1.1. Penalization of barycenters in the Wasserstein space
In this paper we are interested in the penalization of barycenters in the Wasserstein space, which is a minimization problem of the form
where is a subset of
, which is the Wasserstein space of probability measures on
with finite second moments;are given probability measures in ; is the -Wasserstein distance between two probability measures in (cf. Section 2), and is an entropy functional. Finally is a given regularization/penalization parameter; are given non-negative numbers (weights) satisfying .
1.2. Literature review
Problem (1) for has been studied intensively in the literature. It was first studied by Knott and Smith  for Gaussian measures. In , Agueh and Carlier studied the general case proving, among other things, the existence and uniqueness of a minimizer provided that one of vanishes on small sets (i.e., sets whose Hausdorff dimension is at most ). The minimizer is called the barycenter of the measures with weights extending a classical characterization of the Euclidean barycenter. The article  has sparked off many research activities from both theoretical and computational aspects over the last years. Wasserstein barycenters in different settings, such as over Riemannian manifolds and over discrete data, have been investigated [19, 4]. Connections between Wasserstein barycenters and optimal transports have been explored [27, 18]. Several computational methods for the computation of the barycenter have been developed [12, 2, 21, 28]
. Recently Wasserstein barycenters has found many applications in statistics, image processing and machine learning[29, 23, 31]. We refer the reader to the mentioned papers and references therein for a more detailed account of the topic.
The case has been studied in the recent paper  where the existence, uniqueness and stability of a minimizer, which is called the penalized barycenter, has been established. The regularization parameter was proved to provide smooth barycenters especially when the input probability measures are irregular which is useful for data analysis [7, 30]. In addition, the penalized barycenter problem also resembles the discretization formulation of Wasserstein gradient flows for dissipative evolution equations [17, 3, 10] and the fractional heat equation  at a given time step where represent discretized solutions at the previous steps and is proportional to the time-step parameter.
Gaussian measures play an important role in the study of Wasserstein barycenter problem since in this case an useful characterization of the barycenter exists [1, 6] which gives rise to efficient computational algorithms such as the fixed point approach  and the gradient projection method . Our aim in this paper is to seek for a large class of probability measures so that the penalized barycenter can be explicitly characterized and computed similarly to the case of Gaussian measures. We will study the penalization problem (1) for an important classes of probability measures, namely -exponential measures, where the entropy functional is the Tsallis entropy functional respectively. The class of -exponential measures significantly enlarges that of Gaussian measures and containing also -Gaussian measures as special cases, cf. Section 1.3 below. To state our main results, we briefly recall the definition of -exponential measures; more detailed will be given in Section 2.
1.3. -exponential distributions
Let be an increasing, positive, continuous function on , the -logarithmic is defined by 
which is increasing, concave and on . Let and be respectively the infimum and the supremum of , that is
The function has the inverse function, which is called the -exponential function, and is defined on . This inverse function can be extended to the whole as
which is on .
Let () be the set of symmetric (positive definite, respectively) matrices of order . Let
be a given vector andbe a given symmetric positive definite matrix. The -exponential measure with mean and covariance matrix is the probability measure on with Lebesgue density
where , and are continuous functions on playing the role of normalization constants. Two important examples of -exponential measures include Gaussian measures and -Gaussian measures corresponding to and respectively. The -exponential measures play an important role in statistical physics, information geometry and in the analysis of nonlinear diffusion equations [26, 25, 32, 33]. More information about -exponential measures will be reviewed in Section 2.
1.4. Main results of the paper
As already mentioned, in this paper we study the penalization problem (1) for Gaussian measures and -exponential measures, where the entropy functional is the (negative) Boltzmann entropy functional and the Tsallis entropy functional respectively. Main results of the present paper are explicit characterizations of the minimizer of (1) and properties of the objective functions that can be summarized as follows.
Suppose are either Gaussian measures or -Gaussian measures or -exponential measures (in this case, ) with mean zero. Then the minimization problem (1) has a unique minimizer whose variance solves the nonlinear matrix equation (20) or (30) or (48) respectively. Furthermore, the objective function is strictly convex.
The gradient function of the objective function is Lipschitz continuous.
