Entropic regularization of Wasserstein distance between infinite-dimensional Gaussian measures and Gaussian processes

This work studies the entropic regularization formulation of the 2-Wasserstein distance on an infinite-dimensional Hilbert space, in particular for the Gaussian setting. We first present the Minimum Mutual Information property, namely the joint measures of two Gaussian measures on Hilbert space with the smallest mutual information are joint Gaussian measures. This is the infinite-dimensional generalization of the Maximum Entropy property of Gaussian densities on Euclidean space. We then give closed form formulas for the optimal entropic transport plan, entropic 2-Wasserstein distance, and Sinkhorn divergence between two Gaussian measures on a Hilbert space, along with the fixed point equations for the barycenter of a set of Gaussian measures. Our formulations fully exploit the regularization aspect of the entropic formulation and are valid both in singular and nonsingular settings. In the infinite-dimensional setting, both the entropic 2-Wasserstein distance and Sinkhorn divergence are Fr\'echet differentiable, in contrast to the exact 2-Wasserstein distance, which is not differentiable. Our Sinkhorn barycenter equation is new and always has a unique solution. In contrast, the finite-dimensional barycenter equation for the entropic 2-Wasserstein distance fails to generalize to the Hilbert space setting. In the setting of reproducing kernel Hilbert spaces (RKHS), our distance formulas are given explicitly in terms of the corresponding kernel Gram matrices, providing an interpolation between the kernel Maximum Mean Discrepancy (MMD) and the kernel 2-Wasserstein distance.


Introduction
In this work, we study the entropic regularization formulation of the 2-Wasserstein distance in the Hilbert space setting, with a particular focus on Gaussian measures and covariance operators on Hilbert space. This is the infinite-dimensional generalization of recent work on the entropic 2-Wasserstein distance between Gaussian measures on R n , as reported in [55,45,24]. Our work is along the direction of entropic regularization in optimal transport, which has recently attracted much attention in various fields, in particular machine learning and statistics [20,83,30,35,36,38,56,75,77], with applications in computer vision, density functional theory, and inverse problems (e.g. [36,37,53,69]). This direction of research is also closely connected with the Schrödinger bridge problem [81], which has been studied extensively [11,18,21,31,78,91,33,52,79,80].
Our focus in the Gaussian setting stems not only from its use in elucidating various aspects of the abstract theory, since many quantities of interest admit closed form formulas, but also from numerous applications utilizing Gaussian measures and covariance matrices/operators. These include brain imaging [3,27], computer vision [88,89,87], and brain computer interfaces [16]. Many distances/divergences have been studied and employed in practice, including the affine-invariant Riemannian metric [71], corresponding to the Fisher-Rao distance between centered Gaussians, the Alpha Log-Determinant divergences [13], corresponding to Rényi divergences between centered Gaussians, the Log-Euclidean metric [4], and recent work attempting to unify them [2,15,86,63].
Infinite-dimensional setting. The generalization of distances/divergences for Gaussian measures and covariance matrices on R n to the infinite-dimensional setting of Gaussian measures and covariance operators on Hilbert spaces has been carried out by various authors. In general, the infinite-dimensional formulations are substantially more complex than the finite-dimensional ones and regularization is often necessary. This is the case for the affine-invariant Riemannian distance [51], the Log-Hilbert-Schmidt metric [60], the Alpha and Alpha-Beta Log-Determinant divergences [59,61,58,64,65]. The settings for these distances/divergences are the sets of positive definite unitized trace class/ Hilbert-Schmidt operators, which are positive trace class/Hilbert-Schmidt operators plus a positive scalar multiple of the identity operator so that operations such as inversion, logarithm, and determinant, are well-defined. A particular advantage of the 2-Wasserstein distance compared to the above distances/divergences is that the finite and infinite-dimensional distance formulas [34,19] are the same and no regularization is necessary.
Reproducing kernel Hilbert space (RKHS) setting. From the computational and practical viewpoint, this setting is particularly interesting since many quantities of interest admit closed forms via kernel Gram matrices which can be efficiently computed. Examples include the kernel Maximum Mean Discrepancy (MMD) [41] and the RKHS covariance operators, the latter resulting in powerful nonlinear algorithms with substantial improvements over finite-dimensional covariance matrices, see e.g. [93,43,60,66,92] for examples of applications in computer vision.
Contributions of this work.
1. We generalize the Maximum Entropy property of Gaussian densities in R n to the Hilbert space setting, namely the Minimum Mutual Information of joint measures of two Gaussian measures on Hilbert space.
2. For two Gaussian measures on a Hilbert space H, we provide closed form formulas for the optimal entropic transport plan, the entropic 2-Wasserstein distance and the Sinkhorn divergence, generalizing results in [55,45,24]. 3. For a set of Gaussian measures, we show a new Sinkhorn barycenter equation, with always a unique non-trivial solution. In contrast, we show that the finite-dimensional barycenter equation for the entropic 2-Wasserstein distance fails to generalize to the Hilbert space setting. 4. In the RKHS setting, we present closed form formulas for the distances via the finite kernel Gram matrices, providing an interpolation between the kernel MMD [41] and kernel Wasserstein distance [92,62]. 5. Our proofs and results fully exploit the regularization aspect of the entropic formulation and are valid both in singular and non-singular settings. This is novel also in the finite-dimensional setting (compared to [55,45,24]). 6. As we discuss in detail, many properties of the Gaussian case have not been proved in the general theory, due to (i) dim(H) = ∞, (ii) c(x, y) = ||x−y|| 2 is unbounded on H, (iii) the support of Gaussian measures is unbounded .
The Banach space Tr(H) of trace class operators on H is defined by (see e.g. [76]) Some key differences between the finite and infinite-dimensional settings are 1. On R n , for two random variables X, Y with joint and marginal measures µ XY , µ X , µ Y , having densities f (x, y), f X (x), f Y (y), respectively, with respect to the Lebesgue measure, their mutual information is defined as [17] I(X; Y ) = where H(X) = − R n log[f X (x)]f X (x)dx is the differential entropy of X.
The classical Maximum Entropy of Gaussian densities property ( [43], Theorem 9.6.5) states that if X has mean zero and covariance matrix C, then H(X) ≤ 1 2 log(2πe) n det(C), with equality if and only if X ∼ N (0, C).
Thus if both X and Y have Gaussian densities, then I(X; Y ) is minimum if and only if their joint density is Gaussian, so that for c(x, y) = ||x − y|| 2 , and ν 0 , ν 1 being Gaussian, a minimizing γ in (4) is necessarily a joint Gaussian measure of ν 0 , ν 1 . When C is a covariance operator on H with dim(H) = ∞, the quantity det(C) is no longer well-defined. However, I(X; Y ) = KL(µ XY ||µ X ⊗ µ Y ) is well-defined and finite whenever µ XY is absolutely continuous with respect to µ X ⊗ µ Y . In the following, we show that the above Minimum Mutual Information property of joint Gaussian measures generalizes to the infinite-dimensional setting.

If
A is a strictly compact operator on H, then A −1 is unbounded when dim(H) = ∞. Thus finite-dimensional methods that utilize matrix inversion extensively, e.g. in [55,45,24], are not applicable when dim(H) = ∞. Instead, we fully exploit the regularization aspect of problem (4) and invert operators of the form γI + A > 0, thus our proofs fully resolve this issue and are valid in the general setting when A can be singular.
3. The identity operator I is not trace class when dim(H) = ∞. This leads to the breakdown in the entropic barycenter problem (Theorem 12) and has consequences for the analysis of the existence of solutions of the barycenter equations (detail given in Section 11).

