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Foundations of Computational Mathematics
Erschienen in:

17.10.2023

Convex Analysis on Hadamard Spaces and Scaling Problems

verfasst von: Hiroshi Hirai

Erschienen in: Foundations of Computational Mathematics | Ausgabe 6/2024

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Abstract

In this paper, we address the bounded/unbounded determination of geodesically convex optimization on Hadamard spaces. In Euclidean convex optimization, the recession function is a basic tool to study the unboundedness and provides the domain of the Legendre–Fenchel conjugate of the objective function. In a Hadamard space, the asymptotic slope function (Kapovich et al. in J Differ Geom 81:297–354, 2009), which is a function on the boundary at infinity, plays a role of the recession function. We extend this notion by means of convex analysis and optimization and develop a convex analysis foundation for the unbounded determination of geodesically convex optimization on Hadamard spaces, particularly on symmetric spaces of nonpositive curvature. We explain how our developed theory is applied to operator scaling and related optimization on group orbits, which are our motivation.
Vorheriger Artikel A Construction of Conforming Finite Element Spaces in Any Dimension
Nächster Artikel Approximation of Deterministic Mean Field Games with Control-Affine Dynamics

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Fußnoten
1
To see the consistency with his formulation, use the relation \(b_{\lambda \cdot \mathcal{E}}(gg^{\dagger }) = - \log \det (\mathop {\textrm{diag}}\lambda , g^{\dagger }g)\) for any upper triangular matrix g, where \(\det (\mathop {\textrm{diag}}\lambda , g^{\dagger }g)\) is the relative determinant in the sense of [19].
 
2
From \(\frac{\textrm{d}}{{\textrm{d}}t} \mid _{t=0} \langle \pi (e^{tH_0})u, \pi (e^{tH_0})v\rangle =0\) for \(H_0 \in \mathfrak {u}\), we have \(\langle \Pi (H_0)u,v\rangle + \langle u, \Pi (H_0)v\rangle =0\). Thus, \(\Pi (H_0)^{\dagger } = - \Pi (H_0)\). For \(H = H_0+i H_1\) with \(H_0,H_1 \in \mathfrak {u}\), we have \(\Pi (H)^\dagger = \Pi (H_0)^{\dagger } - i \Pi (H_1)^{\dagger } = - \Pi (H_0) + i \Pi (H_1) = \Pi (-H_0+ i H_1) = \Pi (H^{\dagger })\).
 
3
By \(\pi (k^{\dagger }) = \pi (k^{-1}) = \pi (k)^{-1} = \pi (k)^\dagger \) for \(k \in K\) and polar decomposition \(g= k e^{iH}\) for \(k \in K\) and \(H \in \mathfrak {u}\), we have \(\pi (g^{\dagger }) = e^{- i \Pi (H^{\dagger })} \pi (k^{\dagger }) = e^{-i\Pi (H)^{\dagger }} \pi (k)^{\dagger } = (\pi (k)e^{i\Pi (H)})^{\dagger } = \pi (g)^{\dagger }\).
 
4
This fact can be seen from Proposition 2.33 and the fact that the moment map \(\mu : \pi (G)v {\setminus } \{0\} \rightarrow i \mathfrak {u} (=T_I)\) in [15] is written as \(\mu (\pi (g)v) = g^\dagger df_v(gg^\dagger ) g\).
 
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Metadaten
Titel
Convex Analysis on Hadamard Spaces and Scaling Problems
verfasst von
Hiroshi Hirai
Publikationsdatum
17.10.2023
Verlag
Springer US
Erschienen in
Foundations of Computational Mathematics / Ausgabe 6/2024
Print ISSN: 1615-3375
Elektronische ISSN: 1615-3383
DOI
https://doi.org/10.1007/s10208-023-09628-5

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