POSITIVE DEFINITE MATRICES AND THE S-DIVERGENCE

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages S (XX) POSITIVE DEFINITE MATRICES AND THE S-DIVERGENCE SUVRIT SRA (Communicated by ) Abstract. Hermitian positive definite (hpd) matrices form a self-dual convex cone whose interior is a Riemannian manifold of nonpositive curvature. The manifold view is endowed with a geodesically convex distance function but the convex view is not. Drawing motivation from convex optimization, we introduce the S-Divergence, a distance-like function on the convex cone of hpd matrices. We study basic properties of the S-divergence that connect it intimately with the Riemannian distance. In particular, we show that (i) its square-root is a distance; and that (ii) it exhibits several (nonpositive-curvature-like) properties akin to the Riemannian distance.. Introduction Hermitian positive definite (hpd) matrices form a manifold of nonpositive curvature [, Ch.0], [6, Ch.6], whose closure is a self-dual convex cone. The manifold is important in various applications [], while the conic view is fundamental to convex optimization [0, 0] and nonlinear Perron-Frobenius theory [8], for instance. The manifold view comes with a natural distance function, while the conic view does not. Drawing motivation from convex optimization we introduce on the hpd cone a distance-like function: the S-Divergence. We prove several results connecting the S-Divergence to the Riemannian distance. Our main result is that the squareroot of this divergence is actually a distance; our remaining results explore geometric and analytic properties common to the S-Divergence and the Riemannian distance... Setup. Let H n denote the set of n n Hermitian matrices. A matrix A H n is called positive definite if (.) x, Ax > 0 for all x 0, also written as A > 0. The set of positive definite (henceforth positive) matrices in H n is denoted by P n. We say A is positive semidefinite if x, Ax 0 for all x and write A 0. The inequality A B means A B 0. The Frobenius norm of a matrix X is X F := tr(x X), while X denotes the operator norm. Let f be an analytic function on C, and let A have the eigendecomposition A = UΛU with unitary U, then f(a) = Uf(Λ)U with f(λ) equal to the diagonal matrix Diag[f(λ ),..., f(λ n )]. The set P n is a well-studied differentiable Riemannian manifold, with the Riemannian metric given by the differential form ds = A / daa / F. This metric 00 Mathematics Subject Classification. Primary: 5A45; 5A99; 47B65; 65F60. This work was done when author was with the MPI for Intelligent Systems, Tübingen, Germany. A small fraction of this work was presented at the Neural Information Processing Systems (NIPS) Conference 0 see [3]. c XXXX American Mathematical Society

2 SUVRIT SRA induces the Riemannian distance (see e.g., [6, Ch. 6]): (.) δ R (X, Y ) := log(y / XY / ) F for X, Y > 0. The focus of this paper is on a complement to (.), namely, the S-Divergence: (.3) δs(x, Y ) := log det ( ) X+Y log det(xy ) for X, Y > 0. This divergence was proposed as a numerical alternative to the Riemannian distance δ R in [4] there, it was used in an application to image-search, primarily for its empirical benefits. This initial empirical success of S-Divergence motivated us to investigate δ S more closely in this paper. Contributions. The present paper goes substantially beyond our initial conference version [3]. The main differences are: (i) Theorems 4., 4.5, 4.6, 4.7, and 4.9, which establish several new geometric and analytic similarities between δ S and δ R ; (ii) the joint geodesic convexity of δs (Prop. 4.3, Theorem 4.4); and (iii) new conic contraction results for δ S and δ R that uncover properties akin to those exhibited Hilbert s projective metric and Thompson s part metric [8] (Prop. 4., Corollary 4., Theorem 4.4, Theorem 4.5, and Corollary 4.6). Related work. While our paper was under preparation (in 0), we became aware of a concurrent paper of Chebbi and Moakher (CM) [] who considered a single parameter family of divergences that generalize (.3). Our work differs from CM in several key aspects. () CM prove δ S to be a distance for commuting matrices only. We note in passing that the commuting case is essentially equivalent to the scalar case. The noncommuting case is much harder, and was also conjectured by []. We prove the noncommuting case