Main Results
We first state the following generalization of the Maximum Entropy of Gaussian densities in R n . To the best of our knowledge, this property has not been explicitly and rigorously presented in the literature in the infinite-dimensional setting. It states that among all joint measures, with the same covariance operators, of two Gaussian measures µ X , µ Y on two separable Hilbert spaces H 1 , H 2 , the ones with the minimum Mutual Information are precisely the joint Gaussian measures on H 1 × H 2 . In the following, Gauss(H) denotes the set of all Gaussian measures on H and Gauss(µ X , µ Y ) denotes the set of joint Gaussian measures having marginals µ X and µ Y .
Theorem 1 (Minimum Mutual Information of Joint Gaussian Measures) Let H 1 , H 2 be two separable Hilbert spaces. Let µ X = N (m X , C X ) ∈ Gauss(H 1 ), µ Y = N (m Y , C Y ) ∈ Gauss(H 2 ), ker(C X ) = ker(C Y ) = {0}. Let γ ∈ Joint(µ X , µ Y ), γ 0 ∈ Gauss(µ X , µ Y ), γ 0 is equivalent to µ X ⊗ µ Y . Assume that γ and γ 0 have the same covariance operator Γ and that µ X ⊗ µ Y has covariance operator Γ 0 . Then Equality happens if and only if γ = γ 0 . Here V is the unique bounded linear operator satisfying V ∈ HS(H 2 , H 1 ), ||V || < 1, such that Γ = Γ 1/2 0 The operator V in Theorem 1 is defined in Section 7. It links the covariance operators C X and C Y with the cross-covariance operator C XY via the relation Y . In Eq. (9), det refers to the Fredholm determinant (see e.g. [82]). Let A ∈ Tr(H), then the Fredholm determinant of I + A is given by det(I + A) = ∞ j=1 (1 + λ j ), where {λ j } j∈N are the eigenvalues of A. Theorem 1 in turn follows from the following more general result on the KL divergence on Hilbert space. It states in particular that if µ is a Gaussian measure on H, then among all probability measures with the same mean and covariance operator, KL(γ||µ) is minimum if and only if γ is Gaussian.
In Theorem 2, det 2 refers to the Hilbert-Carleman determinant (see e.g. [82] A direct consequence of Theorem 1 is that if C 0 , C 1 are nonsingular, then a minimizer of problem (13) is necessarily a joint Gaussian measure of µ 0 and µ 1 . We show that this holds in the general setting, i.e. in both nonsingular and singular cases. The following result gives the explicit formula for this minimizer, which is unique for any ǫ > 0. It is proved in Section 8, using two different methods: (i) by directly solving the optimization (13), and (ii) by solving the corresponding Schödinger system.
For each fixed ǫ > 0, problem (13) has a unique minimizer γ ǫ , which is the Gaussian measure where The Radon-Nikodym derivative of γ ǫ with respect to µ 0 ⊗ µ 1 is given by where the functions α ǫ : R → R and β ǫ : R → R take the form The constants a, b ∈ R and the operators A, B : H → H are given by Here det is the Fredholm determinant and M ǫ ij : H → H, are defined by Remark 1 The operator M ǫ ij as defined in Eq. (20) can be rewritten as from which it follows that M ǫ ij ∈ Sym + (H) ∩ Tr(H). Thus for the operator the Fredholm determinant det I + 1 2 M ǫ ij is well-defined and positive.
We remark that in the case the cost function c(x, y) is bounded, much more can be said about the duality formulation, see [25].
is convex in each argument. In particular, let C 0 be fixed, then the function Theorem 6 (Lower bound for Entropic 2-Wasserstein distances between general probability measures on Hilbert space) Let µ 0 , µ 1 be two probability measures on H, with means m 0 , m 1 and covariance operators C 0 , C 1 , respectively. Then Equality happens if and only if µ 0 , µ 1 are Gaussian measures.
In [30], the Sinkhorn divergence was proved to be convex in each variable for either a compact metric space X or for measures with bounded support on R n , with cost function c(x, y) = ||x − y|| p , p = 1, 2. In [44], this was shown for sub-Gaussian measures on R n , c(x, y) = ||x − y|| 2 . The following shows strict convexity for Gaussian measures on H with c(x, y) = ||x − y|| 2 . This property is crucial for guaranteeing the uniqueness of the barycenter problem below.
In [30], positivity of the Sinkhorn divergence was proved for either a compact metric space X or for measures with bounded support on R n , with cost function c(x, y) = ||x − y|| p , p = 1, 2. We now show that in the Gaussian case, with c(x, y) = ||x − y|| 2 , this holds in the much more general Hilbert space setting.
Furthermore, F S is strictly convex and F E is convex on Ω, with strict convexity if C 0 is strictly positive.
The expressions for OT ǫ d 2 and S ǫ d 2 in Theorems 4 and 7 are not intuitively close to the exact OT formula, which is the same as in the finite-dimensional setting [34,19]. Theorem 11 below gives equivalent formulas that better express the connections between the exact and regularized settings. .
Given N probability measures µ i ∈ P(H), i = 1, 2, .., N , the entropic barycenterμ with weights w i > 0, The entropic barycenter problem illustrates clearly the bias effect of the entropic regularization term, as analyzed in the finite-dimensional case, e.g. [44,45]. This bias effect is sharp when dim(H) = ∞, with the finite-dimensional barycenter equation failing to generalize to this case. In the following, we call a barycenter trivial if it is a Dirac delta measure in H.

Theorem 12 (Entropic Barycenter of Gaussians)
.., N be a set of Gaussian measures on H. Then a minimizer of problem (38) in P 2 (H) is necessarily Gaussian. Assume at least one of the C ′ i s is strictly positive. Then on Gauss(H), problem (38) is strictly convex and the first order minimality condition is 1. If ǫI ≥ 2 N i=1 w i C i , the unique barycenter is the Dirac delta measure centered atm = N i=1 w i m i . 2. If dim(H) < ∞, a necessary condition for the existence of a non-trivial barycenter is A sufficient condition for the existence of a non-trivial barycenter is ∃α ∈ R, α > 0 such that C i ≥ αI, 1 ≤ i ≤ N, and 0 < ǫ < 2α. (41) In this case, the barycenter is unique and is the Gaussian measure N (m,C), whereC > 0 is the unique solution of Eq. (39). Equivalently,C is the unique strictly positive solution of the following equation 3. If dim(H) = ∞, then Eq. (39) has no solution in Sym + (H). If ǫ 2 N i=1 w i C i , then the barycenter, if it exists, is not the Dirac measure centered atm.
Discussion of results. Eq.(39) is the first order optimality condition for the strictly convex problem (38). It has sharply different behavior when dim(H) = ∞ compared with the case dim(H) < ∞, as we stated.
Conditions (40) and (41) show that, even in the finite-dimensional setting, a non-trivial entropic barycenter of Gaussian measures exists if and only if ǫ is sufficiently small. Condition (40), namely 0 < ǫI < 2 Under condition (41), Eq.(39) has a unique solution, which is strictly positive. Then Eq.(42) also has a unique strictly positive solution, but it also has the trivial solution X 0 = 0 and uncountably infinitely many positive solutions, which are singular (see Theorem 14 and Proposition 7).

