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Bounds for the Wasserstein mean with applications to the Lie-Trotter mean
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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Bounds for the Wasserstein mean with applications to the Lie-Trotter mean Jinmi Hwang, Sejong Kim ∗ Department of Mathematics, Chungbuk National University, Cheongju 28644, Republic of Korea
a r t i c l e
i n f o
Article history: Received 13 April 2018 Available online xxxx Submitted by L. Molnar Keywords: Wasserstein mean Cartan mean Lie-Trotter mean
a b s t r a c t Since barycenters in the Wasserstein space of probability distributions have been introduced, the Wasserstein metric and the Wasserstein mean of positive definite Hermitian matrices have been recently developed. In this paper, we explore some properties of Wasserstein mean of positive definite matrices and find bounds for the Wasserstein mean with respect to operator norm and Loewner order. Using bounds for the Wasserstein mean, we verify that the Wasserstein mean is the multivariate Lie-Trotter mean. © 2019 Elsevier Inc. All rights reserved.
1. Introduction The open convex cone Pm of all m × m positive definite Hermitian matrices equipped with Riemannian trace metric δ(A, B) = log A−1/2 BA−1/2 2 gives us a Cartan-Hadamard manifold (a simply connected complete Riemannian manifold with non-positive sectional curvature), also called a Hadamard space. The natural and canonical mean on a Hadamard space is the least squares mean, called the Cartan mean or Riemannian mean. For a positive probability vector ω = (w1 , . . . , wn ), the Cartan mean of A1 , . . . , An ∈ Pm is defined as Λ(ω; A1 , . . . , An ) = arg min
n
wj δ 2 (X, Aj ).
(1.1)
X∈Pm j=1
Computational and theoretical approaches about the Cartan mean have been widely studied: see references [8,11,12,14,15,17] and therein. One of the important properties of the Cartan mean is the arithmeticgeometric-harmonic mean inequalities: * Corresponding author. E-mail addresses: [email protected] (J. Hwang), [email protected] (S. Kim). https://doi.org/10.1016/j.jmaa.2019.03.049 0022-247X/© 2019 Elsevier Inc. All rights reserved.
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⎡ ⎤−1 n n ⎣ ⎦ wj A−1 ≤ Λ(ω; A , . . . , A ) ≤ wj A j . 1 n j j=1
(1.2)
j=1
Using (1.2) it has been verified in [10] that the Cartan mean is the multivariate Lie-Trotter mean (see Section 4 for the definition of multivariate Lie-Trotter mean): for any differentiable curves γ1 , . . . , γn on Pm with γi (0) = I for all i, 1/s
lim Λ(ω; γ1 (s), γ2 (s), . . . , γn (s))
s→0
= exp
n
wi γi (0)
.
i=1
Motivated from barycenters in the Wasserstein space of probability distributions [1,2], Bhatia, Jain, and Lim [6] recently developed the Wasserstein metric and the Wasserstein mean on Pm . For given A, B ∈ Pm , the Wasserstein metric d(A, B) is given by 1/2
A+B 1/2 1/2 1/2 − tr(A BA ) d(A, B) = tr . 2
(1.3)
As the least squares mean for the Wasserstein metric, the Wasserstein mean denoted by Ω(ω; A1 , . . . , An ) is defined as Ω(ω; A1 , . . . , An ) = arg min
n
wj d2 (X, Aj ),
(1.4)
X∈Pm j=1
and it coincides with the unique solution X ∈ Pm of the equation X=
n
wj (X 1/2 Aj X 1/2 )1/2 ,
(1.5)
j=1
alternatively, I=
n
wj (Aj #X −1 ).
(1.6)
j=1
Note that Ω(1−t, t; A, B) = (1−t)2 A+t2 B+t(1−t)[(AB)1/2 +(BA)1/2 ] =: At B is the geodesic passing from A to B with respect to Wasserstein metric, and the Wasserstein mean satisfies the arithmetic-Wasserstein mean inequality. On the other hand, the Wasserstein mean does not satisfy the monotonicity and the Wasserstein-harmonic mean inequality: see [6, Section 5]. So it is a natural question whether the Wasserstein mean is the multivariate Lie-Trotter mean. The main goals of this paper are to provide some properties of the Wasserstein mean and to verify that the Wasserstein mean is the multivariate Lie-Trotter mean by finding a lower bound for Wasserstein mean. We recall in Section 2 the Wasserstein metric with geodesic and see the Wasserstein distance between A t B and A t C for A, B, C ∈ Pm and t ∈ [0, 1]. Moreover, we explore some interesting properties for the Wasserstein mean of positive definite matrices. We find bounds for the Wasserstein mean in Section 3, and verify in Section 4 that the Wasserstein mean is the multivariate Lie-Trotter mean. Finally, in Section 5 we show that the equation (1.5) for two positive invertible operators A and B and the positive probability vector ω = (1 − t, t) has the unique solution X = A t B. This naturally gives an open problem to extend the Wasserstein mean of positive definite matrices to positive invertible operators.
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2. Wasserstein metric and Wasserstein mean Let Mm be the set of all m × m matrices with complex entries. Let Hm be the real vector space of all m × m Hermitian matrices, and let Pm ⊂ Hm be the open convex cone of all positive definite matrices. For A, B ∈ Hm we denote as A ≤ B if and only if B − A is positive semi-definite, and as A < B if and only if B − A is positive definite. The Frobenius norm · 2 gives rise to the Riemannian structure on the open convex cone Pm with X, Y A = Tr(A−1 XA−1 Y ), where A ∈ Pm and X, Y ∈ TA (Pm ) = Hm . Then Pm is a Cartan-Hadamard Riemannian manifold, a simply connected complete Riemannian manifold with non-positive sectional curvature (the canonical 2-tensor is non-negative). The Riemannian trace metric between A and B is given by δ(A, B) = log A−1/2 BA−1/2 2 , and the unique (up to parametrization) geodesic curve on Pm connecting from A to B is [0, 1] t → A#t B := A1/2 (A−1/2 BA−1/2 )t A1/2 , which is called the weighted geometric mean of A and B. Note that A#B = A#1/2 B is the unique midpoint of A and B with respect to the Riemannian trace metric, and is the unique solution X ∈ Pm of the Riccati equation XA−1 X = B. See [4] for more information. We list a few properties of the weighted geometric mean that we use in the following. Lemma 2.1. Let A, B, C, D ∈ Pm and let t ∈ [0, 1]. Then the following are satisfied. (1) (2) (3) (4) (5)
A#t B = B#1−t A. X(A#t B)X ∗ = (XAX ∗ )#t (XBX ∗ ) for any nonsingular matrix X. (A#t B)−1 = A−1 #t B −1 . det(A#t B) = det(A)1−t det(B)t . [(1 − t)A−1 + tB −1 ]−1 ≤ A#t B ≤ (1 − t)A + tB.