too, and note that our consideration of it was agnostic of CM []. () We establish several theorems that uncover geometric and analytic similarities between δs and δ R. (3) A question closely related to δ S being a distance is whether the matrix [det(x i + X j ) β ] m i,j= is positive semidefinite for arbitrary matrices X,..., X m P n, every integer m, and every scalar β 0. CM considered special cases of this question. We provide a complete characterization of β necessary and sufficient for the above matrix to be semidefinite.. The S-Divergence Consider a differentiable strictly convex function f : R R; then, f(x) f(y) + f (y)(x y), with equality if and only if x = y. The difference between the two sides of this inequality is called a Bregman Divergence: (.) D f (x, y) := f(x) f(y) f (y)(x y). The scalar divergence (.) readily extends to Hermitian matrices. Specifically, if f is differentiable and strictly convex on R, and X, Y H n are arbitrary. Then, the matrix Bregman Divergence is defined as (.) D f (X, Y ) := tr f(x) tr f(y ) tr ( f (Y )(X Y ) ). It can be verified that D f is nonnegative, strictly convex in X, and zero if and only if X = Y. It is typically asymmetric. For example, if f(x) = x, then for X H n, tr f(x) = tr(x ) and (.) becomes the squared Frobenius norm X Y F. If It is a divergence because although nonnegative, definite, and symmetric, it is not a metric. Over vectors, these divergences have been well-studied; see e.g., []. Although not distances, they often behave like squared distances, in a sense that can be made precise for certain f [3].

3 POSITIVE DEFINITE MATRICES AND THE S-DIVERGENCE 3 f(x) = x log x x on (0, ), then tr f(x) = tr(x log X X) and (.) yields the (unnormalized) von Neumann Divergence of quantum information theory. The asymmetry of Bregman divergences can be sometimes undesirable. This has led researchers to consider symmetric divergences, among which the most popular is the Jensen-Shannon / Bregman divergence ( (.3) S f (X, Y ) := Df (X, X+Y ) + D f ( X+Y, Y ) ). Divergence (.3) may also be written in the more revealing form: ) ( (.4) S f (X, Y ) = ( tr f(x) + tr f(y ) tr f X+Y The S-Divergence (.3) can be obtained from (.4) by setting f(x) = log x, so that f(x) = log det(x), the barrier function for the positive definite cone [0]. The S-Divergence may be also be viewed as the Jensen-Bregman divergence between two multivariate gaussians [5], or as the Bhattacharyya distance between them [8]. The following basic properties of S may be easily verified. Proposition.. Let λ(x) be the vector of eigenvalues of X, and Eig(X) the diagonal matrix with λ(x) on its diagonal. Let A, B, C P n. Then, (i) δ S (I, A) = δ S (I, Eig(A)); (ii) δ S (A, B) = δ S (P AP, P BP ), where P GL n (C); (iii) δ S (A, B) = δ S (A, B ); (iv) δs (A B, A C) = nδ S (B, C). 3. The δ S distance In this section we present our main result: the square-root δ S of the S-Divergence is actually a distance. Previous authors [, 4] conjectured this result; both appealed to classical ideas from harmonic analysis [3, Ch. 3] to establish the commutative case. But the noncommutative case requires a different approach, as we show below. Theorem 3.. Let δ S be defined by (.3). Then, δ S is a metric on P n. The proof of Theorem 3. depends on several results, some of which we prove below. Definition 3. ([3, Def..]). Let X be a nonempty set. A function ψ : X X R is said to be negative definite if for all x, y X, ψ(x, y) = ψ(y, x), and the inequality n i,j= c ic j ψ(x i, x j ) 0, holds for all integers n, and subsets {x i } n i= X, {c i} n i= R with n i= c i = 0. Theorem 3.3 ([3, Prop. 3., Ch. 3]). Let ψ : X X R be negative definite. There is a Hilbert space H R X and a mapping x ϕ(x) from X H such that (3.) ϕ(x) ϕ(y) H = (ψ(x, x) + ψ(y, y)) ψ(x, y). Moreover, negative definiteness of ψ is necessary for such a mapping to exist. Theorem 3.3 helps prove the triangle inequality for the scalar case. 3 Lemma 3.4. Let δ s be the scalar version of δ S, i.e., δ s (x, y) := log[(x + y)/( xy)], x, y > 0. Then, δ s is a metric on (0, ). 3 The idea of invoking Schoenberg s theorem (Theorem 3.3) for for establishing the commutative case (which actually is essentially just the scalar case of Lemma 3.4) was also used by []; we arrived at it independently following the classic route of harmonic analysis [3, Ch. 3]. ).