Remark 3
In general, for ǫ > 0, the entropic barycenter problem (38) does not make much sense. Consider the following one-dimensional scenario, where Thus if ǫ ≥ 2σ 2 , then Equation (39) has no positive solution. The above solutionC is also obtained for the case N = 1, w 1 = 1, C 1 = σ 2 > 0, in which case a sensible solution should beC = C 1 , i.e. the barycenter of a set of one point should the point itself. This seemingly pathological behavior is not necessarily surprising, since OT ǫ d 2 is neither a distance nor a divergence.
Sinkhorn barycenter of Gaussian measures. We now consider the barycenter problem with respect to the Sinkhorn divergence. For a set of probability In the current work, we consider barycenter of the N Gaussian measures in the set of all Gaussian measures on H µ = arg min In contrast to the entropic barycenter problem, the debiased Sinkhorn barycenter problem has a consistent generalization to the infinite-dimensional setting, with a unique solution that is valid in both singular and nonsingular cases.
Furthermore,C is strictly positive if and only if Under the additional hypothesis thatC > 0,C is equivalently the unique strictly positive solution of the following equation Define the following map F : Sym + (H) → Sym + (H) by Then the unique solution of Eq.(45) is the unique fixed point of F .
Limiting cases. When ǫ → 0, both Eqs. (42) and (47) become In the finite-dimensional setting, this is the barycenter equation for the exact 2-Wasserstein distance [1], assuming thatC > 0. As of the current writing, to the best of our knowledge, a rigorous proof for the infinite-dimensional case has not yet been established. We note that the proof given in [54], which uses the transport map in [19] to compute gradients, is only applicable in the case dim(H) < ∞, since the transport map is generally unbounded when dim(H) = ∞, see also the discussion in [57].

Exact
Without the condition N i=1 w i C i > 0, this equation always has at least one positive, nonzero singular solution.

Entropic Wasserstein barycenter
3. Sinkhorn barycenter (second version), Without the condition Eq.(47) has the same form as the finite-dimensional version reported in [55] and [45]. It is, however, only applicable for finding the barycenter in the case it is strictly positive, since it is derived under this explicit assumption. If the solution of Eq.(45) is strictly positive, then it is also the unique strictly positive solution of Eq. (47). Eq.(47), however, always has the trivial solution X = 0. Furthermore, if dim(H) ≥ 2 and at least one of the C i 's is positive, then it has uncountably infinitely many positive solutions, which are singular (Proposition 14). The same phenomenon happens for the barycenter equation (50) in the exact, unregularized setting [1], i.e. when ǫ = 0.
As we discuss in detail in Section 11, it is not straightforward to extend the approach in [1] for Eq.(50) and [45] for Eq. (47) in the finite-dimensional setting, which requires all C i 's to be strictly positive for the existence ofC > 0, to the infinite-dimensional setting. This is because it is no longer possible to uniformly lower bound the C i 's by αI for some α > 0 and it is not clear whether this lower bound can be replaced by another strictly positive operator.
We also remark on our condition N i=1 w i C i > 0 for the strict positivity ofC, which is more general than requiring all C i 's to be strictly positive (e.g. [45]).
In fact, we can have N i=1 w i C i > 0, guaranteeingC > 0, with all C i 's being singular (see Section 10 for an example).
The RKHS setting. We now apply the abstract Hilbert space setting above to the reproducing kernel Hilbert space (RKHS) setting. In this case, we obtain an interpolation between Kernel Maximum Mean Discrepancy (MMD) and Kernelized L 2 -Wasserstein Distance. The RKHS formulas are expressed explicitly in terms of the kernel Gram matrices, which are readily computable.
Let X be a complete separable metric space. Let K be a continuous positive definite kernel on X × X . Then the reproducing kernel Hilbert space (RKHS) H K induced by K is separable ( [84], Lemma 4.33). Let Φ : X → H K be the corresponding canonical feature map, so that K(x, y) = Φ(x), Φ(y) HK ∀(x, y) ∈ X × X . Let ρ be a Borel probability measure on X such that Then the RKHS mean vector µ Φ ∈ H K and covariance operator C Φ : H K → H K induced by the feature map Φ are both well-defined and are given by Here the rank-one operator u ⊗ v is defined by (u ⊗ v)w = v, w HK u, u, v, w ∈ H K . Then C Φ is a positive trace class operator on H K (see e.g. [66]).
Let X = [x 1 , . . . , x m ],m ∈ N , be a data matrix randomly sampled from X according to a Borel probability distribution ρ, where m ∈ N is the number of observations. The feature map Φ on X defines the bounded linear operator Φ(X) : The corresponding empirical mean vector and covariance operator for Φ(X) are defined to be where J m = I m − 1 m 1 m 1 T m , 1 m = (1, . . . , 1) T ∈ R m , is the centering matrix, with J 2 m = J m and AJ m is the matrix obtained from the (possibly infinite) matrix A by subtracting the mean column.
Remark 4 To keep our expressions simple, we have assumed that the number of data points in are the same, i.e. m = n. The extension to the case m = n is straightforward.

From Gaussian measures to Gaussian processes
Let us discuss the translation of the results for Gaussian measures on an abstract Hilbert space H into the setting of Gaussian processes, see also [68,32,73,54,57]. Consider the following correspondence between Gaussian measures and Gaussian processes with paths in a Hilbert space, as established in [74]. Let (Ω, F , P ) be a probability space. Let T be an index set. Let (T, A, ν) be a measurable space, ν nonnegative, σ-finite, such that L 2 (T, A, ν) = L 2 (T, ν) is separable (e.g. T ⊂ R n measurable, A = B(T ), ν is the Lebesgue measure). Let ξ = (ξ t ) t∈T = (ξ(t, ω)) t∈T be a real F /A-measurable Gaussian process on (Ω, F , P ), with mean m(t) and covariance function K(s, t), denoted by GP(m, K). The sample paths ξ(·, ω) ∈ H = L 2 (T, ν) almost P -surely, i.e. T ξ 2 (t, ω)dν(t) < ∞, almost P -surely, (63) if and only if ( [74], Theorem 2 and Corollary 1) Then ξ induces the following Gaussian measure P ξ on (H, B(H)) with mean m ∈ H and covariance operator C K : H → H, defined by Conversely, let µ be a Gaussian measure on (H = L 2 (T, A, ν), B(H)). Then there is an F /A-measurable Gaussian process ξ = (ξ t ) t∈T on (Ω, F , P ) with sample paths in H, such that the induced probability measure is P ξ = µ.
Correspondence between covariance function and covariance operator via Mercer Theorem. Covariance functions, being positive definite kernels, can be fully expressed via their induced covariance operators, as follows.
In the following, let T be a σ-compact metric space, that is T = ∪ ∞ i=1 T i , where T 1 ⊂ T 2 ⊂ · · · , with each T i compact. Let ν be a positive, non-degenerate Borel measure on T , i.e. ν(B) > 0 for any open U ⊂ T , with ν(T i ) < ∞∀i ∈ N.
Theorem 16 (Mercer Theorem -version in [85]) Let T be a σ-compact metric space and ν a positive, non-degenerate Borel measure on T . Let K : T × T → R be continuous, positive definite. Assume furthermore that T K(s, t) 2 dν(t) < ∞, ∀s ∈ T. (67) Then C K is Hilbert-Schmidt, self-adjoint, positive. Let {λ k } ∞ k=1 be the eigenvalues of C K , with corresponding orthonormal eigenvectors {φ k } ∞ k=1 . Then where the series converges absolutely for each pair (s, t) ∈ T ×T and uniformly on each compact subset of T × T .
Mercer Theorem thus describes the covariance function K(s, t) fully and explicitly via its covariance operator C K . Since K is positive definite, K(s, t) 2 ≤ K(s, s)K(t, t) ∀s, t ∈ T × T . Thus the condition T K(t, t)dν(t) < ∞ in (64) implies both conditions (67) and (68) in Mercer Theorem and from (69) We now generalize ideas in [68,32,73,54,57], using the fact that Gaussian processes are fully determined by their mean and covariance functions. Most importantly, the following incorporates Mercer Theorem to quantify the correspondence GP(m, K) ⇐⇒ N (m, C K ).
Definition 1 (Divergence between Gaussian processes) Let T be a σ-compact metric space, ν a positive, non-degenerate Borel measure on T . Let H = L 2 (T, B(T ), ν). Let ξ i = GP(m i , K i ), i = 1, 2, be two Gaussian processes with mean m i ∈ H, covariance function K i continuous, and Let D be a divergence function on Gauss(H)×Gauss(H). The corresponding divergence D GP between ξ 1 and ξ 2 is defined to be Mercer Theorem immediately implies the following.
Theorem 17 Assume the hypothesis in Definition 1. Then Remark 5 In our current context, we can immediately apply Definition 1 and Theorem 17 to the exact Wasserstein distances and Sinkhorn divergences. Definition 1 also can also be extended to cover the entropic OT ǫ c distances, however since they are not metrics/divergences, Theorem 17 no longer holds.

