. Then there exists a well-defined projection map defined by the relation
This projection is Lipschitz-continuous in a sense that for .
Given a probability space and a metric space , a strongly measurable function is called square-integrable if and only if
for one (and then all) . The space of square-integrable functions factorized by the relation of being equal almost surely is endowed with a metric
In case is a global NPC space, it is well-known that inherits the Hadamard property. Moreover, given a subalgebra , it is easy to show that , the subspace of corresponding to -measurable square-integrable maps, is closed and convex. In view of Proposition 2 one thus obtains the following Definition:
Definition 2 ().
Suppose is a probability space, and let be a square-integrable random variable with separable image in the Hadamard space . Given a subalgebra , there is a -measurable minimizing the functional
among all -measurable square-integrable random variables. Any other minimizer coincides with almost surely. One refers to as the conditional expectation of given and uses the notation .
Otherwise put, following the notation of Proposition 2, . This definition follows the principle that in the real-valued case, the conditional expectation as introduced by Kolmogorov restricts to as the orthogonal projection to the space of -measurable -functions.
The space is, as in the case , defined to be the space of strongly measurable functions such that modulo equality almost surely. Again comes with a metric of the form
The conditional expectation as defined above is continuous in a sense that for each two and ,
In particular, extends continuously to .
It is inherent in the existence statement provided by Definition 2 that each integrable random variable with values in possesses an expected value defined by
which is the unique minimizer of the functional
Recall the elementary smoothing property of conditional expectations for random variables with values in a linear space: Given an ordered pair of subalgebras , one has
almost surely. Not surprisingly, this associativity property is violated in the nonlinear case, see Example 3.2 in . Actually a generalization of this feature would render the theory of nonlinear subdivision schemes of type (3) obsolete, as the discussion following Theorem 5 below illustrates.
It is for the reason of lacking associativity that K.-T. Sturm in his treatise  defines:
Suppose is a sequence of subalgebras. Furthermore assume . Then one defines the filtered conditional expectation of given as
Let us briefly recall the basic construction of a Markov chain, beginning with a foundational definition. Recall that a family of maps
parametrized by nonnegative integers , is called Markov transition kernel if the following are fulfilled:
is a probability measure for all
The Chapman-Kolmogorov equations are fulfilled: for each one has
We use the notation .
Let denote the space of sequences on the grid , endowed with the infinite power of the discrete sigma algebra on . Consider the filtration
Choose an initial distribution on , and introduce a probability measure on via
and the standard extension theorems. Whenever equals , the point measure on , we will write . Expected values of integrable random variables with respect to are, as usual, written as
Finally, the discrete stochastic process
constitutes a Markov chain associated to the transition kernel , meaning that the linear Markov property holds true: For any nonnegative one has
In particular, this linear Markov property allows for the interpretation of as the transition probability .
In view of the convergence analysis of barycentric subdivision schemes, it is of particular interest to gain a deeper understanding of principle of conditioning in case the filtration stems from a Markov chain. A nonlinear Markov property analogous to (9), see [10, Theorem 5.2], leads to a representation of the conditional expectation explicit enough for our purposes. We provide a short proof adapted to our setting, beginning with an auxiliary result which can be found e.g. in :
Suppose is a Markov chain in associated to the transition kernel . Choose an initial distribution . Furthermore assume is -measurable, and let . Then for nonnegative and measurable and we have
Proposition 6 (Nonlinear Markov property).
Let be a Markov chain as in Lemma 5, and suppose , with Hadamard. Choose with . Then
By the linear Markov property (9),
Clearly , as a measurable function of , is -measurable. Thus, in order to verify that is indeed the conditional expectation of given , it remains to show that for each -measurable function the inequality holds true, cf. Definition 2. For this sake, define
By construction of we have . Thus, Lemma 5 implies
Proposition 6 implies that the expression actually is independent of the initial distribution . Therefore we will omit in the following and simply write .
