SCIENCE CHINA Information Sciences, Volume 63 , Issue 5 : 152201(2020) https://doi.org/10.1007/s11432-019-9911-5

Application of gradient descent algorithms based on geodesic distances

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  • ReceivedApr 20, 2019
  • AcceptedMay 30, 2019
  • PublishedMar 26, 2020



Xiaomin DUAN was supported by National Science and Technology Major Project of China (Grant No. 2016YFF02030012), National Natural Science Foundation of China (Grant No. 61401058) and Natural Science Foundation of Liaoning Province (Grant No. 20180550112). Huafei SUN was partially supported by National Natural Science Foundation of China (Grant No. 61179031). Linyu PENG was supported by JSPS Grant-in-Aid for Scientific Research (Grant No. 16KT0024), the MEXT “Top Global University Project", Waseda University Grant for Special Research Projects (Grant Nos. 2019C–179, 2019E–036), and Waseda University Grant Program for Promotion of International Joint Research.


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  • Figure 1

    (Color online) Positive definite Hermitian matrix system.

  • Figure 11

    (Color online) Comparison of two algorithms.


    Algorithm 1 Riemannian gradient descent algorithm for positive definite Hermitian matrix systems

    1. Set $u_0=(u_0^1,~u_0^2,\ldots~,u_0^m)$ as an initial input. Choose a fixed learning rate $\eta$ for simplicity and a desired tolerance $\varepsilon>0$.

    2. At time $k$, calculate $A(u_k)$ using Theorem 1 and $d({A(u_k)},B)$.

    3. If $d({A(u_k)},B)<\varepsilon$ then stop. Otherwise, move to step $4$.

    4. Make $k$ plus one, and then run step $2$.


    Algorithm 2 Natural gradient algorithm for positive definite Hermitian matrix systems

    1. Set $u_0=(u_0^1,~u_0^2,\ldots~,u_0^m)$ as an initial input. Choose a fixed learning rate $\eta$ and a desired tolerance $\varepsilon>0$.

    2. At time $k$, calculate $u_k$ using Theorem 2 and ${\nabla}J(u_k)$.

    3. If $\|{\nabla}J(u_k)\|_F<\varepsilon$, stop. Otherwise, move to step $4$.

    4. Make $k$ plus one, and then run step $2$.


    Algorithm 3 Natural gradient algorithm for the Karcher mean of $N$ matrices on manifold ${\rm~Sym}(n,\mathbb{C})$

    1. Take the arithmetic mean $\frac{1}{N}\sum_{i=1}^NR^i$ as the initial point $\theta_0$. Choose a learning rate $\eta$ and a desired tolerance $\varepsilon>0$.

    2. At time $k$, calculate $\theta_k$ using Theorem 3 and ${\nabla}L(\theta_k)$.

    3. If $\|{\nabla}L(\theta_k)\|<\varepsilon$, stop. Otherwise, move to step $4$.

    4. Make $k$ plus one, and then run step $2$.