Kullback-Leibler divergence between Gaussian measures
In this section, we briefly review the KL divergence between Gaussian measures on Hilbert space and prove Theorem 2. For a separable Hilbert space H, consider the set P(H) of probability measures on (H, B(H)), where B(H) denotes the Borel σ-algebra on H. We focus on the subset P 2 (H) defined by For µ ∈ P 2 (H), its mean vector m ∈ H and covariance operator C : H → H are well-defined and are given by In particular C is a self-adjoint, positive, and trace class operator on H.
We recall that for two measures µ and ν on a measure space (Ω, F ), with µ σ-finite, ν is said to be absolutely continuous with respect to µ, denoted by ν << µ, if for any A ∈ F , µ(A) = 0 ⇒ ν(A) = 0. In this case, the Radon-Nikodym derivative dν dµ ∈ L 1 (µ) is well-defined. The Kullback-Leibler (KL) divergence between ν and µ is defined by If µ << ν and ν << µ, then we say that µ and ν are equivalent, denoted by µ ∼ ν. We say that µ and ν are mutually singular, Theorem [29], [42], which states that two Gaussian measures µ = N (m 1 , Q) and ν = N (m 2 , R) are either mutually singular or equivalent. The necessary and sufficient conditions for the equivalence of the two Gaussian measures ν and µ are given by the following.
2. There exists S ∈ Sym(H) ∩ HS(H), without the eigenvalue 1, such that We now recall results on Kullback-Leibler divergences between two Gaussian If µ ∼ ν, then we have the following result.
For two equivalent Gaussian measures µ, ν on H, the Radon-Nikodym derivative involves only the means and covariance operators ( [65], Theorem 11). This motivates Theorem 2, which seeks to extend the validity of Eq.(78) by generalizing the expression H log ( dν dµ )dν to H log ( dν dµ )dγ where γ ∈ P 2 (H) is any probability measure with the same mean and covariance operator as ν.
To prove Theorem 2, in the following we utilize the concept of white noise mapping, see e.g. [22,23].
For any pair z 1 , z 2 ∈ Q 1/2 (H), we have by definition of the covariance operator Thus the map W : Since ker(Q) = {0}, the subspace Q 1/2 (H) is dense in H and the map W can be uniquely extended to all of H, as follows. For any z ∈ H, let {z n } n∈N be a sequence in Q 1/2 (H) with lim n→∞ ||z n − z|| H = 0. Then {z n } n∈N is a Cauchy sequence in H, so that by isometry, {W zn } n∈N is also a Cauchy sequence in L 2 (H, µ), thus converging to a unique element in L 2 (H, µ). Thus we can define by the following unique limit in L 2 (H, µ) The map W : H → L 2 (H, µ) is called the white noise mapping associated with the measure µ = N (m, Q). W z can be expressed explicitly in terms of the finite-rank orthogonal projections where {e k } k∈N are the orthonormal eigenvectors of Q. For any z ∈ H, we have Thus Q −1/2 P N z is always well-defined ∀z ∈ H. Furthermore, for all x, y ∈ H, The operator The Radon-Nikodym derivative between two equivalent Gaussian measures on H is expressed explicitly via the white noise mapping, as follows.
Let {α k } k∈N be the eigenvalues of S, with corresponding orthonormal eigenvectors {φ k } k∈N . Let W be the white noise mapping induced by µ. The Radon-Nikodym derivative dν dµ is given by where for each k ∈ N, The series ∞ k=1 Φ k converges in L 1 (H, µ) and L 2 (H, µ).
In general, for z ∈ H, in the By the assumption ν << µ, we have supp(ν) ⊂ supp(µ). The sequence {W PN z } N ∈N is a Cauchy sequence in L 2 (H, µ) converging to W z , with a subsequence converging pointwise µ-almost everywhere, hence ν-almost everywhere. This subsequence is also a Cauchy sequence in L 2 (H, ν), converging to a unique limit, with a subsubsequence converging pointwise ν-almost everywhere. Thus this limit must be W z .
For the first expression, for any z ∈ H, N ∈ N, Since ν is a probability measure, we also have lim N →∞ ||W PN z −W z || L 1 (H,ν) = 0 by Hölder Inequality. Thus This completes the proof. ⊓ ⊔ We note that if Φ is an orthonormal basis for L 2 (X, µ), then the hypothesis of Proposition 1 becomes f ∈ L 2 (X, µ) and the conclusion is immediate.
Proof Let S Φ = span{φ k } k∈N be the closed Hilbert subspace of L 2 (X, µ) with orthonormal basis Φ. We show that the following linear functional is well-defined and bounded. First, by assumption, It follows that A f is well-defined on the dense subspace span{φ k } k∈N , with

then by linearity
Using the expression for W φj , W φ k L 2 (H,ν) from Lemma 1, we have Thus g ∈ L 2 (H, γ). Furthermore, for any N ∈ N and Since γ is a probability measure, by Hölder Inequality, we have g ∈ L 1 (H, γ) Letting gives the final answer. ⊓ ⊔ We are now ready to prove Theorem 2.

Mutual information of Gaussian measures on Hilbert space
In this section, we prove Theorem 1. For completeness, we give a new, shorter proof of the mutual information between two Gaussian measures [7]. We start by reviewing joint measures and cross-covariance operators on Hilbert space.
Joint measures. Following [6], let (H 1 , , 1 ) and (H 2 , , 2 ) be two real separable Hilbert spaces with corresponding Borel σ-algebras B(H 1 ) and B(H 2 ). Let H 1 ×H 2 be the Hilbert space with inner product defined by (u 1 , ). Let µ X and µ Y be its marginal probability measures defined on (H 1 , B(H 1 )) and (H 2 , B(H 2 )), respectively. Assume further that µ XY ∈ P 2 (H 1 × H 2 ). Under this assumption, the mean vector m XY ∈ H 1 × H 2 and covariance operator it follows that µ XY ∈ P 2 (H 1 ×H 2 ) if and only µ X ∈ P 2 (H 1 ) and µ Y ∈ P 2 (H 2 ). Subsequently, throughout the paper, we assume that Cross-covariance operators. Let (m X , C X ) and (m Y , C Y ) be the means and covariance operators of µ X and µ Y , respectively. Since µ XY ∈ P 2 (H 1 × H 2 ), the following linear functional G : By the Riesz Representation Theorem, there exist single-valued, bounded, linear maps C XY : Clearly, C Y X = C * XY . In operator tensor product notation, C XY is called the cross-covariance operator of µ XY . It is closely related to the covariance operators of the marginals µ X and µ Y via the following.
Theorem 21 ( [6]) Consider the projection operators P X : Then C XY admits the following representation where V : H 2 → H 1 is a unique bounded linear operator such that ||V || ≤ 1 and V = P X V P Y .
In column vector notation, the mean vector of the joint measure µ XY is given In operator-valued matrix notation, the covari- Mutual information of Gaussian measures. We now consider joint Gaus- is an orthonormal basis for H 1 ×H 2 and 0 V V * 0 Proof By Lemma 3, the operator 0 V V * 0 is self-adjoint and Hilbert-Schmidt on H 1 ×H 2 , thus admits a countable sequence of eigenvalues, with corresponding eigenvectors forming an orthonormal basis for The nonzero eigenvalues of the last operator are precisely the nonzero eigenvalues of V V * : Combining these two equations, we obtain V V * a = λ 2 a and V * V b = λ 2 b. Thus a and b are necessarily eigenvectors of V V * and V * V , respectively, under the same eigenvalue λ 2 . Due to the square factor, it follows that −λ is an eigenvalue By definition of the Hilbert-Carleman determinant, By applying Formula (78) in Theorem 19 and Lemma 4 to our setting, we obtain the expression for KL(µ XY ||µ X ⊗ µ Y ), first proved in [7] (Proposition 2) by a direct approach. The case H 1 = H 2 = H was proved in [5]. As we now show, by employing the more general result on the KL divergence between Gaussian measures in Theorem 19, this result is obtained immediately.
Proof (of Theorem 22) The condition for µ XY ∼ µ X ⊗ µ Y follows from Lemmas 3,4 and Theorem 18. Assume that µ XY ∼ µ X ⊗ µ Y . Since the mean vector for µ XY and µ X ⊗ µ Y is the same, namely m X m Y , the first term in Eq. (78) is equal to zero. From Eqs. (106) and (78), we have where the last equality follows from Lemma 4.