4 4 SUVRIT SRA Proof. To verify that ψ(x, y) = log((x + y)/) is negative definite, by [3, Thm.., Ch. 3] we may equivalently show that e βψ(x,y) = ( ) x+y β is positive definite for any β > 0 and x, y > 0. This in turn follows from the inner-product representation (3.) (x + y) β = Γ(β) e t(x+y) t β dt = f 0 x, f y, where f x (t) = e tx t β L ([0, )). Corollary 3.5. Let x, y, z R n ++; and let p. Then, ( ) /p ( ) /p ( ) /p (3.3) i δp s(x i, y i ) i δp s(x i, z i ) + i δp s(y i, z i ). Corollary 3.6. Let X, Y, Z > 0 be diagonal matrices. Then, (3.4) δ S (X, Y ) δ S (X, Z) + δ S (Y, Z) Proof. For diagonal X, Y, δ S (X, Y ) = i δ s(x ii, Y ii ); now use Corollary 3.5. Next, we recall an important determinantal inequality for positive matrices. Theorem 3.7 ([5, VI.7]). Let A, B > 0. Let λ (X) denote the vector of eigenvalues of X arranged in decreasing order; define λ (X) likewise. Then, (3.5) n i= (λ i (A) + λ i (B)) det(a + B) n i= (λ i (A) + λ i (B)). Corollary 3.8. Let A, B > 0. Let Eig (X) denote the diagonal matrix with λ (X) as its diagonal; define Eig (X) likewise. Then, (3.6) δ S (Eig (A), Eig (B)) δ S (A, B) δ S (Eig (A), Eig (B)). Proof. In (3.5), dividing by n det(a) det(b) we obtain n i= (λ i ( A ) + λ i ( B )) det ( ) A+B n i= (λ i ( A ) + λ i ( B )). det(a) det(b) det(a) det(b) det(a) det(b) Since determinants are invariant to permutation of eigenvalues, we can rearrange the leftmost and rightmost terms so that upon taking logarithms, (3.6) follows. The final result we need is a classic lemma from linear algebra. Lemma 3.9. If A > 0, and B is Hermitian, then there is a matrix P such that (3.7) P AP = I, and P BP = D, where D is diagonal. Accoutered with the above results, we are ready to prove Theorem 3.. Proof. (Theorem 3.). We need to show that δ S is symmetric, nonnegative, definite, and that it satisfies the triangle inequality. Symmetry and nonnegativity are obvious, while definiteness follows from strict convexity of log det(x), the seed function that generates the S-divergence. The only difficulty is posed by the triangle inequality. Let X, Y, Z > 0 be arbitrary. From Lemma 3.9 we know that there is a matrix P such that P XP = I and P Y P = D. Since Z > 0 is arbitrary, and congruence preserves positive definiteness, we may write just Z instead of P ZP. Also, since δ S (P XP, P Y P ) = δ S (X, Y ) (see Prop..), proving the triangle inequality reduces to showing that (3.8) δ S (I, D) δ S (I, Z) + δ S (D, Z).

