Abstract
This paper provides rates of convergence for empirical (generalised) barycenters on compact geodesic metric spaces under general conditions using empirical processes techniques. Our main assumption is termed a variance inequality and provides a strong connection between usual assumptions in the field of empirical processes and central concepts of metric geometry. We study the validity of variance inequalities in spaces of non-positive and non-negative Aleksandrov curvature. In this last scenario, we show that variance inequalities hold provided geodesics, emanating from a barycenter, can be extended by a constant factor. We also relate variance inequalities to strong geodesic convexity. While not restricted to this setting, our results are largely discussed in the context of the 2-Wasserstein space.
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Acknowledgements
We would like to thank the associate editor, as well as two anonymous referees, for valuable comments that helped improve significantly the original version of the manuscript. We also express our gratitude to Philippe Rigollet for stimulating conversations and useful remarks on a preliminary version of the paper.
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T. Le Gouic and Q. Paris: This work has been funded by the Russian Academic Excellence Project ‘5–100’.
A metric geometry
A metric geometry
1.1 A.1 Geodesic spaces
Let \((M,d)\) be a metric space. We call path in \(M\) a continuous map \(\gamma :I\rightarrow M\) defined on an interval \(I\subset {\mathbb {R}}\). The length \(L(\gamma )\in [0,+\infty ]\) of a path \(\gamma :I\rightarrow M\) is defined by
where the supremum is taken over all \(n\ge 1\) and all \(t_0\le \dots \le t_{n}\) in I. A path is called rectifiable if it has finite length. Two paths \(\gamma _1\) and \(\gamma _2\) are said to be equivalent if \(\gamma _1\circ \varphi _1=\gamma _2\circ \varphi _2\) for non-decreasing and continuous functions \(\varphi _1\) and \(\varphi _2\). In this case, \(\gamma _1\) is said to be a reparametrisation of \(\gamma _2\) and we check that \(L(\gamma _1)=L(\gamma _2)\). A path \(\gamma :[a,b]\rightarrow M\) is said to have constant speed if for all \(a\le s\le t\le b\),
where \(\gamma _{[s,t]}\) denotes the restriction of \(\gamma \) to [s, t].
Proposition A.1
Any rectifiable path has a constant speed reparametrisation \(\gamma :[0,1]\rightarrow M\).
Given \(x,y \in M\), a path \(\gamma :[a,b]\rightarrow M\) is said to connect x to y if \(\gamma (a)=x\) and \(\gamma (b)=y\). By construction of the length function L, \(d(x,y)\le L(\gamma )\) for any path \(\gamma \) connecting x to y. The space \(M\) is called a length space if, for all \(x,y\in M\),
where the infimum is taken over all paths \(\gamma \) connecting x to y. A length space is said to be a geodesic space if, for all \(x,y\in M\), the infimum on the right hand side of (A.2) is attained.
Definition A.2
In a geodesic space, we call geodesic between x and y any constant speed reparametrisation \(\gamma :[0,1]\rightarrow M\) of a path attaining the infimum in (A.2).
For a geodesic \(\gamma \), it follows from its minimising properties that
for all \(0\le s\le t\le 1\). In particular, (A.1) translates in this case as
for all \(0\le s\le t\le 1\). We end by a general characterization of geodesic spaces.
Proposition A.3
Let (M, d) be a metric space.
- (1):
If \(M\) is a geodesic space, then any two points \(x,y\in M\) admit a midpoint, i.e. a point \(z\in M\) such that
$$\begin{aligned} d(x,z)=d(y,z)=\frac{1}{2}d(x,y). \end{aligned}$$- (2):
Conversely, if \(M\) is complete and if any two points in M admit a midpoint, then \(M\) is a geodesic space.
1.2 A.2 Model spaces
Given a real number \(\kappa \in {\mathbb {R}}\), a geodesic space of special interest is the (complete and simply connected) 2-dimensional Riemannian manifold with constant sectional curvature \(\kappa \). For given \(\kappa \in {\mathbb {R}}\), this metric space \((M^2_{\kappa },d_{\kappa })\) is unique up to an isometry, and modelled as follows.