⊓ ⊔
We are now ready to prove Theorem 1, which we restate here for clarity.
Assume that γ and γ 0 have the same covariance operator Γ and that µ X ⊗ µ Y has covariance operator Γ 0 . Then Equality happens if and only if γ = γ 0 . Here V is the unique bounded linear op- we also have γ << γ 0 and the Radon-Nikodym derivatives dγ dγ0 and dγ d(µX ⊗µY ) are both well-defined. Since γ, γ 0 , and µ X ⊗µ Y all have the same mean, namely m X m Y , and γ, γ 0 have the same covariance operators, we have by Theorem 2, The nonsingular case. We first solve (110) for c(x, y) = ||x − y|| 2 under the nonsingular Gaussian setting. Let µ X , m Y ∈ P 2 (H), with means m X , m Y and covariance operators C X , C Y , respectively, then C X , C Y ∈ Tr(H) and Then C XY ∈ Tr(H) and for any A, B ∈ L(H), Proof By Theorem 21, are both Hilbert-Schmidt. Thus it follows that C XY is trace class. Recall that C XY : H → H is given by By Lebesgue Dominated Convergence Theorem, Ax, e j By, e j dµ XY (x, y) If γ ǫ is the minimizer of problem (13), then necessarily γ ǫ ∈ Gauss(µ 0 , µ 1 ) and γ ǫ ∼ µ 0 ⊗ µ 1 .

Lemma 6 Let A : H → H be a compact operator. Then
Proof This is a special case of Lemma 10 and Corollary 2 in [61].
The following result is then immediate.
Lemma 8 Let C ∈ Sym + (H), X ∈ Sym + (H) be fixed. For any constant a ∈ R, a = 0, Proof The desired equality is This last equality is valid as a consequence of Lemma 6, which gives Together with the left hand side of the previous expression, this gives the desired equality.

⊓ ⊔
We apply the following result on the log-concavity of the Fredholm determinant from [59], which is a generalization of Ky Fan's inequality for the log-concavity of the determinant on the set of symmetric positive definite matrices [28].

⊓ ⊔
Let us now compute the Radon-Nikodym density of the optimal entropic transport plan γ ǫ in Theorem 24 with respect to µ 0 ⊗ µ 1 . We note that the optimal V in Theorem 24 is trace class. Consequently, we can apply the following result (see Proposition 1.3.11 in [23] or Corollary 2 in [65]).
where, in the L 1 (H, µ) sense, The following result can be obtained by direct verification.
Eq.(134) is equivalent to Eq. (16) in Theorem 3 via the identity Proof For ||V || < 1, (I − V V * ) is invertible and by Lemma 10, For V = c ǫ (I +(I +c 2 ǫ C 1/2 ǫ , we have V ∈ Tr(H). Similar to the proof of Theorem 24, Thus for any x, y ∈ H and N ∈ N, Combining this with det(I − S) and Proposition 4 gives the desired result. ⊓ ⊔ The general case. We note that while Theorem 24 and Proposition 5 are proved under the hypothesis that ker(C 0 ) = ker(C 1 ) = {0}, the optimal solution obtained is clearly mathematically valid without this assumption. We now confirm that this is indeed the case via a different approach.
Let k(x, y) = exp − c(x,y) ǫ , ǫ > 0. It has been shown that, see e.g. [11,18,25,38,78], problem (110) has a unique minimizer γ ǫ if and only if there exist functions α ǫ , β ǫ satisfying the Schrödinger system In this case, the unique minimizer γ ǫ is the probability measure whose Radon-Nikodym derivative with respect to µ 0 ⊗ µ 1 is given by Motivated by Proposition 5, we now solve the Schrödinger system (136) when c(x, y) = ||x − y|| 2 on H, leading to another proof of Theorem 3, which is valid in the general setting, where C 0 and C 1 can be singular.
We make use of the following results on Gaussian integrals on Hilbert spaces.
The following result then follows immediately Proof (of Theorem 3: optimal entropic transport plan -the general case) We first have The Schrödinger system (136) then becomes Thus for the system of equations (140) to hold ∀x, y ∈ H, we must have We claim that the following A ǫ and B ǫ solve the system of equations (141) Let us verify the first equation in (141) (the second one is analogous). We have Thus in order to have B ǫ = − 2 Replacing the right hand side with the expression in Eq.(143), this is This is precisely Lemma 8 with a = 4 ǫ . Similarly, we have Finally, for the expression of exp(a + b), we note that it is clear that from Eq.(142) that C Here we have used the fact that the eigenvalues of C 1/2 are the same as those of C 1/2 ⊓ ⊔ Lemma 11 Let µ ∈ P 2 (H) with mean m and covariance operator C. Let A ∈ L(H). Then Proof It suffices to consider the case m = 0.
Proof For the optimal Recall that It follows that Here we have invoked Lemma 11. For the first trace term, we have Similarly, the second term is Combining all the previous expressions, we obtain Here we have used the fact that the nonzero eigenvalues of C It follows that D(ϕ ǫ , ψ ǫ ) = ǫE µ0 [log α ǫ ] + ǫE µ1 [log β ǫ ] = OT ǫ d 2 (µ 0 , µ 1 ), as in the proof of Corollary 4.

Corollary 5 (Sinkhorn divergence between Gaussian measures on Hilbert space)
Proof By definition of the Sinkhorn divergence and Theorem 4, This completes the proof.
Using these expressions, we see that the formulas for OT ǫ d 2 and S ǫ d 2 coincide with those in Theorems 4 and 7. Furthermore, it is immediately clear that By L'Hopital's rule, we have ∀x ≥ 0, lim ǫ→0 ǫ log( 1 2 + 1 2 (1 + 16 Combining these limits give the limiting behavior of OT ǫ d 2 and S ǫ d 2 . ⊓ ⊔

Proof of Theorem 15 -RKHS setting. Let λ(A) denote the set of eigenvalues of a compact operator A.
Then Combining these with the expressions for OT ǫ d 2 and S ǫ d 2 in Theorems 4 and 7 gives the desired results.