5 POSITIVE DEFINITE MATRICES AND THE S-DIVERGENCE 5 Consider now the diagonal matrices D and Eig (Z). Corollary 3.6 asserts (3.9) δ S (I, D ) δ S (I, Eig (Z)) + δ S (D, Eig (Z)). Prop..(i) implies that δ S (I, D) = δ S (I, D ) and δ S (I, Z) = δ S (I, Eig (Z)), while Corollary 3.8 shows that δ S (D, Eig (Z)) δ S (D, Z). Combining these inequalities, we immediately obtain (3.8). We now turn our attention to a related connection that enjoys importance in some applications: kernel functions arising from δ S. 3.. Hilbert space embedding. Since δ S is a metric, which for scalars embeds isometrically into Hilbert space (Lemma 3.4), it is natural to ask whether δ S (X, Y ) also admits such an embedding. But as we already noted, such an embedding does not exist. Theorem 3.3 implies that a Hilbert space embedding exists if and only if δs (X, Y ) is a negative definite kernel; equivalently, iff the map (cf. Lemma 3.4) e βδ S (X,Y ) = det(x)β det(y ) β det((x + Y )/) β, is a positive definite kernel for β > 0. It suffices to check whether the matrix (3.0) H β = [h ij ] = [ det(x i + X j ) β], i, j m, is positive definite for every m and arbitrary positive matrices X,..., X m P n. Unfortunately, a quick numerical experiment reveals that H β can be indefinite. This leads us to the weaker question: for what choices of β is H β 0? Theorem 3.0 answers this question for H β formed from symmetric real positive definite matrices. Theorem 3.0. Let X,..., X m be real symmetric matrices in P n. The m m matrix H β defined by (3.0) is positive definite, if and only if β satisfies (3.) β { j : j N, and j (n )} { γ : γ R, and γ > (n )}. Proof. Please refer to the longer version of this paper []. Theorem 3.0 shows that e βδ S is not always a kernel, while for commuting matrices e βδ S is always a positive definite kernel. This raises the following: Open problem. Determine necessary and sufficient conditions on a set X P n, so that e βδ S (X,Y ) is a kernel function on X X for all β > Geometric and analytic similarities with δ R 4.. Geometric mean. We begin by studying an object that connects δ R and δ S most intimately: the matrix geometric mean. For positive matrices A and B, the matrix geometric mean (MGM) is denoted by A B, and is given by the formula (4.) A B := A / (A / BA / ) / A /. The MGM (4.) has a host of attractive properties see for instance the classic paper []. The following variational characterization is important [7]: (4.) A B = argmin X>0 δ R(A, X) + δ R(B, X), and δ R (A, A B) = δ R (B, A B). Surprisingly, the MGM enjoys a similar characterization even under δ S.

6 6 SUVRIT SRA Theorem 4.. Let A, B > 0. Then, (4.3) A B = argmin X>0 [ h(x) := δ S (X, A) + δ S(X, B) ]. Moreover, A B is equidistant from A and B, i.e., δ S (A, A B) = δ S (B, A B). Proof. If A = B, then clearly X = A minimizes h(x). Assume therefore, that A B. Ignoring the constraint X > 0 for the moment, we see that any stationary point of h(x) must satisfy h(x) = 0. This condition translates into h(x) = ( X+A ) + ( X+B ) X = 0 = B = XA X. The last equation is a Riccati equation whose unique positive solution is X = A B [6, Prop..3]. We now show that the stationary point A B is actually a local minimum. Consider the Hessian h(x) = X X [ (X + A) (X + A) + (X + B) (X + B) ]. Writing P = (X + A), Q = (X + B), and using h(x) = 0 we obtain h(x) = (Q P ) + (P Q) > 0. Thus, X = A B is a strict local minimum of h(x). This local minimum is the global minimum as h(x) = 0 has a unique positive solution and h goes to + at the boundary. Equidistance follows easily from A B = B A