If \(\kappa <0\), \((M^2_{\kappa },d_{\kappa })\) is the hyperbolic plane with metric multiplied by \(1/\sqrt{-\kappa }\).
If \(\kappa =0\), \((M^2_{0},d_{0})\) is the Euclidean plane equipped with its Euclidean metric.
If \(\kappa >0\), \((M^2_{\kappa },d_{\kappa })\) is the Euclidean sphere in \({\mathbb {R}}^3\) of radius \(1/\sqrt{\kappa }\) with the angular metric.
The diameter \(\varpi _{\kappa }\) of \(M^{2}_{\kappa }\) is
For \(\kappa \in {\mathbb {R}}\), there is a unique geodesic connecting x to y in \((M^2_{\kappa },d_{\kappa })\) provided \(d_{\kappa }(x,y)<\varpi _{\kappa }\). By convention, we call triangle in \(M^2_{\kappa }\) any set of three distinct points \(\{p,x,y\}\subset M^2_{\kappa }\), with perimeter
Side lengths of triangle \(\{p,x,y\}\) are the numbers \(d_{\kappa }(p,x)\), \(d_{\kappa }(p,y)\) and \(d_{\kappa }(x,y)\). Given \(a,b,c>0\) satisfying the triangle inequality and such that \(a+b+c<2\varpi _{\kappa }\), there exists a unique (up to an isometry) triangle \(\{p,x,y\}\) in \(M^2_{\kappa }\) such that \(d_{\kappa }(p,x)=a\), \(d_{\kappa }(p,y)=b\) and \(d_{\kappa }(x,y)=c\). The angle \(\sphericalangle ^{\kappa }_{p}(x,y)\) at p in \(\{p,x,y\}\subset M^2_{\kappa }\) is defined by
where \(a=d_{\kappa }(p,x)\), \(b=d_{\kappa }(p,y)\), \(c=d_{\kappa }(x,y)\) and \(c_{\kappa }:=s'_{\kappa }\) with
We end by observing that the angle is constant along geodesics in the model space \((M^2_{\kappa },d_{\kappa })\).
Proposition A.4
Let \(\kappa \in {\mathbb {R}}\) and \(\{p,x,y\}\subset (M^2_{\kappa },d_{\kappa })\) be a triangle. If \(\gamma _{x}\) and \(\gamma _{y}\) are geodesics from p to x and from p to y respectively, then for all \((s,t)\in (0,1]^2\),
1.3 A.3 Curvature
In this section, we describe the notion of curvature bounds of metric spaces. Curvature bounds in general metric spaces are defined by comparison arguments involving the model surfaces \((M^2_{\kappa },d_{\kappa })\) discussed in the previous section. The fundamental device allowing for this comparison is that of a comparison triangle. Given a metric space \((M,d)\), we define a triangle in \(M\) as any set of three points \(\{p,x,y\}\subset M\). For \(\kappa \in {\mathbb {R}}\), a comparison triangle for \(\{p,x,y\}\) in \(M^2_{\kappa }\) is an isometric embedding of \(\{p,x,y\}\) in \(M^2_{\kappa }\), i.e. a set \(\{p_{\kappa },x_{\kappa },y_{\kappa }\}\subset M^2_{\kappa }\) such that
Such a comparison triangle always exists (and is unique up to an isometry) provided
We are now in position to define curvature bounds for geodesic spaces.
Definition A.5
Let \(\kappa \in {\mathbb {R}}\) and \((M,d)\) be a geodesic space.
- (1):
We say that \(\mathrm{curv}(M)\ge \kappa \) if for any triangle \(\{p,x,y\}\subset M\) satisfying \(\mathrm{peri}\{p,x,y\}< 2\varpi _{\kappa }\), any comparison triangle \(\{p_{\kappa },x_{\kappa },y_{\kappa }\}\subset M^2_{\kappa }\), any geodesic \(\gamma \) joining x to y in \(M\) and any geodesic \(\gamma _{\kappa }\) joining \(x_{\kappa }\) to \(y_{\kappa }\) in \(M^2_{\kappa }\), we have for all \(t\in [0,1],\)
$$\begin{aligned} d(p,\gamma (t))\ge d_{\kappa }( p_{\kappa },\gamma _{\kappa }(t)). \end{aligned}$$(A.4)- (2):
We say that \(\mathrm{curv}(M)\le \kappa \) if the above definition holds with opposite inequality in (A.4).