In recent years many researchers from a variety of fields have been interested in the study of Wasserstein space: see [1–3,16]. In particular, Bhatia, Jain, and Lim [6] have developed the Wasserstein barycenters with Wasserstein metric (1.3) on Pm . Using the Riemannian geometry with submersion, they have shown that the Wasserstein metric is derived from the Riemannian structure on Pm, and the geodesic connecting A and B is given by γ(t) = (1 − t)2 A + t2 B + t(1 − t)[A(A−1 #B) + (A−1 #B)A] = (1 − t)2 A + t2 B + t(1 − t)[(AB)1/2 + (BA)1/2 ].
(2.7)
We denote γ(t) =: A t B for t ∈ [0, 1]. Since γ(t) is the natural parametrization of the geodesic joining A and B, it satisfies the affine property of parameters: for any s, t, u ∈ [0, 1] (A s B) u (A t B) = A (1−u)s+ut B Lemma 2.2. For any A, B, C ∈ Pm and t ∈ [0, 1]
d(A t B, A t C) ≤ t where λ := λ1 (A) is the largest eigenvalue of A.
mλ −1 A #B − A−1 #C2 , 2
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Proof. Note that A t B = Z(t)Z(t)∗ , where Z(t) = (1 − t)A1/2 + tB 1/2 U for 0 ≤ t ≤ 1 with a unitary matrix U = B 1/2 A1/2 (A1/2 BA1/2 )−1/2 . So Z(t) = (1 − t)A1/2 + t(A−1 #B)A1/2 ∈ F(A t B), since U = B 1/2 A1/2 (A−1/2 B −1 A−1/2 )1/2 = B 1/2 (A#B −1 )A−1/2 , and so B 1/2 U = B(A#B −1 )A−1/2 = (A#B −1 )−1 A1/2 = (A−1 #B)A1/2 by the Riccati equation and Lemma 2.1 (3). Similarly, At C = Y (t)Y (t)∗ , where Y (t) = (1 − t)A1/2 + t(A−1 #C)A1/2 ∈ F(A t C) for 0 ≤ t ≤ 1. Therefore, from Theorem 1 in [6] 1 t d(A t B, A t C) ≤ √ Z(t) − Y (t)2 = √ (A−1 #B)A1/2 − (A−1 #C)A1/2 2 2 2 t ≤ √ A1/2 2 · A−1 #B − A−1 #C2 2
mλ −1 A #B − A−1 #C2 . ≤t 2 The second inequality follows from the sub-multiplicative property of the Frobenius norm: see [9, Section 5.6], m and the last inequality follows from the fact that A1/2 22 = i=1 λi (A) ≤ mλ1 (A), where λ1 (A), . . . , λm (A) are positive eigenvalues of A in the decreasing order. 2 n , and let ω = (w1 , . . . , wn ) ∈ Δn , the simplex of all positive probability Let A = (A1 , . . . , An ) ∈ Pm vectors in Rn . From [1,6], the least squares mean for the Wasserstein metric exist uniquely in Pm , and we call it the Wasserstein mean denoted by Ω(ω; A). We see some interesting properties of the Wasserstein n mean. For given A = (A1 , . . . , An ) ∈ Pm , any permutation σ on {1, . . . , n}, and any M ∈ GLm , we denote as
n Aσ = (Aσ(1) , . . . , Aσ(n) ) ∈ Pm , n , M AM ∗ = (M A1 M ∗ , . . . , M An M ∗ ) ∈ Pm nk , Ak = (A1 , . . . , An , . . . , A1 , . . . , An ) ∈ Pm
where the number of blocks in the last expression is k. For given ω = (w1 , . . . , wn ) ∈ Δn , we also denote as ωσ = (wσ(1) , . . . , wσ(n) ) ∈ Δn , ωk =
1 (w1 , . . . , wn , . . . , w1 , . . . , wn ) ∈ Δnk . k
n Proposition 2.3. Let A = (A1 , . . . , An ) ∈ Pm , and let ω = (w1 , . . . , wn ) ∈ Δn . Then the following are satisfied.
(1) (2) (3) (4) (5)
(Homogeneity) Ω(ω; αA) = αΩ(ω; A) for any α > 0. (Permutation invariancy) Ω(ωσ ; Aσ ) = Ω(ω; A) for any permutation σ on {1, . . . , n}. (Repetition invariancy) Ω(ω k ; Ak ) = Ω(ω; A) for any k ∈ N. (Unitary congruence invariancy) Ω(ω; U AU ∗ ) = U Ω(ω; A)U ∗ for any U ∈ Um . n (Determinantal inequality) det Ω(ω; A) ≥ j=1 (det Aj )wj .
Proof. Items (1)-(4) simply follow from definition (1.4) of the Wasserstein mean and (1.6). Item (5) follows from the arithmetic-Cartan mean inequality and the fact that the determinant of the Cartan mean of positive definite matrices is the geometric mean of their determinants. 2
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3. Bounds for the Wasserstein mean n In the following we let A = (A1 , . . . , An ) ∈ Pm and let ω = (w1 , . . . , wn ) ∈ Δn . The Wasserstein mean satisfies the arithmetic-Wasserstein mean inequality:
Ω(ω; A) ≤
n
wj A j .
(3.8)
j=1
Proposition 3.1. For an operator norm · ⎛ Ω(ω; A) ≤ ⎝
n
⎞2 wj Aj 1/2 ⎠ .
j=1
Proof. Let X = Ω(ω; A). Then by (1.5), by the triangle inequality for the operator norm, by the fact that At = At for any A ∈ Pm and t ≥ 0, and by the sub-multiplicativity for the operator norm in [9, Section 5.6] n n 1/2 1/2 1/2 1/2 1/2 1/2 X = wj X A j X ≤ wj X Aj X j=1 j=1 =
n
n 1/2 wj X 1/2 Aj X 1/2 ≤ wj X1/2 Aj 1/2 .