and Prop Geodesic convexity. The above derivation concludes optimality of the MGM from first principles. In this section, we show that δs is actually jointly geodesically convex, hereafter g-convex, (see (4.5)), a property also satisfied by δ R. Before proving Theorem 4.4, we recall two results; the first implies the second 4. Theorem 4. ([6]). The GM of A, B P n is given by the variational formula A B = max { X H n [ ] } A X 0. Proposition 4.3 (Joint-concavity (see e.g. [6])). Let A, B, C, D > 0. Then, (4.4) (A B) + (C D) (A + C) (B + D). Theorem 4.4. The function δs (X, Y ) is jointly g-convex for X, Y > 0. Proof. Since δ S is continuous, it suffices to show that for X, X, Y, Y > 0, (4.5) δ S(X X, Y Y ) δ S(X, Y ) + δ S(X, Y ). From Prop. 4.3 it follows that X X + Y Y (X + Y ) (X + Y ). Since log det is monotonic and determinants are multiplicative, it then follows that ( ) ( ) log det X X +Y Y log det (X+Y ) (X +Y ), which when combined with the identity log det( (X X )(Y Y ) ) = 4 log det(x Y ) 4 log det(x Y ) yields inequality (4.5), establishing joint g-convexity Basic contraction results. In this section we show that δ S and δ R share several contraction properties. We state our results either in terms of δ S or of δ S, depending on whichever appears more elegant. 4 It is a minor curiosity to note that the mixed-mean inequality for matrix geometric and arithmetic means proved in [9, Thm. ] is a special case of Prop X B

7 POSITIVE DEFINITE MATRICES AND THE S-DIVERGENCE Power-contraction. The metric δ R satisfies (e.g., [6, Exercise 6.5.4]) (4.6) δ R (A t, B t ) tδ R (A, B), for A, B > 0 and t [0, ]. Theorem 4.5 shows that S-Divergence satisfies the same relation. Theorem 4.5. Let A, B > 0, and let t [0, ]. Then, (4.7) δ S(A t, B t ) tδ S(A, B). Moreover, if t, then the inequality gets reversed. Proof. Recall that for t [0, ], the map X X t is operator concave. Thus, (At + B t ) ( ) A+B t; by monotonicity of the determinant it then follows that δ S(A t, B t ) = log det ( (At + B t ) ) det(a t B t ) / log det ( (A + B)) t = tδ det(ab) S(A, B). t/ The reverse inequality for t, follows by considering δ S (A/t, B /t ) Contraction on geodesics. The curve (4.8) γ(t) := A / (A / BA / ) t A /, for t [0, ], parameterizes the unique geodesic between the positive matrices A and B on the manifold (P n, δ R ) [6, Thm. 6..6]. On this curve δ R satisfies δ R (A, γ(t)) = tδ R (A, B), t [0, ]. The S-Divergence satisfies a similar, albeit slightly weaker result. Theorem 4.6. Let A, B > 0, and γ(t) be defined by (4.8). Then, (4.9) δ S(A, γ(t)) tδ S(A, B), 0 t. Proof. The proof follows upon observing that δ S(A, γ(t)) = δ S(I, (A / BA / ) t ) (4.7) tδ S(I, A / BA / ) = tδ S(A, B) A power-monotonicity property. We show below that on matrix powers, δ S and δ R exhibit a similar monotonicity property reminiscent of a power-means inequality. Theorem 4.7. Let A, B > 0. Let scalars t and u satisfy t u <. Then, (4.0) (4.) t δ R (A t, B t ) u δ R (A u, B u ) t δ S(A t, B t ) u δ S(A u, B u ). To our knowledge, inequality (4.0) is also new. Before proving Theorem 4.7 we first state a power-means determinantal inequality (which follows from the monotonicity theorem of [4] on power means; see [] for an independent proof). Proposition 4.8. Let A, B > 0; let scalars t, u satisfy t u <. Then, (4.) det /t( ) A t +B t det /u( ) A u +B u. Proof (Theorem 4.7). (i): Note that δ R (X, Y ) = log E (XY ) F. We must show t log E (A t B t ) F u log E (A u B u ) F. Equivalently, for vectors of eigenvalues we may prove (4.3) log λ /t (A t B t ) log λ /u (A u B u ).