Theorem 1.1 summarizes Theorem 3.1 (for Gaussian measures), Theorem 4.1 (for -Gaussian measures) and Theorem 5.1 (for general -exponential measures). Theorem 1.2 summarizes Theorem 6.2 (for Gaussian measures) and Theorem 6.3 (for -Gaussian measures).
where denotes a Gaussian measure with mean and covariance matrix . Therefore, since the Wassertein distance between Gaussian measures can be computed explicitly, the objective functional in (1) can also be computed explicitly in terms of the variances and (1) becomes a minimization problem over the space of symmetric positive definite matrices. We then prove the strict convexity of the objective function and the existence of solutions to the optimality equation using matrix analysis tools as in . Theorems 3.1, 4.1 and 5.1 establish the existence and uniqueness of a minimizer and provide an explicit characterization of the minimizer in terms of nonlinear matrix equations for the variance generalizing the characterization of the Wasserstein barycenter for Gaussian measures in [1, 6] to the penalized Wasserstein barycenter for Gaussian measures and -exponential measures. Theorem 6.2 and Theorem 6.3 prove the Lipschitz continuity of the gradient of the objective function providing an explicit upper bound for the Lipschitz constant generalizing the results of  for the barycenter for Gaussian measures to our setting. We also perform numerical experiments to show the affect of the parameter and a stability property of the algorithm under small perturbation of the data, cf. Section 7.
1.5. Organization of the paper
The rest of the paper is organized as follows. In Section 2 we review relevant knowledge that will be used in subsequent sections on the Wasserstein metric and the Wasserstein geometry of Gaussian and -exponential distributions. Then we study the penalization of barycenters for Gaussian measures in Section 3 and extend these results to -Gaussian and -exponential measures in Section 4 and Section 5. In Section 6 we describe a gradient projection method for the computation of the minimizer and prove that the gradient function is Lipschitz continuous. Finally, in Section 7, we numerically show affect of parameters to the minimizer and stability of the algorithm under small perturbation of data.
2. Wasserstein metric, Gaussian measures and -exponential measures
In this section, we summarize relevant knowledge that will be used in subsequent sections on the Wasserstein metric and the Wasserstein geometry of Gaussian and -exponential distributions.
2.1. Wasserstein metric
We recall that is the space of probability measures in with finite second moments, that is
Let and be two probability measures belonging to . The -Wasserstein distance, , between and is defined via
where denotes the set of transport plans between and , i.e., the set of all probability measures on having and as the first and the second marginals respectively. More precisely,
for all Borel measurable sets . It has been proved that, under rather general conditions (e.g., when and are absolutely continuous with respect to the Lesbegue measure), an optimal transport plan in (7) uniquely exists and is of the form for some convex function where denotes the push forward [9, 15].
The Wasserstein distance is an instance of a Monge-Kantorevich optimal transportation cost functional and plays a key role in many branches of mathematics such as optimal transportation, partial differential equations, geometric analysis and has been found many applications in other fields such as economics, statistical physics and recently in machine learning. We refer the reader to the celebrated monograph for a great exposition of the topic.
We now consider two important classes of probability measures, namely Gaussian measures and -exponential measures, for which there is an explicit expression for the Wasserstein distance between two members of the same class. Although Gaussian measures are special cases of -exponential measures, but we consider them separately since many proofs for the former are much simplified than those for the latter.
2.2. Wasserstein distance of Gaussian measures
2.3. The entropy of Gaussian measures
The (negative) Boltzmann entropy functional of a probability measure is defined by
Using Gaussian integral, the (negative) Boltzmann entropy of a Gaussian measure can be computed explicitly [11, Theorem 9.4.1]:
We now consider the second class of probability measures: -exponential measures.
2.4. -exponential measures and Wassertein distance
We recall that for a given increasing, positive and continuous function on , the -logarithmic function and the -exponential function are respectively defined in (2) and (4). Two important classes of -exponential functions are:
: the -logarithmic function and the -exponential function become the traditional logarithmic and exponential functions: .
for some : the -logarithmic function and the -exponential function become the -logarithmic and -exponential functions respectively
where and by convention . The -logarithmic function satisfies the following property
For any , we define to be the set of all increasing, continuous function on such that where
where , is a probability density on with mean and covariance matrix , which is called a -exponential distribution. Note that, in the above expression, and
are enough to define only at the identity matrix, not on all . We define the space of all -exponential distribution measures by
Above is the Lesbesgue measure on . Two important cases:
, reduces to the class of Gaussian measures with mean and covariance matrix .