j=1
j=1
Hence, by a simple calculation we obtain the desired inequality. 2 Remark 3.2. By the arithmetic-Wasserstein mean inequality (3.8) and the triangle inequality for operator n norm, we have Ω(ω; A) ≤ wj Aj . That is, the arithmetic mean of operator norms of A1 , . . . , An is j=1
one of the upper bounds for the Wasserstein mean. On the other hand, ⎛ ⎞2 n n 1/2 ⎠ ⎝ Ω(ω; A) ≤ wj Aj ≤ wj Aj , j=1
j=1
since the square map R t → t2 ∈ [0, ∞) is convex. Thus, one can see that Proposition 3.1 gives a sharp upper bound of the Wasserstein mean for the operator norm. Remark 3.3. By Proposition 3.1 and Theorem 2 in [5] with p = ∞, we get the interesting inequalities 2 ⎛ ⎞2 ⎞2 ⎛ n n n ⎝ 1/2 1/2 ωj Aj ωj Aj ⎠ ≤ Ω(ω; A) ≤ ⎝ ωj Aj 1/2 ⎠ . = j=1 j=1 j=1 Unfortunately, the Wasserstein mean does not satisfy the Wasserstein-harmonic mean inequality: see Section 5 in [6]. However, we give the lower bound for the Wasserstein mean together with another upper bound.
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Theorem 3.4. The Wasserstein mean satisfies the following for the Loewner order:
2I −
n
⎡ ⎣ wj A−1 j ≤ Ω(ω; A) ≤ 2I −
j=1
n
⎤−1 wj A j ⎦
,
j=1
where the second inequality holds if 2I −
n
wj Aj is invertible.
j=1
Proof. Let X = Ω(ω; A). By the arithmetic-geometric-harmonic mean inequalities in Lemma 2.1 (5), we have n
wj
j=1
A−1 j +X 2
−1 ≤I=
n
1 1 wj Aj + X −1 . 2 j=1 2 n
wj Aj #X −1 ≤
j=1
Solving the second inequality for X, we obtain the upper bound for the Wasserstein mean. Taking inverse on both sides of the first inequality and applying the arithmetic-harmonic mean inequality, we have ⎡ −1 ⎤−1 −1 n A + X 1 j ⎣ ⎦ ≥ I. wj A−1 wj j + X ≥ 2 j=1 2 2 j=1 n 1
Solving this for X, we obtain the lower bound for the Wasserstein mean. 2 Remark 3.5. Note that
2I −
A + A−1 ≥ A#A−1 = I for any A ∈ Pm . So we have 2
n
wj A−1 j
j=1
This means that 2I −
n
⎡ ⎡ ⎤−1 ⎤−1 n n n ⎣ ⎦ ⎦ ≤⎣ wj A−1 and 2I − w A ≥ wj A j . j j j j=1
j=1
j=1
⎡ wj A−1 is a new lower bound for the Wasserstein mean, but ⎣2I − j
j=1
n
⎤−1 wj A j ⎦
j=1
is not a sharper upper bound. 4. Applications to the Lie-Trotter mean We see in this section some applications of the lower bound of the Wasserstein mean in Theorem 3.4 to the n notion of Lie-Trotter means. A weighted n-mean Gn on Pm for n ≥ 2 is a map Gn (ω; ·) : Pm → Pm that is n idempotent, in the sense that Gn (ω; X, . . . , X) = X for all X ∈ Pm . A weighted n-mean Gn (ω; ·) : Pm → Pm is called a multivariate Lie-Trotter mean if it is differentiable and satisfies lim Gn (ω; γ1 (s), γ2 (s), . . . , γn (s))1/s = exp
s→0
n
wi γi (0) ,
(4.9)
i=1
where for > 0, γi : (−, ) → Pm are differentiable curves with γi (0) = I for all i = 1, . . . , n. See [10] for more details and information.
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n Lemma 4.1. Let Ωω := Ω(ω; ·) : Pm → Pm be the Wasserstein mean for given probability vector ω = (w1 , . . . , wn ). Then it is differentiable at I = (I, . . . , I) with
DΩω (I)(X1 , . . . , Xn ) =
n
wj Xj .
j=1
Proof. Let X1 , . . . , Xn ∈ Hm . If X1 = · · · = Xn = 0, then the statement holds obviously. Without loss of generality, we assume that at least one of X1 , . . . , Xn is not zero. Set ρ := max σ(Xj ), where σ(X) is the 1≤j≤n
spectral radius of X. Then ρ > 0. Define f (t) = 2I −
n
wj (I + tXj )−1
j=1
1 1 on the open interval − 2ρ , 2ρ . Then the map f is a differentiable function to Pm with f (0) = I. Indeed, λi (I + tXj ) = 1 + tλi (Xj ) ≥ 1 − |t| · |λi (Xj )| ≥ 1 − ρ|t| > λi (I + tXj )−1 ≤
1 2
> 0, and
1 < 2, 1 − ρ|t|
1 1 where λi (X) denotes any eigenvalue of X. So f (t) > 0 for any t ∈ − 2ρ , 2ρ . Since the derivative of the map t → (tX + I)−1 at t = 0 is −X, by the linearity property of differentiation n wj Xj . Then by (3.8) and Theorem 3.4, we have f (0) = j=1
[2I −
n j=1
wj (I + tXj )−1 ] − I t
Ωω (ω; I + tX1 , . . . , I + tXn ) − I t n n w (I + tX j) − I j=1 j = ≤ wj Xj t j=1 ≤
for any sufficiently small t > 0. Since Ωω (I, . . . , I) = I, we have Ωw (ω; I + tX1 , . . . , I + tXn ) − Ωw (I, . . . , I) = wj Xj . t j=1 n
lim+
t→0
By the similar argument for t < 0, we obtain the conclusion. 2 Lemma 4.2. For > 0, let γ : (−, ) → Pm be a continuous map with γ(0) = I. Then there exists a δ > 0 such that γ(s) > 12 I for all s ∈ (−δ, δ). Proof. Since γ : (−, ) → Pm is a continuous map with γ(0) = I, there exists a δ > 0 such that γ(s) ∈ B1/2 (I) = {A ∈ Hm : A − I < 12 } for all s ∈ (−δ, δ). Then |λi (γ(s)) − 1| = |λi (γ(s) − I)| ≤ γ(s) − I <
1 , 2
since γ(s) − I ∈ Hm . It implies that λi (γ(s)) > 1/2, and thus, γ(s) > (1/2)I. 2
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Theorem 4.3. The Wasserstein mean is the multivariate Lie-Trotter mean, that is, for given ω = (w1 , . . . , wn ) ∈ Δn lim Ω(ω; γ1 (s), γ2 (s), . . . , γn (s))1/s
s→0
⎡ ⎤ n = exp ⎣ wj γj (0)⎦ , j=1
where for > 0, γj : (−, ) → Pm are differentiable curves with γj (0) = I for all j = 1, . . . , n. Proof. Let ω = (w1 , . . . , wn ) ∈ Δn and let γ1 , . . . , γn : (−, ) → Pm be differentiable curves with γj (0) = I for all j. By Lemma 4.2 there exists a sufficiently small δ > 0 so that γj (s) > 12 I for all j and s ∈ (−δ, δ). n Then wj γj (s)−1 < 2I for any s ∈ (−δ, δ). j=1
By (3.8) and Theorem 3.4, we have 2I −
n
wj γj (s)−1 ≤ Ω(ω; γ1 (s), . . . , γn (s)) ≤
j=1
n
wj γj (s).