Remark 6
In Theorem 15, terms of the form AJ m and J m AJ m give the same answer, since they have the same eigenvalues, leading to the same trace and determinant. However, for implementation, the choice of which term to use is important. For computing the log det terms, we usually use the Cholesky decomposition, which requires a symmetric positive definite matrix and hence one needs to use the term J m AJ m , which is symmetric.
We now prove Theorem 6, which we restate as Theorem 26 below.

then we show in Theorems 3 and 4 that
Hence equality happens if and only if µ 0 , µ 1 are Gaussians. ⊓ ⊔

Entropic 2-Wasserstein barycenter
In this section, we prove Theorem 5 on the convexity of OT ǫ d 2 and Theorem 12 on the OT ǫ d 2 -based barycenter. We first recall the concept of the Fréchet derivative on Banach spaces (see e.g. [46]). Let V, W be Banach spaces and L(V, W ) be the Banach space of bounded linear maps between V and W . Assume that f : Then the map f is said to be Fréchet differentiable at x 0 ∈ Ω if there exists a bounded linear map Df (x 0 ) : The map Df (x 0 ) is called the Fréchet derivative of f at x 0 . Let now W = R. If x 0 is a local minimizer for f , then necessarily (see e.g. Theorem 1.33 in [72]) Furthermore, if f is Fréchet differentiable, then (Proposition 3.11 in [72]) for Ω ⊂ V open and convex, Under this identification, In If, furthermore, X 0 is invertible and X ∈ Sym(H) ∩ Tr(H), then In general, let f : L(H) → L(H) be such that f (X 0 ) and X 0 commute, then Proof i) For the function sq(X) = X 2 , we have Let f (X) = X = sq(sqrt(X)), then by the chain rule This gives us the first identity.
ii) For the second identity, we note that Taking the trace operation on both sides gives which gives us the desired expression.
As in the proof of Lemma 18,

Thus it follows from Lemma 17 that
Df (X 0 )(X) = c 2 2 Tr  ||f We remark that the condition that B and C commute is crucial in Lemma 23. It can be verified numerically that, without this condition, the stated inequality is generally false even if A is also positive.
Proof By the assumption that B and C commute, If in addition Y 0 , Z 0 are invertible, then equality happens if and only if X = 0.
Proof By Lemma 17, for any X ∈ Sym(H) ∩ Tr(H), Pre-and post-multiplying both sides by Dsqrt(Z 0 )(X)Y 0 and Y 0 , respectively, Taking trace on both sides and applying Lemma 23 gives The Fréchet derivative Df (X 0 ) : Sym(H) → L(L(H), R) is given by For c = 0, equality happens if and only C 1/2 XC 1/2 = 0. If C is strictly positive, then equality happens if and only if X = 0.
Proof Let h(X) = (I+c 2 C 1/2 XC 1/2 ) 1/2 , g(X) = (I+h(X)) −1 , and f (X)(Y ) = Tr[C 1/2 g(X)C 1/2 Y ]. By the chain rule and Lemma 21, In particular, for Y = X, by Lemma 24, Since Z 0 and (I+Z Then f is convex on Ω. Furthermore, f is strictly convex if C is strictly positive. Proof For any X 0 ∈ Ω, X ∈ Sym(H), by Lemmas 19 and 18, Thus we have the map Df : Ω → L(L(H), R), with Differentiating this map gives the second-order Fréchet derivative where Z 0 = I + c 2 C 1/2 X 0 C 1/2 , by Lemma 25. In particular, for Y = X, Equality happens if and only if C 1/2 XC 1/2 = 0. If C is strictly positive, then equality happens if and only if X = 0 by Lemma 26. Thus f is convex on Ω and furthermore, it is strictly convex if C is strictly positive. ⊓ ⊔

Proof (of Theorem 5 -Convexity of entropic Wasserstein distance)
By the strict convexity of the square Hilbert norm || || 2 , the function m → ||m − m 0 || 2 is strictly convex in m. For the covariance part, by Theorem 4, Since Tr(X) is linear in X and the remaining part is convex in Sym + (H) ∩ Tr(H) by Proposition 6, F is convex in X ∈ Sym + (H) ∩ Tr(H), with strict convexity if C 0 is strictly positive. ⊓ ⊔ Then f is strictly convex. For the minimum of f , there are two scenarios Equivalently, x * is the unique positive solution of the equation which also has the solution x 0 = 0. Proof is a strictly increasing function on [0, ∞).
We have the following three scenarios In this case f ′ (0) > 0 and thus f ′ (x) > 0 ∀x > 0 and thus the minimum In this case f ′ (0) < 0 and thus there exists a unique x * > 0 such that f ′ (x * ) = 0. This is the unique global minimizer of f and We note the last equation also has the solution x = 0. ⊓ ⊔ Proof (of Theorem 12 -Infinite-dimensional setting) As a direct consequence of Theorem 6, the entropic barycenter of a set of Gaussian measures in P 2 (H), if it exists, must be another Gaussian measure on H. By Theorem 4, OT ǫ d 2 (N (m 0 , C 0 ), N (m 1 , C 1 )) decomposes into the squared Euclidean distance ||m 0 − m 1 || 2 and the distance OT ǫ d 2 (N (0, C 0 ), N (0, C 1 )). It follows that we can compute the barycentric mean and covariance operator separately.

The barycentric mean is obviously that the Euclidean meanm =
On Ω, F is Fréchet differentiable. Since we assume that at least one of the C ′ i s is strictly positive, F is strictly convex. Thus a minimizer X 0 ∈ Ω of F must necessarily be unique and satisfy DF (X 0 ) = 0.
(ii) Assume now that ǫI 2 N i=1 w i C i . We show that X 0 = 0 is not a global minimum of F in Sym + (H) ∩ Tr(H). For any X 0 ∈ Sym + (H) ∩ Tr(H), The assumption ǫI 2 N i=1 w i C i means that ∃u ∈ H, ||u|| = 1, such that According to Lemma 27, the property 0 < ǫ < 2 N i=1 w i u, C i u implies that there exists a unique x * > 0 such that Thus X 0 = 0 is not a global minimum of F in Sym + ∩ Tr(H).

⊓ ⊔
Proof (of Theorem 12 -Finite-dimensional setting) When dim(H) < ∞, if we impose the additional condition that ker(X 0 ) = {0}, that is X 0 is invertible, then the identity (180) is equivalent to By Lemma 8, on the left hand side,

Substituting into the previous equation, we obtain
This gives Eq. (42). Define the following map F : Sym + (H) → Sym + (H) by then we have by the monotonicity of the square root function Thus it is clear that (39) has the solution X 0 = 0, which is necessarily unique due to the strict convexity of the entropic OT distance.
(iii) The condition 0 < ǫI < 2 N i=1 w i C i is thus necessary if Eq.(39) is to have a solution X 0 ≥ 0, X = 0. By Proposition 9, if C i ≥ αI, 1 ≤ i ≤ N , and 0 < ǫ < 2α, then Eq.(39) has a strictly positive solution, which is necessarily unique due to the strict convexity of the entropic OT distance.
2. Let u ∈ R n , ||u|| = 1. If 0 < ǫ < 2 N i=1 w i u, C i u , then X u = x u (u ⊗ u) = x u uu T is a fixed point of G, where x u is the unique positive solution of the one-dimensional fixed point equation Hence under these assumptions, the map G as defined in Eq.(183), has uncountably infinitely many fixed points, which are positive but singular.

