Cyclic projections in Hadamard spaces

The method of cyclic projections is a classical algorithm seeking an intersection point of a finite family $C_{1},\dots,C_{k}$ of closed convex subsets in a Hilbert space $X$. Denote by $P_{i}$ the closest-point projection $X\to C_{i}$; it sends a point $x\in X$ to the (necessarily unique) point $P_{i}(x)$ in $C_{i}$ that minimizes the distance to $x$. Given a point $x\in X$ consider the sequence $x_{n}=P^{n}(x)$, where $P$ is the cyclic composition of projections $P=P_{1}\circ\dots\circ P_{k}$. The method of cyclic projections analyzes the sequence $(x_{n})$, tries to find a limit point $x_{\infty}$, to show $x_{\infty}\in C_{1}\cap\dots\nobreak\cap C_{k}$, and to understand the rate of convergence.

Let us list some results in the area. If the intersection $C_{1}\cap\dots\cap C_{k}$ is non-empty, then $(x_{n})$ always converges weakly to some point in $C_{1}\cap\dots\cap C_{k}$ [14]. However, this convergence does not need to be strong [19]. If, in addition, $C_{i}$ are linear subspaces, then the convergence is strong [23, 18]. If the intersection $C_{1}\cap\dots\cap C_{k}$ is not assumed to be non-empty, the analysis of the sequence $(x_{n})$ is more complicated. However, in [11] it has been established that the cyclic product $P=P_{1}\circ\dots\circ P_{k}$ is asymptotically regular; by definition, this means, that for any starting point $x\in X$, we have $|x_{n}-x_{n+1}|\to 0$ as $n\to\infty$. The rates of convergence, respectively, the rates of asymptotic regularity have been investigated in several works, see, for instance [12, 20]. For further reference, see [12, 17, 7, 8, 4].

More recently, the method of cyclic projections has been investigated beyond the setting of Hilbert spaces in so-called Hadamard spaces (also known as CAT(0) spaces, or globally non-positively curved spaces in the sense of Alexandrov). This class of metric spaces includes hyperbolic spaces, metric trees, as well as complete simply-connected Riemannian manifolds of non-positive curvature; it has played an important role in many areas of mathematics in the last decades. We assume some familiarity with Hadamard spaces, refer the reader to [10, 9, 16, 15, 3, 2] as general references on this subject. For the introduction and applications of the method of cyclic projections in Hadamard spaces, see [6], [7, Section 6.8], and the references therein.

Hadamard spaces are defined (loosely speaking) by the property that their distance function is at least as convex as the distance function on a Hilbert space. In particular, Hadamard spaces contain a huge variety of convex subsets; closest point projections to closed convex subsets are well-defined and $1$-Lipschitz, and the questions discussed above about cyclic projections are absolutely meaningful in a Hadamard space $X$.

Many results discussed above have been transferred from the linear setting of Hilbert spaces to general Hadamard spaces. For instance, if the subsets $C_{i}$ have a non-empty intersection, then the cyclic product of projection $P$ is asymptotically regular and, for any initial point $x\in X$, the sequence $x_{n}=\nobreak P^{n}(x)$ converges weakly to a point $x_{\infty}\in C_{1}\cap\dots\cap C_{k}$ [5, 8]. (The weak topology on Hadamard spaces is discussed in [7, 6, 22]). The rate of convergence in this setting has been studied in [21].

Therefore it is somewhat surprising, that the fundamental result of Heinz Bauschke [11] for (possibly) non-intersecting convex subsets $C_{i}$ does not admit a generalization to the setting of Hadamard spaces. The following main result of this paper provides a negative answer to the question of Miroslav Bačák [7, Problem 6.13].

We provide an explicit example with $X$ being a product of two trees, proving the theorem for $k=3$. Setting $C_{3}=\dots=C_{k}$ defines examples for any $k\geqslant 3$.

In this example, all subsets $C_{i}$ are isometric to the unit interval, the projections $P_{i}$ map all of these segments isometrically onto $C_{i}$ and the composition $P=\nobreak P_{1}\circ P_{2}\circ P_{3}$ maps $C_{1}$ to itself isometrically but exchanges the endpoints of this interval. A stronger version of the theorem is proved in the appendix; it requires a somewhat deeper understanding of the geometry of Hadamard spaces. It seems possible, but would require some non-trivial technical work, to adapt the example from the appendix so that the Hadamard space becomes a smooth Hadamard manifold.