j=1
Taking logarithms, using the operator monotonicity of the logarithm map, and multiplying all terms by 1/s for s > 0, we get ⎡ log ⎣2I −
n
⎡
⎤1/s wj γj (s)−1 ⎦
≤ log Ω(ω; γ1 (s), . . . , γn (s))1/s ≤ log ⎣
j=1
n
⎤1/s wj γj (s)⎦
.
j=1
Note that ⎡ ⎤ n 1 lim log ⎣2I − wj γj (s)−1 ⎦ s→0+ s j=1 ⎡ = lim+ ⎣2I − s→0
n
⎤−1 ⎡ wj γj (s)−1 ⎦
j=1
⎣−
n
wj
⎤ n wj γj (0). −γj (s)−1 γj (s)γj (s)−1 ⎦ =
j=1
j=1
Taking the limit as s → 0+ and using from [10] that the arithmetic mean is the multivariable Lie-Trotter mean together with the above consequence, we obtain lim+ log Ω(ω; γ1 (s), . . . , γn (s))1/s =
s→0
n
wj γj (0).
j=1
Since the logarithm map log : Pm → Hm is diffeomorphic, we get the desired identity. By the similar argument for t < 0, we obtain the conclusion. 2 Taking γj (s) = Asj for each j and some Aj ∈ Pm , we obtain from Theorem 4.3 Corollary 4.4. Let A1 , . . . , An ∈ Pm and ω = (w1 , . . . , wn ) ∈ Δn . Then lim Ω(ω; As1 , . . . , Asn )1/s
s→0
⎡ ⎤ n = exp ⎣ wj log Aj ⎦ . j=1
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5. Final remarks It is a natural question if the Wasserstein mean can be defined on the setting P of positive definite operators. Since one can not have the Wasserstein metric on P , the definition (1.4) may not be available. One possible approach to define the operator Wasserstein mean is to show the existence and uniqueness of the solution of the equation (1.6). On the other hand, one can not find the explicit form of the solution of the nonlinear equation (1.6), but we have seen that the solution of (1.6) for two positive definite matrices A and B coincides with the geodesic γ(t) = A t B in (2.7) with respect to the Wasserstein metric. We directly solve the nonlinear equation (1.6) for n = 2 by using the properties of geometric mean of positive definite operators. For positive definite operators A, B ∈ P and t ∈ [0, 1] the weighted geometric mean of A and B is defined by A#t B = A1/2 (A−1/2 BA−1/2 )t A1/2 . Note that A#B = A#1/2 B is the unique positive definite solution X ∈ P of the Riccati equation XA−1 X = B. Moreover, it satisfies most of all properties in Lemma 2.1 except the determinantal identity in Lemma 2.1 (4). See [7,13] for more details. Theorem 5.1. Let A, B ∈ P and t ∈ [0, 1]. Then the nonlinear equation I = (1 − t)(A#X −1 ) + t(B#X −1 )
(5.10)
has a unique positive definite solution X = A t B. Proof. Taking the congruence transformation by A−1/2 in the equation (5.10), we get A−1 = (1 − t)(A−1/2 X −1 A−1/2 )1/2 + t(A−1/2 BA−1/2 )#(A−1/2 X −1 A−1/2 ). Let Y = A−1/2 X −1 A−1/2 and Z = A−1/2 BA−1/2 . Then A−1 = (1 − t)Y 1/2 + t(Z#Y ). By using the Riccati equation, we have 1 −1 [A − (1 − t)Y 1/2 ]Y −1 [A−1 − (1 − t)Y 1/2 ] = Z. t2 2 Pre- and post-multiplying all terms by A yields Y −1/2 − (1 − t)A = t2 AZA, so Y −1/2 = (1 − t)A + t(AZA)1/2 . By assumption, we have (A1/2 XA1/2 )1/2 = (1 − t)A + t(A1/2 BA1/2 )1/2 . Taking square on both sides and applying the congruence transformation by A−1/2 , we obtain X = A−1/2 [(1 − t)A + t(A1/2 BA1/2 )1/2 ]2 A−1/2 = A t B. 2 Open question. For positive definite operators A1 , . . . , An and a positive probability vector (w1 , . . . , wn ), the nonlinear equation I=
n
wj (Aj #X −1 ),
j=1
has a unique positive definite solution X in the setting P of positive definite operators? This is an interesting and challangeable problem, and Theorem 5.1 gives us a partial positive answer.