The previous definition has a natural geometric interpretation: if \(\mathrm{curv}(M)\ge \kappa \) (resp. \(\mathrm{curv}(M)\le \kappa \)) a triangle \(\{p,x,y\}\) looks thicker (resp. thiner) than a corresponding comparison triangle \(\{p_{\kappa },x_{\kappa },y_{\kappa }\}\) in \(M^2_{\kappa }\). In the context of \(\kappa =0\), the above definition may be given an alternative form of practical interest.
Proposition A.6
Let \((M,d)\) be a geodesic space. Then \(\mathrm{curv}(M)\ge 0\) if, and only if, for any points \(p,x,y\in M\) and any geodesic \(\gamma \) joining x to y, we have
We have \(\mathrm{curv}(M)\le 0\) if, and only if, the same statement holds with opposite inequality.
The proof follows immediately from Definition A.5 by exploiting the geometry of the Euclidean plane. Note indeed that, whenever \(\{p,x,y\}\subset {\mathbb {R}}^2\) and \({\mathbb {R}}^2\) is equipped with the Euclidean metric \(\Vert .-.\Vert \), the unique geodesic from x to y is \(\gamma (t)=(1-t)x+ty\) and, for all \(t\in [0,1]\),
For \(\kappa \ne 0\), an equivalent formulation of Definition A.5, given only in terms of the ambient metric d, is given in the next subsection using the notion of angle. A (complete) geodesic space \((M,d)\) with \(\mathrm{curv}(M)\le \kappa \) for some \(\kappa \ge 0\) is sometimes called a CAT(\(\kappa \)) space in reference to contributions of E. Cartan, A.D. Alexandrov and V.A. Toponogov. A CAT(0) space is also referred to as an NPC (non positively curved) space or an Hadamard space. Similarly, \(M\) is also called an PC (positively curved) space if \(\mathrm{curv}(M)\ge 0\). If \((M,d)\) is a Riemannian manifold (complete for instance) with sectional curvature lower (resp. upper) bounded by \(\kappa \) at every point, then \(\mathrm {curv}(M)\ge \kappa \) (resp \(\le \kappa \)) in the sense of Definition A.5. It is worth noting that the previous definitions are of global nature as they require comparison inequalities to be valid for all triangles (that admit a comparison triangle in the relevant model space). Some definitions of curvature require the previous comparison inequalities to hold only locally. The local validity of these comparison inequalities is known, under suitable conditions depending on the value of \(\kappa \), to imply their global validity. Results in this direction are known as globalisation theorems.
1.4 A.4 Angles and space of directions
Angles, as defined below, allow to provide alternative characterisations of curvature bounds. Let \((M,d)\) be a metric space and let \(\kappa \in {\mathbb {R}}\). Given a triangle \(\{p,x,y\}\) in M with \(\text {peri}\{p,x,y\}<2\varpi _{\kappa }\), we define the comparison angle \(\sphericalangle ^{\kappa }_p(x,y)\in [0,\pi ]\) at p by
where \(c_{\kappa }\) and \(s_{\kappa }\) are as in (A.3). In other words, given any comparison triangle \(\{p_{\kappa },x_{\kappa },y_{\kappa }\}\) of \(\{p,x,y\}\) in \( M^2_{\kappa }\),
This allows to give an equivalent definition of curvature lower bounds that has the advantage of making sense on arbitrary metric spaces, not necessarily geodesic.
