Log-Determinant Divergences Between Positive De nite Hilbert-Schmidt Operators

07/11/2017
Auteurs : Hà Quang Minh
Publication GSI2017
OAI : oai:www.see.asso.fr:17410:22586
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Résumé

The current work generalizes the author's previous work on the infinite-dimensional Alpha Log-Determinant (Log-Det) divergences and Alpha-Beta Log-Det divergences, de ned on the set of positive definite unitized trace class operators on a Hilbert space, to the entire Hilbert manifold of positive de nite unitized Hilbert-Schmidt operators. 
This generalization is carried out via the introduction of the extended Hilbert-Carleman determinant for unitized Hilbert-Schmidt operators, in addition to the previously introduced extended Fredholm determinant for unitized trace class operators. The resulting parametrized family of Alpha-Beta Log-Det divergences is general and contains many divergences between positive de nite unitized Hilbert-Schmidt operators as special cases, including the infinite-dimensional generalizations of the affine-invariant Riemannian distance and symmetric Stein divergence.

Log-Determinant Divergences Between Positive Denite Hilbert-Schmidt Operators

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Log-Determinant Divergences Between Positive Denite Hilbert-Schmidt Operators
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This generalization is carried out via the introduction of the extended Hilbert-Carleman determinant for unitized Hilbert-Schmidt operators, in addition to the previously introduced extended Fredholm determinant for unitized trace class operators. The resulting parametrized family of Alpha-Beta Log-Det divergences is general and contains many divergences between positive de nite unitized Hilbert-Schmidt operators as special cases, including the infinite-dimensional generalizations of the affine-invariant Riemannian distance and symmetric Stein divergence.
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Log-Determinant Divergences Between Positive Definite Hilbert-Schmidt Operators Hà Quang Minh Pattern Analysis and Computer Vision (PAVIS) Istituto Italiano di Tecnologia (IIT), Genova, Italy minh.haquang@iit.it Abstract. The current work generalizes the author’s previous work on the infinite-dimensional Alpha Log-Determinant (Log-Det) divergences and Alpha-Beta Log-Det divergences, defined on the set of positive def- inite unitized trace class operators on a Hilbert space, to the entire Hilbert manifold of positive definite unitized Hilbert-Schmidt operators. This generalization is carried out via the introduction of the extended Hilbert-Carleman determinant for unitized Hilbert-Schmidt operators, in addition to the previously introduced extended Fredholm determi- nant for unitized trace class operators. The resulting parametrized family of Alpha-Beta Log-Det divergences is general and contains many diver- gences between positive definite unitized Hilbert-Schmidt operators as special cases, including the infinite-dimensional generalizations of the affine-invariant Riemannian distance and symmetric Stein divergence. 1 Introduction The current work is a continuation and generalization of the author’s previous work [9], [7], which generalizes the finite-dimensional Log-Determinant diver- gences to the infinite-dimensional setting. We recall that for the convex cone Sym++ (n) of symmetric, positive definite (SPD) matrices of size n×n, n ∈ N, the Alpha-Beta Log-Determinant (Log-Det) divergence between A, B ∈ Sym++ (n) is a parametrized family of divergences defined by (see [3]) D(α,β) (A, B) = 1 αβ log det  α(AB−1 )β + β(AB−1 )−α α + β  , α > 0, β > 0, (1) along with the limiting cases (α > 0, β = 0), (α = 0, β > 0), and (α = 0, β = 0). This family contains many distance-like functions on Sym++ (n), including 1. The affine-invariant Riemannian distance daiE [1], corresponding to D(0,0) (A, B) = 1 2 d2 aiE(A, B) = 1 2 || log(B−1/2 AB−1/2 )||2 F , (2) where log(A) denotes the principal logarithm of A and || ||F denotes the Frobenius norm, with ||A||F = p tr(A∗A). This is the geodesic distance associated with the affine-invariant Riemannian metric [10, 6, 1, 11]. 