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Acknowledgments We thank to anonymous referee for valuable comments to improve this paper. The work of S. Kim was supported by the National Research Foundation of Korea grant funded by the Korea government (MIST) (No. NRF-2018R1C1B6001394). References [1] M. Agueh, G. Carlier, Barycenters in the Wasserstein space, SIAM J. Math. Anal. Appl. 43 (2011) 904–924. [2] P.C. Álvarez-Esteban, E. del Barrio, J.A. Cuesta-Albertos, C. Matŕan, A fixed point approach to barycenters in Wasserstein space, J. Math. Anal. Appl. 441 (2016) 744–762. [3] L. Ambrosio, N. Gigli, G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edition, Birkhäuser, 2008. [4] R. Bhatia, Positive Definite Matrices, Princeton Series in Applied Mathematics, Princeton, 2007. [5] R. Bhatia, T. Jain, Y. Lim, Inequalities for the Wasserstein mean of positive definite matrices, Linear Algebra Appl. (2018), https://doi.org/10.1016/j.laa.2018.03.017. [6] R. Bhatia, T. Jain, Y. Lim, On the Bures-Wasserstein distance between positive definite matrices, Expo. Math. (2018), https://doi.org/10.1016/j.exmath.2018.01.002. [7] G. Corach, H. Porta, L. Recht, Convexity of the geodesic distance on spaces of positive operators, Illinois J. Math. 38 (1994) 87–94. [8] J. Holbrook, No dice: a determinic approach to the Cartan centroid, J. Ramanujan Math. Soc. 27 (4) (2012) 509–521. [9] R.A. Horn, C.R. Johnson, Matrix Analysis, 2nd edition, Cambridge University Press, 2013. [10] J. Hwang, S. Kim, Lie-Trotter means of positive definite operators, Linear Algebra Appl. 531 (2017) 268–280. [11] H. Karcher, Riemannian center of mass and mollifier smoothing, Comm. Pure Appl. Math. 30 (5) (1977) 509–541. [12] S. Kim, H. Lee, Y. Lim, An order inequality characterizing invariant barycenters on symmetric cones, J. Math. Anal. Appl. 442 (2016) 1–16. [13] F. Kubo, T. Ando, Means of positive linear operators, Math. Ann. 246 (3) (1979/80) 205–224. [14] Y. Lim, M. Pálfia, Matrix power means and the Karcher mean, J. Funct. Anal. 262 (4) (2012) 1498–1514. [15] M. Moakher, A differential geometric approach to the geometric mean of symmetric positive-definite matrices, SIAM J. Matrix Anal. Appl. 26 (3) (2005) 735–747. [16] C. Villani, Optimal Transport: Old and New, vol. 338, Springer Science and Business Media, 2008. [17] T. Yamazaki, The Riemannian mean and matrix inequalities related to the Ando-Hiai inequality and chaotic order, Oper. Matrices 6 (2012) 577–588.
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Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Bounds for the Wasserstein mean with applications to the Lie-Trotter mean Jinmi Hwang, Sejong Kim ∗ Department of Mathematics, Chungbuk National University, Cheongju 28644, Republic of Korea
a r t i c l e
i n f o
Article history: Received 13 April 2018 Available online xxxx Submitted by L. Molnar Keywords: Wasserstein mean Cartan mean Lie-Trotter mean
a b s t r a c t Since barycenters in the Wasserstein space of probability distributions have been introduced, the Wasserstein metric and the Wasserstein mean of positive definite Hermitian matrices have been recently developed. In this paper, we explore some properties of Wasserstein mean of positive definite matrices and find bounds for the Wasserstein mean with respect to operator norm and Loewner order. Using bounds for the Wasserstein mean, we verify that the Wasserstein mean is the multivariate Lie-Trotter mean. © 2019 Elsevier Inc. All rights reserved.
1. Introduction The open convex cone Pm of all m × m positive definite Hermitian matrices equipped with Riemannian trace metric δ(A, B) = log A−1/2 BA−1/2 2 gives us a Cartan-Hadamard manifold (a simply connected complete Riemannian manifold with non-positive sectional curvature), also called a Hadamard space. The natural and canonical mean on a Hadamard space is the least squares mean, called the Cartan mean or Riemannian mean. For a positive probability vector ω = (w1 , . . . , wn ), the Cartan mean of A1 , . . . , An ∈ Pm is defined as Λ(ω; A1 , . . . , An ) = arg min
n
wj δ 2 (X, Aj ).
(1.1)
X∈Pm j=1
Computational and theoretical approaches about the Cartan mean have been widely studied: see references [8,11,12,14,15,17] and therein. One of the important properties of the Cartan mean is the arithmeticgeometric-harmonic mean inequalities: * Corresponding author. E-mail addresses: [email protected] (J. Hwang), [email protected] (S. Kim). https://doi.org/10.1016/j.jmaa.2019.03.049 0022-247X/© 2019 Elsevier Inc. All rights reserved.
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⎡ ⎤−1 n n ⎣ ⎦ wj A−1 ≤ Λ(ω; A , . . . , A ) ≤ wj A j . 1 n j j=1
(1.2)
j=1
Using (1.2) it has been verified in [10] that the Cartan mean is the multivariate Lie-Trotter mean (see Section 4 for the definition of multivariate Lie-Trotter mean): for any differentiable curves γ1 , . . . , γn on Pm with γi (0) = I for all i, 1/s
lim Λ(ω; γ1 (s), γ2 (s), . . . , γn (s))
s→0
= exp
n
wi γi (0)
.
i=1
Motivated from barycenters in the Wasserstein space of probability distributions [1,2], Bhatia, Jain, and Lim [6] recently developed the Wasserstein metric and the Wasserstein mean on Pm . For given A, B ∈ Pm , the Wasserstein metric d(A, B) is given by 1/2
A+B 1/2 1/2 1/2 − tr(A BA ) d(A, B) = tr . 2
(1.3)
As the least squares mean for the Wasserstein metric, the Wasserstein mean denoted by Ω(ω; A1 , . . . , An ) is defined as Ω(ω; A1 , . . . , An ) = arg min
n
wj d2 (X, Aj ),
(1.4)
X∈Pm j=1
and it coincides with the unique solution X ∈ Pm of the equation X=
n
wj (X 1/2 Aj X 1/2 )1/2 ,
(1.5)
j=1
alternatively, I=
n
wj (Aj #X −1 ).
(1.6)
j=1
Note that Ω(1−t, t; A, B) = (1−t)2 A+t2 B+t(1−t)[(AB)1/2 +(BA)1/2 ] =: At B is the geodesic passing from A to B with respect to Wasserstein metric, and the Wasserstein mean satisfies the arithmetic-Wasserstein mean inequality. On the other hand, the Wasserstein mean does not satisfy the monotonicity and the Wasserstein-harmonic mean inequality: see [6, Section 5]. So it is a natural question whether the Wasserstein mean is the multivariate Lie-Trotter mean. The main goals of this paper are to provide some properties of the Wasserstein mean and to verify that the Wasserstein mean is the multivariate Lie-Trotter mean by finding a lower bound for Wasserstein mean. We recall in Section 2 the Wasserstein metric with geodesic and see the Wasserstein distance between A t B and A t C for A, B, C ∈ Pm and t ∈ [0, 1]. Moreover, we explore some interesting properties for the Wasserstein mean of positive definite matrices. We find bounds for the Wasserstein mean in Section 3, and verify in Section 4 that the Wasserstein mean is the multivariate Lie-Trotter mean. Finally, in Section 5 we show that the equation (1.5) for two positive invertible operators A and B and the positive probability vector ω = (1 − t, t) has the unique solution X = A t B. This naturally gives an open problem to extend the Wasserstein mean of positive definite matrices to positive invertible operators.