Proof (of Proposition 7)
For the first part, let γ ∈ R, γ > 0 be such that C i ≤ γI, 1 ≤ i ≤ N . Let β ǫ = α − ǫ 2 > 0 and consider the following set This is a compact, convex set in Sym(n). By Lemma 41, the operator monotonicity of the square root function, and the inequality (1 + a 2 ) 1/2 ≤ 1 + a, a ∈ R, a ≥ 0, we have with c ǫ = 4 ǫ , Thus the continuous map G maps the compact convex set K ǫ into itself. By Brouwer Fixed Point Theorem, G has at least a fixed point in K ǫ .
For the second part, by Lemma 40, Therefore, for X u = x(u ⊗ u), Since (I − u ⊗ u) 2 = (I − u ⊗ u) and (I − u ⊗ u)(u ⊗ u) = 0, by Lemma 40, It follows that Thus the fixed point equation X u = G(X u ) is equivalent to By the assumption 0 < ǫ < 2 N i=1 w i u, C i u , this equation has a unique positive solution by Lemma 27.

Sinkhorn barycenter of Gaussian measures
In this section, we prove Theorem 8 on the strict convexity of the Sinkhorn divergence and Theorem 13 on the barycenter of a set of Gaussian measures on H under the Sinkhorn divergence.
Proof By assumption, lim t→0 This proves the formula for Df (X 0 ). Here we use the fact the Fréchet Differentiability implies continuity. The proof for Dh(X 0 ) is entirely similar.
Proof Let h(X) = (I + c 2 X 2 ) 1/2 and g(X) = (I + h(X)) −1 . By the chain rule, By Lemma 30, with f (X) = g(X)X, Since g(X) and X commute, we also have f (X) = Xg(X) and thus This gives the second, equivalent, expression for Df (X 0 )(X).
Proposition 9 Consider the Gaussian measures N (0, C), N (0, C i ), 1 ≤ i ≤ N . Define the following function F : Then F is well-defined on the larger, open set Ω = {X ∈ Sym(H) ∩ Tr(H) : F is Fréchet differentiable on Ω and the condition DF (X 0 ) = 0 is equivalent to A solution X 0 of Eq.(198) must necessarily satisfy X 0 ∈ Sym + (H) ∩ Tr(H).
Under the additional hypothesis that X 0 > 0, Eq.(198) is equivalent to Proof Let F i (C) = S ǫ d 2 (N (0, C), N (0, C i )). By Theorem 7, Summing over i, 1 ≤ i ≤ N , we obtain the Fréchet derivative for F , namely By Lemma 20, the first order optimality condition DF (X 0 )(X) = 0 ∀X ∈ Sym(H) ∩ Tr(H) is equivalent to DF (X 0 ) = 0, which in turn is equivalent to This gives the first equation. Clearly any solution X 0 of this equation must necessarily satisfy X 0 ∈ Sym + (H) ∩ Tr(H).
Assume now X 0 > 0. We rewrite DF (X 0 ) = 0 as Under the condition X 0 > 0, by Lemma 26, pre-and post-multiplying X 1/2 0 on both sides gives the equivalent expression where the right hand side follows from Lemma 8. This in turn is This completes the proof.

⊓ ⊔
Existence of the Fixed Point. It is clear from Eq.(45) that if it has a solution X 0 , then necessarily X 0 ≥ 0. We now prove that Eq.(45) has at least one solution X 0 , which is then necessarily unique by the strict convexity of the Sinkhorn divergence. This is done via the Schauder Fixed Point Theorem. Let E be a Banach space and M ⊂ E. We recall that a mapping f : M → E is said to be compact if it is continuous and maps bounded subsets into relatively compact subsets of E, that is subsets whose closures are compact. Consider the map F : Sym + (H) → Sym + (H) as defined in Eq. (48). The proof of the existence of a fixed point of F consists of two steps 1. We show that F is compact.
2. Let γ ∈ R, γ > 0 be such that C i ≤ γI, 1 ≤ i ≤ N , and consider the set We show that F maps K into itself. We can then apply Schauder Fixed Point Theorem to obtain the existence of a fixed point of F in K.
Proof If B ≥ 0, then by Lemma 41, we have A * BA ≥ 0. Assume now that A * BA ≥ 0, which means that x, A * BAx = Ax, BAx ≥ 0 ∀x ∈ H. In particular, for y = Ax ∈ Im(A), we have y, By ≥ 0.
It follows that y, By = lim n→∞ y n , By n ≥ 0. Since this holds for all y ∈ H, we have B ≥ 0.
The following result is then immediate.
Corollary 6 Let C p (H) denote the set of pth Schatten class operators on H, 1 ≤ p ≤ ∞. For two operators A, B ∈ Sym + (H) ∩ C p (H), Theorem 28 (Theorem 2.3 in [48]) Let A, B be two positive operators on H and f any operator monotone function with f (0) = 0. Then The following result is then immediate.
Corollary 7 Let A, B be two positive operators on H. Then Proposition 10 Let C ∈ Sym + (H) ∩ Tr(H). The following maps are compact Proof (i) By Corollary 7, we have Thus the map F 1 is continuous on Sym + (H).
Since C 1/2 is a compact operator on H, it maps bounded subsets of H into relatively compact subsets of H. Consider the set Y = {C 1/2 x : x ∈ H, ||x|| ≤ 1} ⊂ H, then Y being relatively compact means that every sequence {y n = C 1/2 x n } n∈N in Y contains a subsequence {y n k = C 1/2 x n k } k∈N that converges in H, that is ∃y ∈ H such that lim k→∞ ||C 1/2 x n k − y|| = 0.
Let {λ k } k∈N be the eigenvalues of C, arranged in decreasing order, with corresponding orthonormal eigenvectors {e k } k∈N . Let N ∈ N be fixed and consider the finite-rank operator C . Then for any ǫ > 0, there exists N (ǫ) ∈ N such that Consequently, for ||X|| ≤ 1, N > N (ǫ), For a fixed N ∈ N, consider the set Z N = {X 1/2 C N } is the desired convergent subsequence, with limit Y N ∈ HS(H). This shows that the set Z N = {X 1/2 C N : ||X|| ≤ 1} is relatively compact ∀N ∈ N in HS(H), so that ∀ǫ > 0, there is a finite ǫ-net in HS(H) such that Consequently, for N > N (ǫ), This shows that the set {X 1/2 C 1/2 : ||X|| ≤ 1} is relatively compact in HS(H), hence in L(H) and thus F 1 is a compact map on Sym + (H). Moreover, each sequence {X where Y ∈ HS(H) is as defined in Part (i). Thus F 2 is a compact map on Sym + (H). (iv) Entirely analogous to F 3 , the map F 4 is compact on Sym + (H).
Proof Let us show that the product map is compact. Let M B be any bounded, non-empty subset of M . By assumption of compactness, each sequence {f (X n ), X n ∈ M B } n∈N contains a convergent subsequence {f (X n1 )} with limit Y 1 in E. Next, the sequence {g(X n1 )} contains a convergent subsequence {g(X n2 )} with limit Y 2 in E. We then have as n 2 → ∞. Thus the set {h 2 (X) = f (X)g(X) : X ∈ M B } is relatively compact. An analogous argument shows that h 2 is continuous, hence h 2 is a compact map on M . ⊓ ⊔ Proof Let c ǫ = 4 ǫ and g(X) = I + I + c 2 ǫ X 2 1/2 1/2 and for 1 ≤ i ≤ N , Using the inequality (1 + a 2 ) 1/2 ≤ 1 + a ∀a ≥ 0, we obtain Applying Corollary 6 twice gives By Corollary 7, Combining Eqs.(214), (215), (217),(216), and (218) gives This shows that F is continuous in the operator norm || ||. In particular, for This completes the proof.
Assume now that X is compact and singular, X = 0. Let {λ k } k∈N be the set of eigenvalues of X, λ k ≥ 0, arranged in decreasing order, with corresponding orthonormal eigenvectors {e k } k∈N forming an orthonormal basis in H. Then λ k e k ⊗ e k and X ≤ γI ⇐⇒ λ 1 ≤ γ.
The barycentric mean is obviously the Euclidean meanm = Then F is strictly convex, since S ǫ d 2 is strictly convex, thus its minimum, if it exists, is unique. By Proposition 9, where F : Sym + (H) → Sym + (H) is the map defined by Eq. (48). By Proposition 12, F is a compact map on Sym + (H). Let γ ∈ R, γ > 0 be such that C i ≤ γI, 1 ≤ i ≤ N and consider the set Then K is a closed, bounded, convex subset of L(H). By Proposition 13, X ∈ K ⇒ F (X) ∈ K. Thus by Schauder Fixed Point Theorem, there exists X 0 ∈ K such that X 0 = F (X 0 ), which must be the unique global minimizer of the strictly convex function F . Clearly, with c ǫ = 4 ǫ , Since 0 ≤ X 0 ≤ γI, 0 ≤ C i ≤ γI, 1 ≤ i ≤ N , Thus it follows that X 0 > 0 ⇐⇒ N i=1 w i C i > 0. ⊓ ⊔

