Least-squares approximation by elements from matrix orbits achieved by gradient flows on compact lie groups

Li, Chi-Kwong; Poon, Yiu‐Tung; Schulte‐Herbrüggen, Thomas

doi:10.1090/s0025-5718-2010-02450-0

Cited by 7 publications

(6 citation statements)

References 38 publications

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“…Similarly, the study of local minimizers of the matrix nearness problem corresponds to the study of local minimizers of Θ (N , S , a) . It is worth pointing out that for the Frobenius norm, local minimizers of the matrix nearness problem arise naturally as stability points of (effective) gradient descent algorithms, as those considered in [16]. Hence, settling Conjecture 4.2 in the affirmative would be a relevant result from an applied point of view.…”

Section: Generalized Frame Operator Distancesmentioning

confidence: 99%

Local Lidskii's theorems for unitarily invariant norms

Massey

Rios

Stojanoff

2018

Linear Algebra and its Applications

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Lidskii's additive inequalities (both for eigenvalues and singular values) can be interpreted as an explicit description of global minimizers of functions that are built on unitarily invariant norms, with domains consisting of certain orbits of matrices (under the action of the unitary group). In this paper, we show that Lidskii's inequalities actually describe all global minimizers of such functions and that local minimizers are also global minimizers. We use these results to obtain partial results related to local minimizers of generalized frame operator distances in the context of finite frame theory.

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Section: Generalized Frame Operator Distancesmentioning

confidence: 99%

Local Lidskii's theorems for unitarily invariant norms

Massey

Rios

Stojanoff

2018

Linear Algebra and its Applications

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“…While some choices of complexity functionals lead to solutions in closed form, we must usually resort to numerical optimization. For compact Lie groups represented on finite dimensional Hilbert spaces there exist some provably powerful gradient flow methods as described in [33,34] but the non-compact and (at least formally) infinite dimensional case that is often of interest to us is more complicated and less well understood. Nonetheless, for specific examples of groups and parametrizations we are able to prove the convexity of some linear and even nonlinear expectation minimization schemes which allows us to employ existing schemes such as generalized Newton's method [35] for obtaining optimal coordinates.…”

Section: Iterative Complexity Reductionmentioning

confidence: 99%

Low-dimensional manifolds for exact representation of open quantum systems

2017

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Weakly nonlinear degrees of freedom in dissipative quantum systems tend to localize near manifolds of quasi-classical states. We present a family of analytical and computational methods for deriving optimal unitary model transformations that reduce the complexity of representing typical states. These transformations minimize the quantum relative entropy distance between a given state and particular quasi-classical manifolds. This naturally splits the description of quantum states into transformation coordinates that specify the nearest quasi-classical state and a transformed quantum state that can be represented in fewer basis levels. We derive coupled equations of motion for the coordinates and the transformed state and demonstrate how this can be exploited for efficient numerical simulation. Our optimization objective naturally quantifies the non-classicality of states occurring in some given open system dynamics. This allows us to compare the intrinsic complexity of different open quantum systems.

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“…Such an approach was considered by N. Strawn [22,23] for the Frobenius norm N . Also, we could study the evolution of solutions of gradient flows as considered in [16].…”

Section: Introductionmentioning

confidence: 99%

Generalized frame operator distance problems

Massey

Rios

Stojanoff

2019

Journal of Mathematical Analysis and Applications

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Let S ∈ M d (C) + be a positive semidefinite d × d complex matrix and let a = (a i ) i∈I k ∈ R k >0 , indexed by I k = {1, . . . , k}, be a k-tuple of positive numbers. Let T d (a) denote the set of familiesis the product of spheres in C d endowed with the product metric. For a strictly convex unitarily invariant norm N in M d (C), we consider the generalized frame operator distance function Θ (N , S , a) defined on T d (a), given byIn this paper we determine the geometrical and spectral structure of local minimizers G 0 ∈ T d (a) of Θ (N , S , a) . In particular, we show that local minimizers are global minimizers, and that these families do not depend on the particular choice of N .

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Least-squares approximation by elements from matrix orbits achieved by gradient flows on compact lie groups

Cited by 7 publications

References 38 publications

Local Lidskii's theorems for unitarily invariant norms

Local Lidskii's theorems for unitarily invariant norms

Low-dimensional manifolds for exact representation of open quantum systems

Generalized frame operator distance problems

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