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2. Wasserstein metric and Wasserstein mean Let Mm be the set of all m × m matrices with complex entries. Let Hm be the real vector space of all m × m Hermitian matrices, and let Pm ⊂ Hm be the open convex cone of all positive definite matrices. For A, B ∈ Hm we denote as A ≤ B if and only if B − A is positive semi-definite, and as A < B if and only if B − A is positive definite. The Frobenius norm · 2 gives rise to the Riemannian structure on the open convex cone Pm with X, Y A = Tr(A−1 XA−1 Y ), where A ∈ Pm and X, Y ∈ TA (Pm ) = Hm . Then Pm is a Cartan-Hadamard Riemannian manifold, a simply connected complete Riemannian manifold with non-positive sectional curvature (the canonical 2-tensor is non-negative). The Riemannian trace metric between A and B is given by δ(A, B) = log A−1/2 BA−1/2 2 , and the unique (up to parametrization) geodesic curve on Pm connecting from A to B is [0, 1] t → A#t B := A1/2 (A−1/2 BA−1/2 )t A1/2 , which is called the weighted geometric mean of A and B. Note that A#B = A#1/2 B is the unique midpoint of A and B with respect to the Riemannian trace metric, and is the unique solution X ∈ Pm of the Riccati equation XA−1 X = B. See [4] for more information. We list a few properties of the weighted geometric mean that we use in the following. Lemma 2.1. Let A, B, C, D ∈ Pm and let t ∈ [0, 1]. Then the following are satisfied. (1) (2) (3) (4) (5)
A#t B = B#1−t A. X(A#t B)X ∗ = (XAX ∗ )#t (XBX ∗ ) for any nonsingular matrix X. (A#t B)−1 = A−1 #t B −1 . det(A#t B) = det(A)1−t det(B)t . [(1 − t)A−1 + tB −1 ]−1 ≤ A#t B ≤ (1 − t)A + tB.
In recent years many researchers from a variety of fields have been interested in the study of Wasserstein space: see [1–3,16]. In particular, Bhatia, Jain, and Lim [6] have developed the Wasserstein barycenters with Wasserstein metric (1.3) on Pm . Using the Riemannian geometry with submersion, they have shown that the Wasserstein metric is derived from the Riemannian structure on Pm, and the geodesic connecting A and B is given by γ(t) = (1 − t)2 A + t2 B + t(1 − t)[A(A−1 #B) + (A−1 #B)A] = (1 − t)2 A + t2 B + t(1 − t)[(AB)1/2 + (BA)1/2 ].
(2.7)
We denote γ(t) =: A t B for t ∈ [0, 1]. Since γ(t) is the natural parametrization of the geodesic joining A and B, it satisfies the affine property of parameters: for any s, t, u ∈ [0, 1] (A s B) u (A t B) = A (1−u)s+ut B Lemma 2.2. For any A, B, C ∈ Pm and t ∈ [0, 1]
d(A t B, A t C) ≤ t where λ := λ1 (A) is the largest eigenvalue of A.
mλ −1 A #B − A−1 #C2 , 2
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Proof. Note that A t B = Z(t)Z(t)∗ , where Z(t) = (1 − t)A1/2 + tB 1/2 U for 0 ≤ t ≤ 1 with a unitary matrix U = B 1/2 A1/2 (A1/2 BA1/2 )−1/2 . So Z(t) = (1 − t)A1/2 + t(A−1 #B)A1/2 ∈ F(A t B), since U = B 1/2 A1/2 (A−1/2 B −1 A−1/2 )1/2 = B 1/2 (A#B −1 )A−1/2 , and so B 1/2 U = B(A#B −1 )A−1/2 = (A#B −1 )−1 A1/2 = (A−1 #B)A1/2 by the Riccati equation and Lemma 2.1 (3). Similarly, At C = Y (t)Y (t)∗ , where Y (t) = (1 − t)A1/2 + t(A−1 #C)A1/2 ∈ F(A t C) for 0 ≤ t ≤ 1. Therefore, from Theorem 1 in [6] 1 t d(A t B, A t C) ≤ √ Z(t) − Y (t)2 = √ (A−1 #B)A1/2 − (A−1 #C)A1/2 2 2 2 t ≤ √ A1/2 2 · A−1 #B − A−1 #C2 2
mλ −1 A #B − A−1 #C2 . ≤t 2 The second inequality follows from the sub-multiplicative property of the Frobenius norm: see [9, Section 5.6], m and the last inequality follows from the fact that A1/2 22 = i=1 λi (A) ≤ mλ1 (A), where λ1 (A), . . . , λm (A) are positive eigenvalues of A in the decreasing order. 2 n , and let ω = (w1 , . . . , wn ) ∈ Δn , the simplex of all positive probability Let A = (A1 , . . . , An ) ∈ Pm vectors in Rn . From [1,6], the least squares mean for the Wasserstein metric exist uniquely in Pm , and we call it the Wasserstein mean denoted by Ω(ω; A). We see some interesting properties of the Wasserstein n mean. For given A = (A1 , . . . , An ) ∈ Pm , any permutation σ on {1, . . . , n}, and any M ∈ GLm , we denote as
n Aσ = (Aσ(1) , . . . , Aσ(n) ) ∈ Pm , n , M AM ∗ = (M A1 M ∗ , . . . , M An M ∗ ) ∈ Pm nk , Ak = (A1 , . . . , An , . . . , A1 , . . . , An ) ∈ Pm
where the number of blocks in the last expression is k. For given ω = (w1 , . . . , wn ) ∈ Δn , we also denote as ωσ = (wσ(1) , . . . , wσ(n) ) ∈ Δn , ωk =
1 (w1 , . . . , wn , . . . , w1 , . . . , wn ) ∈ Δnk . k
n Proposition 2.3. Let A = (A1 , . . . , An ) ∈ Pm , and let ω = (w1 , . . . , wn ) ∈ Δn . Then the following are satisfied.
(1) (2) (3) (4) (5)
(Homogeneity) Ω(ω; αA) = αΩ(ω; A) for any α > 0. (Permutation invariancy) Ω(ωσ ; Aσ ) = Ω(ω; A) for any permutation σ on {1, . . . , n}. (Repetition invariancy) Ω(ω k ; Ak ) = Ω(ω; A) for any k ∈ N. (Unitary congruence invariancy) Ω(ω; U AU ∗ ) = U Ω(ω; A)U ∗ for any U ∈ Um . n (Determinantal inequality) det Ω(ω; A) ≥ j=1 (det Aj )wj .