8 8 SUVRIT SRA The log-majorization relation stated in [5, Theorem IX..9] says log λ /t (A t B t ) log λ /u (A u B u ), to which we apply the map x x immediately obtaining (4.3). Notice, that we have in fact proved the more general result t log E (A t B t ) Φ u log E (A u B u ) Φ, where Φ is a symmetric gauge function (a permutation invariant absolute norm). (ii): To prove (4.) we must show that t log det( (A t +B t )/ ) t log det(at B t ) u log det( (A u +B u )/ ) u log det(au B u ). But this inequality is immediate from Prop. 4.8 and the monotonicity of log Contraction under translation. The last basic contraction result that we prove is an analogue of the following shrinkage property [9, Prop..6]: (4.4) δ R (A + X, A + Y ) α α+β δ R(X, Y ), for A 0, and X, Y > 0, where α = max { X, Y } and β = λ min (A). This result plays a crucial role in deriving contractive maps for certain nonlinear matrix equations [7]. Theorem 4.9. Let X, Y > 0, and A 0, then the function (4.5) g(a) := δ S(A + X, A + Y ), is monotonically decreasing and convex in A. Proof. We must show that if A B, then g(a) g(b). Equivalently, we show that the gradient A g(a) 0, which follows easily since ( ) A g(a) = (A+X)+(A+Y ) (A + X) (A + Y ) 0, as the map X X is operator convex. It remains to prove convexity of g. Consider therefore its Hessian g(a). Let P = (A + X), Q = (B + X), so ( ) ( ) g(a) = (P P + Q Q) P +Q P +Q. Using matrix convexity of X X we obtain g(a) P +Q (P P + Q Q) P +Q = (P Q) (P Q), which is positive definite since by assumption P Q. Corollary 4.0. Let X, Y > 0, A 0, β = λ min (A). Then, (4.6) δ S(A + X, A + Y ) δ S(βI + X, βi + Y ) δ S(X, Y ) Conic contraction. We establish below (Theorem 4.4) a compression property for the S-Divergence, which it shares with the well-known Hilbert and Thompson metrics on convex cones [8, Ch.]. We start with some preparatory results. Proposition 4.. Let P C n k (k n) have full column rank. The function f : P n R X log det(p XP ) log det(x) is operator decreasing.

9 POSITIVE DEFINITE MATRICES AND THE S-DIVERGENCE 9 Proof. It suffices to show that f(x) 0. This amounts to establishing that [ ] (4.7) P (P XP ) P X X P P P 0. XP Inequality (4.7) follows once we note the factorization [ ] [ ] [ ] [ ] X P I 0 X I I 0 P P = XP 0 P. I X 0 P Corollary 4.. Let X, Y > 0. Let A = ( ) X+Y, G = X Y ; and let P C n k (k n) have full column rank. Then, det(p AP ) (4.8) det(p GP ) det(a) det(g). Proof. Since A G, it follows from Prop. 4. that log det(p AP ) log det(a) log det(p GP ) log det(g). Rearranging, and using the fact that P AP P GP, we obtain (4.8). Theorem 4.3 ([, Thm. 3]). Let Π : P n P k be a positive linear map. Then, (4.9) Π(A B) Π(A) Π(B) for A, B P n. We are now ready to prove the main theorem of this section. Theorem 4.4. Let P C n k (k n) have full column rank. Then, (4.0) δ S(P AP, P BP ) δ S(A, B) for A, B P n. Proof. We may equivalently show that (4.) ( P ) (A+B)P det ( ) A+B det det(p. AP ) det(p BP ) det(ab) But Prop. 4.3 shows P (A B)P (P AP ) (P BP ), which implies that = det(p AP ) det(p BP ) det[(p AP ) (P BP )] det(p (A B)P ). Consequently, an invocation of Corollary 4. concludes the argument. Theorem 4.4 relates δ S to the classical Hilbert and Thompson metrics on convex cones, which also satisfy similar results (for the wider class of order-preserving subhomogenous maps [8]); this explains the name conic contraction. Theorem 4.4 also extends to δ R, as noted in Corollary 4.6, which itself follows from Theorem 4.5 (we believe that this theorem must exist in the literature see [] for our proof). Theorem 4.5. If A, B P n, and P C n k (k n) has full column rank. Then, (4.) λ j (P AP (P BP ) ) λ j (AB ) for j k. Corollary 4.6. Let P C n k (k n) have full column rank. Then, (4.3) δ Φ (P AP, P BP ) δ Φ (A, B) := log(b / AB / ) Φ, where Φ is any symmetric gauge function. We conclude by noting a bi-lipschitz-like inequality between δ S and δ R. Theorem 4.7 ([]). Let A, B P n. Let δ T (A, B) = log(b / AB / ) denote the Thompson-part metric [8]. Then, we have the following bounds (4.4) 8δ S(A, B) δ R(A, B) δ T (A, B) ( δ S(A, B) + n log ).

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