Finding a shortest path between two points is a fundamental problem in the general area of optimization with applications ranging from computing the shortest route between two physical locations on a road network (Delling et al., 2009) to path planning in robotics (LaValle, 2006)
. In machine learning, shortest paths are at the core of theIsomap algorithm for manifold learning (Silva and Tenenbaum, 2002; Tenenbaum et al., 2000) and the MDS-MAP algorithm (Shang et al., 2003; Shang and Ruml, 2004) for multidimensional scaling in the presence of missing distances. (See also the work of Kruskal and Seery (1980).)
In this paper we study shortest paths and shortest path distances on a subset of a Euclidean space. The resulting mathematical model is closely related to sampling-based motion planning in robotics and is directly relevant to machine learning tasks such as manifold learning and manifold clustering. Our main interest lies in how well such paths and distances are approximated by their equivalents in a neighborhood graph built on a sample of points from .
An important concept in machine learning where data points are available (typically in a Euclidean space), a neighborhood graph based on these points is a graph with nodes indexing the points themselves and edges between two nodes when their corresponding points are within a certain distance. An edge is weighed by the Euclidean distance between the two underlying points. (Other variants exist.) The shortest paths in the neighborhood graph thus offer a natural discrete analog to the shortest paths on the set, which are continuous by nature. This correspondence has algorithmic implications: one is instinctively led to computing shortest paths in the graph to produce estimates for shortest paths on. This is exactly what Isomap does (Silva and Tenenbaum, 2002). In robotics, sampling-based motion planning algorithms also rely on sampling the environment (here the surface) to discretize the optimization problem (Thrun et al., 2005).
1.1 Shortest paths
Shortest paths are, of course, well-studied objects in mathematics, particularly in metric geometry (Burago et al., 2001). They have also been the object of intensive study in robotics (Latombe, 2012). The approximation of shortest path distances on the surface by shortest path distances in the graph was established by Bernstein et al. (2000) with a view towards providing theoretical guarantees for Isomap. This was done in the context of a geodesically convex surface. The same sort of approximation has also been considered in robotics; see, for example, in (Karaman and Frazzoli, 2011; Kavraki et al., 1998; Janson et al., 2015; LaValle and Kuffner, 2001), and references therein, where the context is that of a Euclidean domain with holes representing the obstacles that the robot needs to avoid. Note that, in this literature, the neighborhood graph is modified to only include collision-free edges.
In Section 3 we establish some basic results on shortest paths and the corresponding metric. We then revisit the results of Bernstein et al. (2000) on the approximation of this metric with the pseudometric on a neighborhood graph, making a connection with the seminal work of Dubins (1957). We also show that it is possible to approximate shortest paths on the surface with shortest paths in a neighborhood graph under much milder regularity assumptions.
1.2 Curvature-constrained shortest paths
We also consider curvature-constrained shortest paths and the corresponding distances. These have long been considered in robotics to model settings where the robot has limited turning radius. Theoretical results date back at least to the seminal work of Dubins (1957). See also (Boissonnat et al., 1992; Reeds and Shepp, 1990). Still in robotics, sampling-based motion planning algorithms designed to satisfy differential motion constraints (including kinematic/dynamical constraints) are studied, for example, in (Li et al., 2015; Karaman and Frazzoli, 2010; Schmerling et al., 2015a, b). Importantly, in the present work, no constraints are placed on the initial and final orientations. Another difference with this literature, where motion is most typically in a Euclidean domain, is that we consider a surface that may be curved, and this forces the shortest paths on to satisfy some nontrivial curvature constraint.
In machine learning, angle and curvature-constrained paths have been recently considered for the task of surface learning where the surface may self-intersect (Babaeian et al., 2015) and for the task of multi-surface clustering (Babaeian et al., 2015).
In Section 4 we derive some theory on curvature-constrained shortest paths and the corresponding (semi-)metric, and in particular establish bounds on approximations by curvature-constrained shortest paths in a neighborhood graph. This requires a notion of discrete curvature applicable to polygonal lines, which we describe in (20).
In this section we set most of the notation for the reminder of the paper, introduce some fundamental concepts in metric geometry, and also list some basic results that will be used later on.
For two vectors, their inner product is denoted and their angle is defined as , where denotes the Euclidean norm. We also define as their wedge product. Recall that is a 2-vector, and the space of 2-vectors can be endowed with an inner product, and the resulting norm — also denoted by — satisfies , which is also the area of the parallelogram defined by and .
