Page 297 - Special Topic Session (STS) - Volume 1

STS430 Eustasio D.B. et al.
which measures the spread of the distributions. Then, to measure closeness to
a deformation model, we take a look at the minimal variation among warped
distributions, a quantity that we could consider as a minimal alignment cost.
Under some mild conditions, a deformation model holds if and only if this
minimal alignment cost is null and we can base our assessment of a
deformation model on this quantity.
As in [16, 26], we provide results (a Central Limit Theorem and bootstrap
versions) that enable to reject that the minimal alignment cost exceeds some
threshold, and hence to conclude that it is below that threshold. Our results
are given in a setup of general, nonparametric classes of warping functions.
We also provide results in the somewhat more restrictive setup where one is
interested in the more classical goodness-of-fit problem for the deformation
model. Note that a general Central Limit Theorem is available for the
Wasserstein distance in [19]. This works is published in a long version in [9].

2. Wasserstein variation and deformation models for distributions
Much recent work has been conducted to measure the spread or the inner
structure of a collection of distributions. In this paper, we define a notion of
variability which relies on the notion of Fréchet mean for the space of
probabilities endowed with the Wasserstein metrics, of which we will recall the

definition hereafter. First, for any integer d ≥ 1, consider the set (ℝ ) of

probabilities with finite rth moment. For and in (ℝ ), we denote by

∏(, ) the set of all probability measures over the product set ℝ × ℝ
with first (respectively second) marginal (respectively ) . The

transportation cost between these two measures is defined as

(, ) = inf ∫‖ − ‖ (, ).

∏(,)

This transportation cost makes it possible to endow the set (ℝ ) with

the metric (, ). More details on Wasserstein distances and their links with

optimal transport problems can be found, e.g., in [27, 35].
Within this framework, we can define a global measure of separation of a

collection of probability measures as follows. Given , … (ℝ ), let

1

1 ⁄

1

( , … , ) = inf { ∑ ( , )}

1

(ℝ )
=1
be the Wasserstein r-variation of , … , or the variance of the s.

1

The special case r = 2 has been studied in the literature. The existence of
2
2
a minimizer of the map ↦ { ( , ) + ⋯ + ( , )}/ is proved in [1], as
2

1
2
well as its uniqueness under some smoothness assumptions. Such a minimizer,
, is called a barycenter or Fréchet mean of , … , . Hence,
1

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