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Convex Analysis on Hadamard Spaces and Scaling Problems

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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.

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Notes

  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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Acknowledgements

The author thanks Hiroyuki Ochiai for helpful discussion and thanks for Zhiyuan Zhan for corrections. The author also thanks Harold Nieuwboer and Michael Walter for discussion on the Legendre–Fenchel duality. The work was partially supported by JST PRESTO Grant No. JPMJPR192A, Japan.

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  1. Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan

    Hiroshi Hirai

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Hirai, H. Convex Analysis on Hadamard Spaces and Scaling Problems. Found Comput Math (2023). https://doi.org/10.1007/s10208-023-09628-5

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