For and , let denote the open ball of with center and radius . For , let , their Minkowski sum. In particular, is the -tubular neighborhood of .
Consider two subsets of a Euclidean space, with compact and open. Then there is such that .
Suppose the statement is not true. In that case, for integer, take . Since is bounded, we may assume WLOG that it converges to some . (Here and elsewhere, is shorthand for the sequence , and when we write we mean that converges to .) Clearly , since for all and is compact. This implies that , and since is open, there is such that . Since , for large enough, , implying , which is a contradiction. ∎
For two sets and , define their Hausdorff distance
By convention, we set for any . Note that
A curve in is a continuous function on an interval . We will often identify the function with its image . We say that a sequence of curves converges uniformly to a curve if these curves can be parameterized in such a way that the convergence as functions is uniform; see (Burago et al., 2001, Def 2.5.13).
We note that, for two parameterized curves and ,
and in particular, for curves, uniform convergence implies convergence in Hausdorff metric.
For , let , which is the length of the polygonal line defined by . This definition is extended to general curves in the usual way (Burago et al., 2001, Def 2.3.1): the length of a curve is
where the supremum is over all increasing sequences . Any curve with finite length admits a unit-speed parameterization (Burago et al., 2001, Def 2.5.7, Prop 2.5.9), meaning that for any such curve there is an interval and a continuous function such that and for all in . All curves will be assumed to be unit-speed (i.e., parameterized by arc length) by default. Note that a unit-speed curve is differentiable almost everywhere with unit norm derivative, meaning for almost all ; in particular, such a curve is 1-Lipschitz, where we say that a function is -Lipschitz for some if for all .
For any curve and any -Lipschitz function , .
Consider a parameterization . Then for any increasing sequence ,
and by taking the supremum of such sequences, we obtain the result, since the right-hand side becomes . ∎
Suppose is a curve and is a closed ball such that . If denotes the metric projection of onto , then .
Assume WLOG that is the closed unit ball and consider a unit-speed parameterization . Let denote the metric projection onto , which has the simple expression for (while, of course, for ). By continuity, there is and a subinterval , with , such that for all . For , the differential of at , denoted , is equal to , where denotes the identity linear function. Hence, has operator norm equal to . By Taylor’s theorem, we thus have
for all such that the line segment joining and does not contain the origin, and this extends to all by continuity. In particular, is -Lipschitz on . Then, by Lemma 2,
by the fact that and by construction. ∎
We say that a unit-speed curve has curvature bounded by if it is differentiable and its derivative is -Lipschitz. Assuming the curve is twice differentiable at , its curvature at is defined as
In that case, has curvature bounded by if and only if .
3 Shortest paths and their approximation
In this section, we consider the intrinsic metric on a subset and its approximation by a pseudo-metric defined based on a neighborhood graph built on a finite sample of points from the surface. In Section 3.1 we define the intrinsic metric on a given subset, and list a few of its properties. In Section 3.2 we define the notion of neighborhood graph and a pseudo-metric based on shortest path distances in that graph. In Section 3.3 we show that this pseudo-metric can be used to approximate the intrinsic metric on a subset. We discuss the results obtained by Bernstein et al. (2000) and make a connection with the classical work of Dubins (1957).
3.1 The intrinsic metric on a surface
The intrinsic metric on a set is the metric inherited from the ambient space . For , it is defined as
When for all , we say that is path-connected (Waldmann, 2014, Sec 2.5). A lot is known about this type of metric (Burago et al., 2001). In particular, when is a smooth submanifold, this is the Riemannian metric induced by the ambient space (Burago et al., 2001, Sec 5.1.3).
If is closed and are such that , the infimum in (5) is attained. If is open and connected, for all .
The intrinsic metric on is in general different from the ambient Euclidean metric inherited from . The two coincide only when is convex. However, it is true that for all , and in particular this implies that the ambient topology is always at least as fine as the intrinsic topology. But there are cases where the two topologies differ.
Example 1 (A set with infinite intrinsic diameter and finite ambient diameter).
Consider a closed spiral with infinite length, for example defined as , where for and one-to-one (decreasing) and such that . The resulting set is compact in , but unbounded for its intrinsic metric since for all . (o denotes the origin.) In particular, if and , then in the ambient topology, while in the intrinsic topology. Suppose we now thicken the spiral and redefine where is decreasing and such that . In that case, is the closure of its interior and, assuming and are , is except at the origin o. And still, has infinite intrinsic diameter.
Example 2 (A set with finite intrinsic diameter having different intrinsic and ambient topologies).