2. The Alpha Log-Det divergences [2], corresponding to D(α,1−α) (A, B), with D(α,1−α) (A, B) = 1 α(1 − α) log  det[αA + (1 − α)B] det(A)α det(B)1−α  , 0 < α < 1, (3) D(1,0) (A, B) = tr(A−1 B − I) − log det(A−1 B), (4) D(0,1) (A, B) = tr(B−1 A − I) − log det(B−1 A). (5) The case α = 1/2 gives the symmetric Stein divergence (also called the Jensen-Bregman LogDet divergence), whose square root is a metric on Sym++ (n) [13], with D(1/2,1/2) (A, B) = 4d2 stein(A, B) = 4[log det(A+B 2 )−1 2 log det(AB)]. Previous work. In [9], we generalized the Alpha Log-Det divergences be- tween SPD matrices [2] to the infinite-dimensional Alpha Log-Determinant di- vergences between positive definite unitized trace class operators on an infinite- dimensional Hilbert space. This is done via the introduction of the extended Fredholm determinant for unitized trace class operators, along with the corre- sponding generalization of the log-concavity of the determinant for SPD matri- ces to the infinite-dimensional setting. In [7], we present a formulation for the Alpha-Beta Log-Det divergences between positive definite unitized trace class operators, generalizing the Alpha-Beta Log-Det divergences between SPD ma- trices as defined by Eq.(1). Contributions of this work. The current work is a continuation and gen- eralization of [9] and [7]. In particular, we generalize the Alpha-Beta Log-Det divergences in [7] to the entire Hilbert manifold of positive definite unitized Hilbert-Schmidt operators on an infinite-dimensional Hilbert space. This is done by the introduction of the extended Hilbert-Carleman determinant for unitized Hilbert-Schmidt operators, in addition to the extended Fredholm determinant for unitized trace class operators employed in [9] and [7]. As in the finite-dimensional setting [3] and in [9], [7], the resulting family of divergences is general and ad- mits as special cases many metrics and distance-like functions between positive definite unitized Hilbert-Schmidt operators, including the infinite-dimensional affine-invariant Riemannian distance in [5].The proofs for all theorems stated in this paper, along with many other results, are given in the arXiv preprint [8]. 2 Positive Definite Unitized Trace Class and Hilbert-Schmidt Operators Throughout the paper, we assume that H is a real separable Hilbert space, with dim(H) = ∞, unless stated otherwise. Let L(H) be the Banach space of bounded linear operators on H. Let Sym++ (H) ⊂ L(H) be the set of bounded, self- adjoint, strictly positive operators on H, that is A ∈ Sym++ (H) ⇐⇒ hx, Axi > 0 ∀x ∈ H, x 6= 0. Most importantly, we consider the set P(H) ⊂ Sym++ (H) of self- adjoint, bounded, positive definite operators on H, which is defined by A ∈ P(H) ⇐⇒ A = A∗ , ∃MA > 0 such that hx, Axi ≥ MA||x||2 ∀x ∈ H. We use the notation A > 0 ⇐⇒ A ∈ P(H). In the following, let Cp(H) denote the set of pth Schatten class operators on H (see e.g. [4]), under the norm || ||p, 1 ≤ p ≤ ∞, which is defined by Cp(H) = {A ∈ L(H) : ||A||p = (tr|A|p )1/p < ∞}, (6) where |A| = (A∗ A)1/2 . The cases we consider in this work are: (i) the space C1(H) of trace class operators on H, also denoted by Tr(H), and (ii) the space C2(H) of Hilbert-Schmidt operators on H, also denoted by HS(H). Extended (unitized) trace class operators. In [9], we define the set of extended (or unitized) trace class operators on H to be TrX(H) = {A + γI : A ∈ Tr(H), γ ∈ R}. (7) The set TrX(H) becomes a Banach algebra under the extended trace class norm ||A + γI||trX = ||A||tr + |γ| = tr|A| + |γ|. For (A + γI) ∈ TrX(H), its extended trace is defined to be trX(A + γI) = tr(A) + γ, with trX(I) = 1. (8) Extended (unitized) Hilbert-Schmidt operators. In [5], the author con- sidered the following set of extended (unitized) Hilbert-Schmidt operators HSX(H) = {A + γI : A ∈ HS(H), γ ∈ R}. (9) The set HSX(H) can be equipped with the extended Hilbert-Schmidt inner prod- uct h , ieHS, defined by hA + γI, B + µIieHS = hA, BiHS + γµ = tr(A∗ B) + γµ. (10) along with the associated extended Hilbert-Schmidt norm ||A + γI||2 eHS = ||A||2 HS + γ2 = tr(A∗ A) + γ2 , with ||I||eHS = 1. (11) Positive definite unitized trace class and Hilbert-Schmidt oper- ators. The set of positive definite unitized trace class operators PC 1(H) ⊂ TrX(H) is defined to be the intersection PC 1(H) = TrX(H) ∩ P(H) = {A + γI > 0 : A∗ = A, A ∈ Tr(H), γ ∈ R}. (12) The set of positive definite unitized Hilbert-Schmidt operators PC 2(H) ⊂ HSX(H) is defined to be the intersection PC 2(H) = HSX(H) ∩ P(H) = {A + γI > 0 : A = A∗ , A ∈ HS(H), γ ∈ R}. (13) We remark that in [9] and [7], we use the notations PTr(H) and Σ(H) to denote PC 1(H) and PC 2(H), respectively. In the following, we refer to elements of PC 1(H) and PC 2(H) as positive definite trace class operators and positive definite Hilbert-Schmidt operators, respectively. In [5], it is shown that the set PC 2(H) assumes the structure of an infinite- dimensional Hilbert manifold and can be equipped with the following Rieman- nian metric. For each P ∈ PC 2(H), on the tangent space TP (PC 2(H)) ∼