In the case , becomes the class of all -Gaussian measures
and are given by
Note that when , . Thus Gaussian measures are special cases of -Gaussian measures.
The -exponential measures play an important role in statistical physics, information geometry and in the analysis of nonlinear diffusion equations [26, 25, 32, 33]. We refer to [25, 32, 13] for further details on -Gaussian measures, -exponential measures and and their properties.
The following result explains why -Gaussian measures and -exponential measures are special. It will play a key role in the analysis of this paper.
For any , the space of -Gaussian measures is convex and isometric to the space of Gaussian measures with respect to the Wasserstein metric.
For any with , the space is convex and isometric to the space of Gaussian measures with respect to the Wasserstein metric.
Let and be two -exponential distributions. Then , where is defined in (9), is the optimal plan in the definition of .
2.5. The Tsallis entropy of a -Gaussian measure
The Tsallis entropy is defined by
The Tsallis entropy of a -Gaussian can also be computed explicitly using the property (13) and similar computations as in the Gaussian case.
It holds that 
3. Penalization of barycenters for Gaussian measures
In this section we study the following penalization of barycenters in the space of Gaussian measures
where the (negative) Boltzmann entropy functional of a probability measure defined in (10) and is a regularization parameter.
We assume that and seek for a Gaussian minimizer . We note that we consider here Gaussian measures with zero mean just for simplicity. The main results of the paper can be easily extended to the case of non-zero mean. From now on, we equip with the Frobenius inner product . The Frobenius norm is defined by . For we write if is positive semidefinite, and if is positive definite. Note that if and only if for all We denote by the Löwner order interval
Assume that . The penalization of barycenters problem (1) has a unique solution where the covariance matrix solves the following nonlinear matrix equation
In particular, in the scalar case (), we obtain
Before proving this theorem, we show the existence of solutions to equation (20).
Equation (20) has a positive definite solution.
Pick so that for all Set
By definition of and ,
for every This shows that the map is a continuous self map on the Löwner order interval By Brouwer’s fixed point theorem, it has a fixed point. ∎
We are now ready to prove Theorem 3.1
Proof of Theorem 3.1.
Thus we can write (1) as a minimization problem in the space of positive definite matrices
It has been proved  that
is strictly convex,
denotes the geometric mean betweenand defined by
which is symmetric in and . According to [22, Proof of Theorem 8, Chapter 10]
is strictly convex. Using Jacobi’s formula for the derivative of the determinant and the chain rule, we get
It follows that is strictly convex. Furthermore, we have
From this we deduce that
where the gradient is with respect to the Frobenius inner product. Hence if and only if
Using the definition (24) of the geometric mean, the above equation can be written as
which is equation (20). By Lemma 3.2 this equation has a positive definite solution. This together with the strict convexity of imply that has a unique minimizer which is a Gaussian measure where solves (20). In the one dimensional case this equation reads
which results in
This completes the proof of the theorem. ∎
4. Penalization of barycenters for -Gaussian measures
In this section we study the following penalization of barycenters in the space of -Gaussian measures
where the Tsallis entropy functional defined by
We assume that and seek for a Gaussian minimizer .
Assume that . Suppose that for all . The penalization of barycenters problem (28) has a unique solution for all if either or and for sufficiently small if . The covariance matrix solves the following nonlinear matrix equation
where is defined by
The following proposition shows that equation (30) possesses a positive definite solution.
Equation (30) has a positive definite solution.
Similarly as the proof of Lemma 3.2 we will also apply Brouwer’s fixed point theorem. We will show that
has a fixed point which is a positive definite matrix. Due to the appearance of the second term on the left-hand side of (30) the proof of this proposition is significantly involved than that of Lemma 3.2. Suppose that for all . Then similarly as in the proof of Lemma 3.2, for (with chosen later), we have
Multiplying this inequality with then adding them together, noting that , we obtain
from which it follows that
To continue we consider two cases.
Case 1: . It follows from (32) that
Since , we have .
Case 1.1: . Consider the following equation
We have and . Since is continuous, it follows that there exists such that , that is
Similarly by considering the function , we deduce that there exists such that
Case 1.2: . Using the same argument as in the previous case for
we can show that there exist such that
Therefore in both Cases 1.1 and 1.2, there exist such that
Substituting these quantities into (33) we obtain