Let with continuous and strictly decreasing and satisfying . Consider a strictly increasing sequence such that . In the cone of defined in polar coordinates by consider a self-avoiding path of length 1 starting at and ending at the origin o. Note that when . Define . Clearly, for all , so that has intrinsic diameter bounded by 2. By construction converges to o in the ambient topology but is not even convergent in the intrinsic topology. (If it were to converge in the intrinsic topology, the limit would have to be o, but for all .) Also, as in Example 1, if we carefully thicken each , the resulting can be made to be the closure of its interior and have border except at the origin o.
Having established that the ambient and intrinsic topologies need not coincide, we will mostly focus on the case where they do, which corresponds to assuming the following.
The intrinsic and ambient topologies coincide on .
The following is well-known to the specialist. We provide a proof for completeness, and also because similar, but more complex arguments will be used later on.
Any smooth submanifold of with empty or smooth boundary satisfies Property 1.
Let be a smooth submanifold of dimension . Assume for contradiction that the topologies do not coincide. Then there is and , and a sequence such that and for all . Let be an open set containing and be a diffeomorphism, where is an open subset of either or — the latter if . Let and such that . Let and define and , where in this instance denotes the usual operator. For sufficiently large we have , in which case we let . For sufficiently large we also have , in which case we let defined on . Then is a curve on joining and , of length
This leads to a contradiction, since while . ∎
3.2 A neighborhood graph and its metric
We approximate the intrinsic metric (shortest-path distance) on with the metric (shortest-path distance) on a neighborhood graph based on a sample from denoted . While such a graph has node set indexing — which we take to be — there are various ways of defining the edges. In what follows, we write when nodes and are neighbors in the graph. We will use the following well-known variant (Maier et al., 2009):
-ball graph: if and only if .
We weigh each edge with the Euclidean distance between and , and set the weight to when , thus working with
In the context of a weighted graph, we can define a path as simply a sequence of nodes, and its length is then the sum of the weights over the sequence of node pairs that defines it. In our context, the length of a path is thus
Equivalently, this is the length of the polygonal line defined by the sequence of points . The shortest-path distance between and is defined as the length of the shortest path joining and , namely
This is the discrete analog of the intrinsic metric on a set defined in (5).
For two sample points, , define
thus defining a metric on the sample .
This can be extended to a pseudo-metric333 A ‘pseudo-metric’ is like a metric except that it needs not be definite. on the surface as follows: for , define
so that indexes the sample points that are nearest to . ( is only a pseudo-metric on since we may have even when .)
The construction of a pseudo-metric on a neighborhood graph is meant here to approximate the intrinsic metric on the surface. The approximation results that follow are based on how dense the sample is on the surface, which we quantify using the Hausdorff distance between and as sets, namely
Comparing with is exactly what Bernstein et al. (2000) did to provide theoretical guarantees for Isomap (Tenenbaum et al., 2000). We extend their result to more general surfaces and also provide a convergence result under very mild assumptions; see Theorem 1 below.
Consider compact and a sample , and let . For , form the corresponding -ball graph. When , we have
This was established in (Bernstein et al., 2000, Th 2) under essentially the same conditions, but we provide a proof for completeness.
If , then and are direct neighbors in the graph and so
We thus turn to the case where . Let and let be parameterized by arc length such that and , which exists by Lemma 4. Let for , where , noting that and . Let be closest to among the sample points, noting that and . In particular, by definition of . By the triangle inequality, for any ,
so that forms a path in the -ball graph. We then have, using the fact that and ,
using the fact that with . ∎
It is possible to tighten the bound in the very special case where is convex (and in particular flat). Indeed, a refinement of the arguments provided above lead to an error term in . We do not know if this extends to the case where is curved beyond the the case where it is isometric to a convex set.
We establish a complementary lower bound in Proposition 2 below under some regularity assumptions on . Before doing so, we use Proposition 1 to derive a qualitative result that states that, under very mild assumptions on , shortest paths in a neighborhood graph can indeed be approximated by shortest paths on . (Proposition 2 will provide a quantitative error bound for this approximation.)
Consider compact and satisfying Property 1. For any , there is such that the following holds. Consider a sample and let . For , form the -ball graph based on . Take any such that both and . If and , then for every shortest path in the graph joining and , seen as a unit-speed polygonal curve in , there is a shortest path on joining and such that .