Comparison of barycenter fixed point equations
We now show that for dim(H) ≥ 2, N i=1 w i C i > 0, Eq.(47) has uncountably infinitely many positive, singular solutions.
Consider first the case H = R and C i = σ 2 i , i = 1, . . . , N , has a unique fixed point x * , which satisfies x * > 0. The function has two fixed points, namely x * and x 0 = 0.
Proof The fixed point equation x = F (x) is equivalent to Consider the left hand side, which is f (x) = 1 + 1 + c 2 Thus f (x) is strictly increasing on [0, ∞) and hence there must exist a unique x * > 0 at which f (x * ) = 0 ⇐⇒ x * = F (x * ). Since x * > 0, by Proposition 9, It is obvious that G has another fixed point x 0 = 0. ⊓ ⊔ Lemma 40 Let u ∈ H, ||u|| = 1. Consider the rank-one operator u ⊗ u : H → H defined by (u ⊗ u)x = u, x u. Then for any A ∈ L(H) and any k ∈ N, If A is self-adjoint, positive, then [(u ⊗ u)A(u ⊗ u)] 1/k = u, Au 1/k (u ⊗ u).
Proof Consider the cases k = 1 and k = 2. For any x ∈ H, (u ⊗ u)A(u ⊗ u)x = (u ⊗ u)A u, x u = u, x u, Au u = u, Au (u ⊗ u)x.
Since (u ⊗ u) 2 = (u ⊗ u), we have For the first expression, the general case then follows by induction.
If A is self-adjoint, positive, then (u ⊗ u)A(u ⊗ u) is self-adjoint, positive, and [(u ⊗ u)A(u ⊗ u)] 1/k is well-defined and unique [12]. We have from which the second identity follows. ⊓ ⊔ Proposition 14 Let C i ∈ Sym + (H), 1 ≤ i ≤ N be fixed. Consider the following map G : Sym + (H) → Sym + (H), defined by Then X 0 = 0 is a fixed point of G. Let u ∈ H, ||u|| = 1 be such that N i=1 w i u, C i u > 0. Then X u = x u (u ⊗ u) is a fixed point of G, where x u is the unique positive solution of the following one-dimensional fixed point equation The condition N i=1 w i u, C i u > 0 is satisfied for at least one u ∈ H, u = 0, since otherwise C 1 = · · · C N = 0. If N i=1 w i C i > 0, then N i=1 w i u, C i u > 0∀u ∈ H, u = 0. Thus under this assumption, for dim(H) ≥ 2, G has uncountably infinitely many fixed points of the form X u = x u (u ⊗ u).
Therefore, for X u = x(u ⊗ u), I + c 2 ǫ X 1/2 u C i X 1/2 u = (I − u ⊗ u) + 1 + c 2 ǫ x u, C i u (u ⊗ u). Since (I − u ⊗ u) 2 = (I − u ⊗ u) and (I − u ⊗ u)(u ⊗ u) = 0, by Lemma 40, Applying the same argument and using the fact that N i=1 w i = 1, we have Thus the fixed point equation X u = G(X u ) becomes As shown in Lemma 39, under the condition N i= w i u, C i u > 0, this onedimensional fixed point equation has a unique positive solution x * u . Thus X u = x * u (u ⊗ u) is a fixed point of G. ⊓ ⊔ When ǫ = 0, the fixed points X u of G in Proposition 14 admit a closed form.
Proposition 15 Let C i ∈ Sym + (H), 1 ≤ i ≤ N be fixed. Consider the following map G : Sym + (H) → Sym + (H), defined by Then X 0 = 0 is a fixed point of G. Let u ∈ H, ||u|| = 1 be such that N i=1 w i u, C i u 1/2 > 0, then the following is a fixed point of G The condition N i=1 w i u, C i u 1/2 > 0 is satisfied for at least one u ∈ H, u = 0, since otherwise C 1 = · · · C N = 0. If N i=1 w i C i > 0, then N i=1 w i u, C i u > 0∀u ∈ H, u = 0. Since w i > 0, i = 1, . . . , N , for each u = 0, there must be at least one i for which u, C i u > 0. This implies that under this assumption, N i=1 w i u, C i u 1/2 > 0 is satisfied for all u ∈ H, u = 0. In this case, when dim(H) ≥ 2, G has uncountably infinitely many fixed points of the form X u .
One can immediately see that some of the solutions of the above equation include X = 0, X = C, X u = u, Cu (u ⊗ u), for any u ∈ H, ||u|| = 1, including X k = λ k (e k ⊗ e k ), k ∈ N, where {λ k } k∈N are the eigenvalues of C, with corresponding orthonormal eigenvectors {e k } k∈N .
Thus the fixed point equation X u = G(X u ) becomes by the assumption N i=1 w i u, C i u 1/2 > 0. Hence Assume that there exist α, β ∈ R, α > 0, β > 0 such that αI ≤ C i ≤ βI, 1 ≤ i ≤ N . Consider the set K 2 = {X ∈ Sym ++ (n) : αI ≤ X ≤ βI}. Then by Lemma 41 α 2 I ≤ αX = X 1/2 αX 1/2 ≤ X 1/2 C i X 1/2 ≤ X 1/2 βX 1/2 = βX ≤ β 2 I. It is not clear, however, whether this proof strategy can be extended to the infinite-dimensional setting. This is because one can no longer assume that there is a uniform lower bound of the form αI for the C i 's and X as above.
One might assume instead that there is an operator C > 0 such that C i ≥ C, 1 ≤ i ≤ N and consider the set K 3 = {X ∈ Sym + (H) : C ≤ X ≤ βI}. This does not help, however, since the condition X ≥ C > 0 does not imply that X 1/2 CX 1/2 ≥ C 2 . The following is a counterexample (232) It can be verified numerically that X ≥ C but X 1/2 CX 1/2 C 2 .
We note also that the condition 0 < αI ≤ C i ≤ β I , 1 ≤ i ≤ N , above is more restrictive that the condition N i=1 w i C i > 0 stated in Theorem 13, which guarantees the existence of a strictly positive solution of Eq. (47). As an example, let H = R N and {e i } N i=1 be an orthonormal basis in H. Define guaranteeing thatC > 0, even though all the C i 's are singular.

Miscellaneous Technical Results
Operator monotone functions. We recall the concept of operator monotone functions. Let A, B be two self-adjoint bounded operators on H, then we say . Some of the well-known operator monotone functions are the following.
The following is the generalization of Proposition V.1.6 in [8] to the infinitedimensional setting, with the additional assumption that A, B be invertible, since if dim(H) = ∞, then A > 0 does not imply that A is invertible.