Proof. Items (1)-(4) simply follow from definition (1.4) of the Wasserstein mean and (1.6). Item (5) follows from the arithmetic-Cartan mean inequality and the fact that the determinant of the Cartan mean of positive definite matrices is the geometric mean of their determinants. 2
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3. Bounds for the Wasserstein mean n In the following we let A = (A1 , . . . , An ) ∈ Pm and let ω = (w1 , . . . , wn ) ∈ Δn . The Wasserstein mean satisfies the arithmetic-Wasserstein mean inequality:
Ω(ω; A) ≤
n
wj A j .
(3.8)
j=1
Proposition 3.1. For an operator norm · ⎛ Ω(ω; A) ≤ ⎝
n
⎞2 wj Aj 1/2 ⎠ .
j=1
Proof. Let X = Ω(ω; A). Then by (1.5), by the triangle inequality for the operator norm, by the fact that At = At for any A ∈ Pm and t ≥ 0, and by the sub-multiplicativity for the operator norm in [9, Section 5.6] n n 1/2 1/2 1/2 1/2 1/2 1/2 X = wj X A j X ≤ wj X Aj X j=1 j=1 =
n
n 1/2 wj X 1/2 Aj X 1/2 ≤ wj X1/2 Aj 1/2 .
j=1
j=1
Hence, by a simple calculation we obtain the desired inequality. 2 Remark 3.2. By the arithmetic-Wasserstein mean inequality (3.8) and the triangle inequality for operator n norm, we have Ω(ω; A) ≤ wj Aj . That is, the arithmetic mean of operator norms of A1 , . . . , An is j=1
one of the upper bounds for the Wasserstein mean. On the other hand, ⎛ ⎞2 n n 1/2 ⎠ ⎝ Ω(ω; A) ≤ wj Aj ≤ wj Aj , j=1
j=1
since the square map R t → t2 ∈ [0, ∞) is convex. Thus, one can see that Proposition 3.1 gives a sharp upper bound of the Wasserstein mean for the operator norm. Remark 3.3. By Proposition 3.1 and Theorem 2 in [5] with p = ∞, we get the interesting inequalities 2 ⎛ ⎞2 ⎞2 ⎛ n n n ⎝ 1/2 1/2 ωj Aj ωj Aj ⎠ ≤ Ω(ω; A) ≤ ⎝ ωj Aj 1/2 ⎠ . = j=1 j=1 j=1 Unfortunately, the Wasserstein mean does not satisfy the Wasserstein-harmonic mean inequality: see Section 5 in [6]. However, we give the lower bound for the Wasserstein mean together with another upper bound.
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Theorem 3.4. The Wasserstein mean satisfies the following for the Loewner order:
2I −
n
⎡ ⎣ wj A−1 j ≤ Ω(ω; A) ≤ 2I −
j=1
n
⎤−1 wj A j ⎦
,
j=1
where the second inequality holds if 2I −
n
wj Aj is invertible.
j=1
Proof. Let X = Ω(ω; A). By the arithmetic-geometric-harmonic mean inequalities in Lemma 2.1 (5), we have n
wj
j=1
A−1 j +X 2
−1 ≤I=
n
1 1 wj Aj + X −1 . 2 j=1 2 n
wj Aj #X −1 ≤
j=1
Solving the second inequality for X, we obtain the upper bound for the Wasserstein mean. Taking inverse on both sides of the first inequality and applying the arithmetic-harmonic mean inequality, we have ⎡ −1 ⎤−1 −1 n A + X 1 j ⎣ ⎦ ≥ I. wj A−1 wj j + X ≥ 2 j=1 2 2 j=1 n 1
Solving this for X, we obtain the lower bound for the Wasserstein mean. 2 Remark 3.5. Note that
2I −
A + A−1 ≥ A#A−1 = I for any A ∈ Pm . So we have 2
n
wj A−1 j
j=1
This means that 2I −
n
⎡ ⎡ ⎤−1 ⎤−1 n n n ⎣ ⎦ ⎦ ≤⎣ wj A−1 and 2I − w A ≥ wj A j . j j j j=1
j=1
j=1
⎡ wj A−1 is a new lower bound for the Wasserstein mean, but ⎣2I − j
j=1
n
⎤−1 wj A j ⎦
j=1
is not a sharper upper bound. 4. Applications to the Lie-Trotter mean We see in this section some applications of the lower bound of the Wasserstein mean in Theorem 3.4 to the n notion of Lie-Trotter means. A weighted n-mean Gn on Pm for n ≥ 2 is a map Gn (ω; ·) : Pm → Pm that is n idempotent, in the sense that Gn (ω; X, . . . , X) = X for all X ∈ Pm . A weighted n-mean Gn (ω; ·) : Pm → Pm is called a multivariate Lie-Trotter mean if it is differentiable and satisfies lim Gn (ω; γ1 (s), γ2 (s), . . . , γn (s))1/s = exp
s→0
n
wi γi (0) ,
(4.9)
i=1
where for > 0, γi : (−, ) → Pm are differentiable curves with γi (0) = I for all i = 1, . . . , n. See [10] for more details and information.
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n Lemma 4.1. Let Ωω := Ω(ω; ·) : Pm → Pm be the Wasserstein mean for given probability vector ω = (w1 , . . . , wn ). Then it is differentiable at I = (I, . . . , I) with
DΩω (I)(X1 , . . . , Xn ) =
n
wj Xj .