Definition A.7
(Quadruple comparison) Let (M, d) be a metric space. Let \(\kappa \in {\mathbb {R}}\). We say that \(\mathrm {curv}(M)\ge \kappa \), if for any four disctinct points \(p,x,y,z\in M\) such that every three points have a perimeter less than \(\varpi _\kappa \),
Proposition A.8
If (M, d) is a geodesic space, then curvature lower bounds as defined in Definition A.5 and Definition A.7 are equivalent. Moreover, even if (M, d) is not geodesic, and satisfies Definition A.7, then, Eq. (A.4) is satisfied for all \(t\in [0,1]\) and \(\gamma (t)\in M\) such that
The next result presents a characterization of curvature bounds in terms of the monotonicity of the comparison angle.
Proposition A.9
(Angle monotonicity) Let \((M,d)\) be a geodesic space and let \(\kappa \in {\mathbb {R}}\). Then \(\mathrm{curv}(M)\ge \kappa \) (resp. \(\mathrm{curv}(M)\le \kappa \)), in the sense of Definition A.5, if and only if, for any triangle \(\{p,x,y\}\) in M and any geodesics \(\gamma _{x}\) and \(\gamma _y\) from p to x and from p to y respectively, the function
is non-increasing (resp. non-decreasing) in each variable when the other is fixed.
In a geodesic space \((M,d)\), if \(p\in M\) and \(\gamma _x\) and \(\gamma _y\) are two geodesics connecting p to x and y respectively, we define
when it exists. It follows from Proposition A.9 that this limit exists provided \(\mathrm{curv}(M)\ge \kappa \) or \(\mathrm{curv}(M)\le \kappa \). It may be shown furthermore that this limit is independent of \(\kappa \). Hence, whenever \(M\) has upper or lower curvature bound, we denote \(\sphericalangle (\gamma _x,\gamma _y)\) the angle between these two geodesics. Given a third geodesic \(\gamma _z:[0,1]\rightarrow S\) such that \(\gamma _z(0)=p\) and \(\gamma _z(1)=z\), we have the triangular inequality
so that \(\sphericalangle \) defines a pseudo metric on the set G(p) of all geodesics emanating from p. Defining the equivalence relation \(\sim \) on G(p) by \(\alpha \sim \beta \Leftrightarrow \sphericalangle (\alpha ,\beta )=0\), the angle \(\sphericalangle \) induces a metric (still denoted \(\sphericalangle \)) on the quotient set \(G(p)/\sim \) and we call space of directions the completion \((\Sigma _p,\sphericalangle )\) of \((G(p)/\sim ,\sphericalangle )\). An element of \(\Sigma _p\) is called a direction.
1.5 A.5 Tangent cones
Metric spaces considered so far have a priori no differentiable structure. In this context, an analog of a tangent space is provided by the notion of a tangent cone. This section shortly reviews this notion. Below, M denotes a geodesic space with lower or upper bounded curvature in the sense of Definition A.5.
Definition A.10
(Tangent cone) Let \(p\in M\). The tangent cone \(T_pM\) at p is the Euclidean cone over the space of directions \((\Sigma _p,\sphericalangle )\). In other words, \(T_pM\) is the metric space:
Whose underlying set consists in equivalent classes in \(\Sigma _p\times [0,+\infty )\) for the equivalence relation \(\sim \) defined by
$$\begin{aligned} (\alpha ,s)\sim (\beta ,t) \Leftrightarrow ((s=0 \text{ and } t=0) \text{ or } (s=t \text{ and } \alpha =\beta )). \end{aligned}$$A point in \(T_pM\) is either the tip of the cone \(o_p\), i.e. the class \(\Sigma _p\times \{0\}\), or a couple \((\alpha ,s)\in \Sigma _p\times (0,+\infty )\) (identified to the class \(\{(\alpha ,s)\}\)).
Whose metric \(d_p\) is defined (without ambiguity) by
$$\begin{aligned} d_p((\alpha ,s),(\beta ,t)):=\sqrt{s^2+t^2-2st\cos \sphericalangle (\alpha ,\beta )}. \end{aligned}$$
For \(u=(\alpha ,s)\) and \(v=(\beta ,t)\in T_pM\), we often denote \(\Vert u-v\Vert _p:=d_p(u,v)\), \(\Vert u\Vert _p:=d_p(o_p,u)=s\) and
Proposition A.11
Let \((M,d)\) be a geodesic space and \(p\in M\) be fixed. If \(\mathrm{curv}(M)\ge \kappa \) for some \(\kappa \in {\mathbb {R}}\), then \(T_pM\) is a metric space with \(\mathrm{curv}(T_pM)\ge 0\) (in the sense of Definition A.7).
