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Frame potentials and the geometry of frames. (English) Zbl 1342.42029

Authors’ abstract: “This paper concerns the geometric structure of optimizers for frame potentials. We consider finite, real or complex frames and rotation or unitarily invariant potentials, and mostly specialize to Parseval frames, meaning the frame potential to be optimized is a function on the manifold of Gram matrices belonging to finite Parseval frames. Next to the known classes of equal-norm and equiangular Parseval frames, we introduce equidistributed Parseval frames, which are more general than the equiangular type but have more structure than equal-norm ones. We also provide examples where this class coincides with that of Grassmannian frames, the minimizers for the maximal magnitude among inner products between frame vectors. These different types of frames are characterized in relation to the optimization of frame potentials. Based on results by Łojasiewicz, we show that the gradient descent for a real analytic frame potential on the manifold of Gram matrices belonging to Parseval frames always converges to a critical point. We then derive geometric structures associated with the critical points of different choices of frame potentials. The optimal frames for families of such potentials are thus shown to be equal-norm, or additionally equipartitioned, or even equidistributed.”
Central for the authors’ main result is the development of a four parameter family of frame potentials, \(\{\Phi^{\alpha,\beta,\delta,\eta}=\Phi_{\mathrm{sum}}^{\eta}+\Phi^{\delta}_{\mathrm{diag}}+\Phi_{\mathrm{ch}}^{\alpha,\beta}\}_{\alpha,\beta,\delta,\eta\in(0,\infty)}\) where the sum potential \(\Phi_{\mathrm{sum}}^{\eta}(G),\) the diagonal potential \( \Phi^{\delta}_{\mathrm{diag}}(g)\), and the chain potential \(\Phi_{\mathrm{ch}}^{\alpha,\beta}(G)\) are defined on the real analytic submanifold of the \(N\times N\) Hermitians, which consists of the Gram matrices of Parseval frames for \(\mathbb{F}^K,\mathbb{F}=\mathbb{R}\) or \(\mathbb{C}\). Each of those potentials is nonnegative and at most quadratic in the elementary one parameter potential function \(E_{x,y}^{\gamma}(G)=e^{\gamma|G_{x,y}|^2}\). Numerical implementations of the optimization problem developed by the authors lead to concrete examples of frames with a special structure.

MSC:

42C15 General harmonic expansions, frames
53B21 Methods of local Riemannian geometry
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