j=1
Proof. Let X1 , . . . , Xn ∈ Hm . If X1 = · · · = Xn = 0, then the statement holds obviously. Without loss of generality, we assume that at least one of X1 , . . . , Xn is not zero. Set ρ := max σ(Xj ), where σ(X) is the 1≤j≤n
spectral radius of X. Then ρ > 0. Define f (t) = 2I −
n
wj (I + tXj )−1
j=1
1 1 on the open interval − 2ρ , 2ρ . Then the map f is a differentiable function to Pm with f (0) = I. Indeed, λi (I + tXj ) = 1 + tλi (Xj ) ≥ 1 − |t| · |λi (Xj )| ≥ 1 − ρ|t| > λi (I + tXj )−1 ≤
1 2
> 0, and
1 < 2, 1 − ρ|t|
1 1 where λi (X) denotes any eigenvalue of X. So f (t) > 0 for any t ∈ − 2ρ , 2ρ . Since the derivative of the map t → (tX + I)−1 at t = 0 is −X, by the linearity property of differentiation n wj Xj . Then by (3.8) and Theorem 3.4, we have f (0) = j=1
[2I −
n j=1
wj (I + tXj )−1 ] − I t
Ωω (ω; I + tX1 , . . . , I + tXn ) − I t n n w (I + tX j) − I j=1 j = ≤ wj Xj t j=1 ≤
for any sufficiently small t > 0. Since Ωω (I, . . . , I) = I, we have Ωw (ω; I + tX1 , . . . , I + tXn ) − Ωw (I, . . . , I) = wj Xj . t j=1 n
lim+
t→0
By the similar argument for t < 0, we obtain the conclusion. 2 Lemma 4.2. For > 0, let γ : (−, ) → Pm be a continuous map with γ(0) = I. Then there exists a δ > 0 such that γ(s) > 12 I for all s ∈ (−δ, δ). Proof. Since γ : (−, ) → Pm is a continuous map with γ(0) = I, there exists a δ > 0 such that γ(s) ∈ B1/2 (I) = {A ∈ Hm : A − I < 12 } for all s ∈ (−δ, δ). Then |λi (γ(s)) − 1| = |λi (γ(s) − I)| ≤ γ(s) − I <
1 , 2
since γ(s) − I ∈ Hm . It implies that λi (γ(s)) > 1/2, and thus, γ(s) > (1/2)I. 2
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Theorem 4.3. The Wasserstein mean is the multivariate Lie-Trotter mean, that is, for given ω = (w1 , . . . , wn ) ∈ Δn lim Ω(ω; γ1 (s), γ2 (s), . . . , γn (s))1/s
s→0
⎡ ⎤ n = exp ⎣ wj γj (0)⎦ , j=1
where for > 0, γj : (−, ) → Pm are differentiable curves with γj (0) = I for all j = 1, . . . , n. Proof. Let ω = (w1 , . . . , wn ) ∈ Δn and let γ1 , . . . , γn : (−, ) → Pm be differentiable curves with γj (0) = I for all j. By Lemma 4.2 there exists a sufficiently small δ > 0 so that γj (s) > 12 I for all j and s ∈ (−δ, δ). n Then wj γj (s)−1 < 2I for any s ∈ (−δ, δ). j=1
By (3.8) and Theorem 3.4, we have 2I −
n
wj γj (s)−1 ≤ Ω(ω; γ1 (s), . . . , γn (s)) ≤
j=1
n
wj γj (s).
j=1
Taking logarithms, using the operator monotonicity of the logarithm map, and multiplying all terms by 1/s for s > 0, we get ⎡ log ⎣2I −
n
⎡
⎤1/s wj γj (s)−1 ⎦
≤ log Ω(ω; γ1 (s), . . . , γn (s))1/s ≤ log ⎣
j=1
n
⎤1/s wj γj (s)⎦
.
j=1
Note that ⎡ ⎤ n 1 lim log ⎣2I − wj γj (s)−1 ⎦ s→0+ s j=1 ⎡ = lim+ ⎣2I − s→0
n
⎤−1 ⎡ wj γj (s)−1 ⎦
j=1
⎣−
n
wj
⎤ n wj γj (0). −γj (s)−1 γj (s)γj (s)−1 ⎦ =
j=1
j=1
Taking the limit as s → 0+ and using from [10] that the arithmetic mean is the multivariable Lie-Trotter mean together with the above consequence, we obtain lim+ log Ω(ω; γ1 (s), . . . , γn (s))1/s =
s→0
n
wj γj (0).
j=1
Since the logarithm map log : Pm → Hm is diffeomorphic, we get the desired identity. By the similar argument for t < 0, we obtain the conclusion. 2 Taking γj (s) = Asj for each j and some Aj ∈ Pm , we obtain from Theorem 4.3 Corollary 4.4. Let A1 , . . . , An ∈ Pm and ω = (w1 , . . . , wn ) ∈ Δn . Then lim Ω(ω; As1 , . . . , Asn )1/s
s→0
⎡ ⎤ n = exp ⎣ wj log Aj ⎦ . j=1
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5. Final remarks It is a natural question if the Wasserstein mean can be defined on the setting P of positive definite operators. Since one can not have the Wasserstein metric on P , the definition (1.4) may not be available. One possible approach to define the operator Wasserstein mean is to show the existence and uniqueness of the solution of the equation (1.6). On the other hand, one can not find the explicit form of the solution of the nonlinear equation (1.6), but we have seen that the solution of (1.6) for two positive definite matrices A and B coincides with the geodesic γ(t) = A t B in (2.7) with respect to the Wasserstein metric. We directly solve the nonlinear equation (1.6) for n = 2 by using the properties of geometric mean of positive definite operators. For positive definite operators A, B ∈ P and t ∈ [0, 1] the weighted geometric mean of A and B is defined by A#t B = A1/2 (A−1/2 BA−1/2 )t A1/2 . Note that A#B = A#1/2 B is the unique positive definite solution X ∈ P of the Riccati equation XA−1 X = B. Moreover, it satisfies most of all properties in Lemma 2.1 except the determinantal identity in Lemma 2.1 (4). See [7,13] for more details. Theorem 5.1. Let A, B ∈ P and t ∈ [0, 1]. Then the nonlinear equation I = (1 − t)(A#X −1 ) + t(B#X −1 )
(5.10)
has a unique positive definite solution X = A t B. Proof. Taking the congruence transformation by A−1/2 in the equation (5.10), we get A−1 = (1 − t)(A−1/2 X −1 A−1/2 )1/2 + t(A−1/2 BA−1/2 )#(A−1/2 X −1 A−1/2 ). Let Y = A−1/2 X −1 A−1/2 and Z = A−1/2 BA−1/2 . Then A−1 = (1 − t)Y 1/2 + t(Z#Y ). By using the Riccati equation, we have 1 −1 [A − (1 − t)Y 1/2 ]Y −1 [A−1 − (1 − t)Y 1/2 ] = Z. t2 2 Pre- and post-multiplying all terms by A yields Y −1/2 − (1 − t)A = t2 AZA, so Y −1/2 = (1 − t)A + t(AZA)1/2 . By assumption, we have (A1/2 XA1/2 )1/2 = (1 − t)A + t(A1/2 BA1/2 )1/2 . Taking square on both sides and applying the congruence transformation by A−1/2 , we obtain X = A−1/2 [(1 − t)A + t(A1/2 BA1/2 )1/2 ]2 A−1/2 = A t B. 2 Open question. For positive definite operators A1 , . . . , An and a positive probability vector (w1 , . . . , wn ), the nonlinear equation I=
n
wj (Aj #X −1 ),
j=1
has a unique positive definite solution X in the setting P of positive definite operators? This is an interesting and challangeable problem, and Theorem 5.1 gives us a partial positive answer.
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Doctopic: Functional Analysis
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