Note that the tangent space is not always a geodesic space, see discussion before Proposition 28 in [63] and the Proposition itself for the proof. Notation \(\Vert .\Vert _p\) and \(\langle .,.\rangle _p\) introduced above is justified by the fact that the cone \(T_pM\) possesses a Hilbert-like structure described as follows. For a point \(u=(\alpha ,t)\) and \(\lambda \ge 0\), we define \(\lambda \cdot u:=(\alpha , \lambda t)\). Then, the sum of points \(u,v\in T_pM\) is defined as the mid-point of \(2\cdot u\) and \(2\cdot v\) as defined in Definition A.3. Finally, it may be checked using the previous definitions that, for any \(u,v\in T_pM\) and any \(\lambda \ge 0\), we get
Next we define logarithmic maps. Fix \(p\in M\). Since \(M\) has upper or lower bounded curvature, the angle monotonicity imposes the following observation. If there are two geodesics \(\gamma ^1_x\) and \(\gamma ^2_x\) connecting p to x such that \(\sphericalangle (\gamma ^1_x,\gamma ^2_x)=0\), then \(\gamma ^1_{x}=\gamma ^2_{x}\). In other words, the set of points \(x\in M\) for which there is not only one equivalence class of geodesics connecting p to x is exactly the set of points x connected to p by at least two distinct geodesics. This set of points is denoted C(p) and called the cut-locus of p. For all \(x\in M\setminus C(p)\), we denote \(\uparrow ^x_{p}\) the direction of the unique geodesic \(\gamma _x:[0,1]\rightarrow M\) connecting p to x.
Definition A.12
(Logarithmic map) Let \((M,d)\) be a geodesic space with curvature bounded from above or below in the sense of Definition A.5 and fix \(p\in M\). We denote
For all \(t\in [0,1]\) and all \(x\in M\setminus C(p)\), one checks in particular that
More generally, if \(\gamma :[0,1]\rightarrow M\) is a geodesic in \(M\) and if we denote \(p=\gamma (t)\) for some \(t\in [0,1]\), then provided \(\gamma (0)=x\) and \(\gamma (1)=y\) both belong to \(M\setminus C(p)\), we check that, for all \(s\in [0,1]\),
Note finally that C(p) is empty for spaces of curvature bounded from above by 0. It is often practical to extend the definition of \(\log _p\) to the cut locus C(p). To that aim, it suffices to select, for any \(x\in C(p)\), a geodesic connecting p to x. Denoting as above \(\uparrow _p^x\) the direction associated to this chosen geodesic, we simply extend \(\log _p\) to M by setting \(\log _p(x)=(\uparrow ^{x}_p,d(p,x))\). This extension depends a priori on the choice of geodesics connecting p to \(x\in C(p)\).
Remark A.13
Throughout the paper, we assume this extension \(\log _p:M\rightarrow T_pM\) can be chosen measurable with respect to the respective Borel \(\sigma \)-algebras.
Next is a fundamental result.
Proposition A.14
Let \((M,d)\) be a geodesic space and \(p\in M\) be fixed. If \(\mathrm{curv}(M)\ge 0,\) then for all \(x,y\in M\),
with equality if \(x=p\) or \(y=p\).
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Ahidar-Coutrix, A., Le Gouic, T. & Paris, Q. Convergence rates for empirical barycenters in metric spaces: curvature, convexity and extendable geodesics. Probab. Theory Relat. Fields 177, 323–368 (2020). https://doi.org/10.1007/s00440-019-00950-0
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DOI: https://doi.org/10.1007/s00440-019-00950-0
Mathematics Subject Classification
- 51F99
- 51K10
- 62G05