We are now in a position to establish a link between nonlinear Markov semigroups and barycentric refinement processes. Suppose is a nonnegative compactly supported -variate sequence such that for all . Define recursively and
defines a Markov transition kernel. This kernel is homogeneous in the sense that for any . We write , denote the associated Markov chain by , and refer to as the characteristic Markov chain for . The central observation of this article is the following consequence of the nonlinear Markov property (10):
Suppose is bounded, where is a Hadamard space. Let be a barycentric refinement scheme acting on data from according to the subdivision rule (3). Let denote the characteristic Markov chain of . Then
This statement is proven by induction over using the following computation and Proposition 6:
The remainder of this article is devoted to analyzing the effects of this representation of the iterates of on the convergence properties of barycentric schemes with nonnegative masks.
2. The convergence problem
Suppose is an -variate compactly supported sequence of nonnegative reals. Define a refinement scheme via
Then a necessary condition for the convergence of on is the basic sum rule (2). In case the mask obeys this rule, we conclude
Moreover, converges if and only if there exists a continuous subject to the functional equations
Due to Equation (11), is referred to as an -refinable function. Given bounded, real-valued input data , the limit function may be written as
In particular, , where denotes the Dirac distribution on the origin.
Assuming that conditional expectations of bounded random variables mapping to the metric space are well-defined in the sense of Definition 2, and in addition satisfy the smoothing property (7), we could deduce from Theorem 4
Note, however, that , the -step transition probabilities of , can be viewed as , where denotes the linear counterpart to , and the Dirac delta on the origin, cf. Theorem 5. Thus, the assumption of associativity would guarantee that every scheme converging for linear input data would converge on as well. Indeed, the limit functions for given input data would satisfy
where , leading to a complete analogy to the linear case. However, nonlinear conditioning is a non-associative operation. The above observations demonstrate that this lack of property (7) constitutes the need for a further discussion of convergence.
The first result of this section is a small, but useful generalization of Theorem 1 in . Although the proof transcribes more or less directly, we give a detailed exposition for the reader’s convenience.
Let be refinement schemes acting on data from a Hadamard space . Then converges under the following assumptions:
(i) There is a function , a nonnegative real number and a positive integer such that
for and .
(ii) is convergent and satisfies
(iii) There is such that
We set and claim that this defines a Cauchy sequence in . Note first that given and , by continuity of respectively , we find and such that
Moreover, due to convergence of , by multiplying both the numerator and the denominator of the number with a power of two if necessary we may assume to be sufficiently large for
to hold for , in addition to (14). This together with (i) and (ii) implies
showing that is a Cauchy sequence. Since is complete, we find a continuous with uniformly. We claim that converges to in the sense of (1). For and , we obtain the inequality
which together with
establishes the claim. ∎
In the following we rely on a nonlinear version of Jensen’s inequality due to K. T. Sturm, cf. :
Lemma 8 (Conditional Jensen’s inequality).
Suppose is a convex, lower semicontinuous function on a Hadamard space , and is a probability space. Moreover suppose is a filtration in . Then for each bounded, -measurable random variable the following holds true:
Suppose the linear scheme associated to converges, and , with bounded, convex and balanced. Denote by the Minkowski functional of . Furthermore define via
Then the barycentric scheme associated to is weakly contractive with respect to .
The Hadamard property implies that for each the function
Recall that the transition kernel of takes the form
Substituting for , we deduce
Since the linear subdivision scheme with mask converges, we find a refinable function satisfying (11) and (12), cf. Theorem 5. Recall that one obtains this refinable function as the limit of the linear scheme acting on the input data , cf. Theorem 5 :
Accordingly, setting , we obtain
Now, if are such that , and , one concludes
Then, since the refinable function is uniformly continuous, the property (12) implies that for large enough,
By boundedness of we also obtain
Thus, for one concludes:
which completes the proof. ∎
Recall that the tensor product of two masks and is defined by