On the other hand, in the case $k=2$, the result of Heinz Bauschke [11] does admit a generalization; in this case, the algorithm is sufficiently simple to be controlled explicitly, even providing an optimal rate of asymptotic regularity. As it was pointed out by an anonymous referee, the following statement follows from [5, Theorem 3.3], under the additional assumption of the existence of a fixed point of the composition $P$.

Moreover, $|x_{n}-x_{n+1}|=o(\frac{1}{\sqrt{n}})$ for any $x\in X$ and $x_{n}=\nobreak P^{n}(x)$.

Here and further we denote by $|x-y|$ the distance between points $x$ and $y$ in any metric space, even without linear structure.

Examples given by the real axis $C_{1}\subset\mathbb{R}^{2}$ and the set

$$C_{2}=\{\,(x,y):x>0,y\geqslant 1+x^{-\varepsilon}\,\}$$

reveal that the convergence rate in Proposition 1 cannot be improved to $O(n^{-\frac{1}{2}-\varepsilon})$ for any $\varepsilon>0$.

This also shows that the optimal rate of asymptotic regularity for cyclic product of projections on two convex subsets is the same for the Euclidean plane and general Hadamard spaces.

Our space $X$ is a product of two trees, thus of two Hadamard spaces. Hence $X$ is a Hadamard space.

Denote by $a$, $b$, $c$ and $u$, $v$, $w$ the sides of $S$ and $T$ respectively.

The following diagram shows 3 isometric copies of $2{\times}2$-square in $X$; they are obtained as the products of two pairs of sides in $S$ and $T$ as labeled.

Consider the segments $C_{1}$, $C_{2}$, and $C_{3}$ shown on the diagram; they all have slope $-1$ and project to each other isometrically. Note that each projection $P_{i}$ reverses the shown orientation. It follows that the composition $P=P_{1}\circ P_{2}\circ P_{3}$ sends the segment $C_{1}$ to itself isometrically and changes the orientation of the segment. In particular, $P$ exchanges the ends of the segment, hence $P$ is not asymptotically regular. (In fact, for an end $e$ of $C_{1}$, and any $n$, we have $|P^{n}(e)-\nobreak P^{n+1}(e)|=1$.)

Finally, setting $C_{3}=\dots=C_{k}$ defines examples for any $k\geqslant 3$.∎

Proof of 1. By definition, $x_{n}\in C_{1}$ for all $n$. Set $y_{n+1}=P_{2}\circ P^{n}(x)$, so $y_{1}=P_{2}(x)$, $x_{1}=P_{1}(y_{1})$, $y_{2}=P_{2}(x_{1})$, and so on. Further set

	$\displaystyle r_{n}$	$\displaystyle:=|x_{n}-x_{n+1}|,$
	$\displaystyle s_{n}$	$\displaystyle:=|y_{n}-y_{n+1}|.$

Since the closest-point projection is nonexpanding, we get

$$s_{1}\geqslant r_{1}\geqslant s_{2}\geqslant r_{2}\geqslant\dots$$

➊

Set

	$\displaystyle a_{n}$	$\displaystyle\mathrel{{:}{=}}|x_{n}-y_{n}|=\operatorname{\rm dist}_{C_{1}}y_{n},$
	$\displaystyle b_{n}$	$\displaystyle\mathrel{{:}{=}}|y_{n+1}-x_{n}|=\operatorname{\rm dist}_{C_{2}}x_{n}.$

Note that

$$a_{1}\geqslant b_{1}\geqslant a_{2}\geqslant b_{2}\geqslant\dots$$

➋

Since $C_{1}$ is convex and $x_{n}\in C_{1}$ lies at the minimal distance from $y_{n}$, we have $\measuredangle[x_{n}\,{}^{x_{n-1}}_{y_{n}}]\geqslant\tfrac{\pi}{2}$. Since $X$ is a Hadamard space,

$$r_{n}^{2}\leqslant b_{n}^{2}-a_{n+1}^{2}.$$

Therefore, ➋ ‣ 3 implies that

$$\sum_{n}r_{n}^{2}\leqslant b_{1}^{2}.$$

By ➊ ‣ 3, $r_{n}$ is non-increasing. Therefore, $r_{n}=o(\tfrac{1}{\sqrt{n}})$. ∎