Fix such that . We first prove that there is such an , but that may depend on and . For this, we reason by contradiction: assuming the statement is false, there exists , and sequence and such that , , and a sample with , as well as a shortest path in the -ball neighborhood graph joining and with the property that for any shortest path on joining and . Note that by Proposition 1. Assume WLOG that and , which in particular implies that for all .
It is thus possible to parameterize each so that it is 1-Lipschitz on . As a family of functions on , is therefore equicontinuous. The family is also uniformly bounded by virtue of the fact that each starts at and as length bounded by . By the Arzelà-Ascoli theorem, there is a subsequence of that converges uniformly as 1-Lipschitz functions on to some 1-Lipschitz function defined on that same interval. Necessarily, and . WLOG assume that this subsequence is itself. We then have
We claim that . If not, by the fact that is closed, there is such that and . As ,
Since the line segments making up have length bounded by , must include at least one sample point when . Therefore, for large enough. Let be such that . By compactness, we may assume WLOG that . We then have by uniform convergence, so that since is closed. So we have a contradiction.
Collecting our findings, we found a curve joining and , with and as , which contradicts our working hypothesis that for all . This proves the first part of the theorem.
We now show that one can choose that works for all . We reason by contradiction exactly as before, except that now are replaced by , and by , thus all possibly changing with . By the fact that is compact, we may assume WLOG that there are such that and . By Property 1, we have that is continuous with respect to the Euclidean metric. In particular, . With this we can now see that the remaining arguments are identical to those backing the first part. ∎
We now turn to proving a bound that complements Proposition 1, that is, a more quantitative (or explicit) version of Theorem 1. Bernstein et al. (2000) obtain such a bound when is a submanifold without intrinsic curvature. Their cornerstone result is the following, which they call the Minimum Length Lemma.
(Bernstein et al., 2000) Let be a unit-speed curve with curvature bounded by . Then for all such that .
Note that the result is sharp in that the inequality is an equality when is a piece of a circle of radius . Of course, we also have for all , since .
While Bernstein et al. (2000) prove this result from scratch, a very short proof of a slightly weaker bound follows from a result in the pioneering work of Dubins (1957) on shortest paths with curvature constraints. Indeed, in the setting of Lemma 6, let denote a unit-speed parametrization of a circle of radius . (Dubins, 1957, Prop 2) says that when , which leads to
when . (The first inequality is due to the fact that .)
Using either Lemma 6 or this weaker bound, it is straightforward to obtain a useful comparison between the intrinsic metric and the ambient Euclidean metric, locally. Note that we still follow the footsteps of Bernstein et al. (2000). The core assumption is the following.
The shortest paths on have curvature bounded by .
This is true when is sufficiently smooth. See Lemma 13 further down.
We start with (11), where only the lower bound is nontrivial. Take and let . Let be a unit-speed shortest path on joining and . By Property 2, has curvature bounded by . Knowing that, we apply Lemma 6 to get
where the last inequality holds if is sufficiently small. In view of that, it suffices to prove that there is such that when satisfy . This is true because Property 1 guarantees that is continuous as a function on the compact set . ∎
We now have all the ingredients to establish a bound that complements Proposition 1. Such a bound is already available in the work of Bernstein et al. (2000) in a somewhat more restricted setting where it is assumed that is a compact and geodesically convex submanifold.
Fix such that , for otherwise there is nothing to prove. Let define a shortest path in the graph joining and , so that , where . Define and for . Since , by Lemma 7, . By assumption, , and this is seen to force , which then implies that . We thus have
4 Curvature-constrained shortest paths and their approximation
In this section, we define the curvature-constrained intrinsic semi-metric on a subset and consider its approximation by a curvature-constrained pseudo-semi-metric based on a neighborhood graph built on a finite sample of points from the surface. In Section 4.1 we define the curvature-constrained semi-metric on a given subset, and list a few of its properties. In Section 4.2 we define a notion of discrete curvature which has useful consistency properties. In Section 4.3 we define a new notion of neighborhood graph and a pseudo-semi-metric based on shortest path distances in that graph. In Section 4.4 we show that this pseudo-metric can be used to approximate its continuous counterpart.
4.1 The curvature-constrained intrinsic semi-metric on a surface
A notion of curvature-constrained semi-metric on a subset is obtained from its intrinsic metric defined in Section 3.1 by adding a curvature constraint. In more detail, for and , define
By convention, if there is no path as in (14), then .
is thus the length of the shortest path on joining and among those with curvature bounded pointwise by .
Compared with the (unconstrained) intrinsic metric (5), we always have, for any subset , and for and any ,