Abstract
In this paper, we propose a new splitting algorithm for dynamical low-rank approximation motivated by the fibre bundle structure of the set of fixed rank matrices. We first introduce a geometric description of the set of fixed rank matrices which relies on a natural parametrization of matrices. More precisely, it is endowed with the structure of analytic principal bundle, with an explicit description of local charts. For matrix differential equations, we introduce a first order numerical integrator working in local coordinates. The resulting algorithm can be interpreted as a particular splitting of the projection operator onto the tangent space of the low-rank matrix manifold. It is proven to be exact in some particular case. Numerical experiments confirm this result and illustrate the robustness of the proposed algorithm.
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Notes
We have been aware of this reference while revising the present paper.
For any \(A \in \mathbb {R}^{n \times m}\), the Moore–Penrose pseudo inverse is given by \(A = (A^TA)^{-1} A^T.\)
Here \(\mathbb {G}_r(\mathbb {R}^p) = \{V_r \subset \mathbb {R}^p : \dim (V_r)=r\}\) denotes the Grassmann manifold.
This means the tangent space to the local parameter space \(\mathbb {R}^{(n-r)\times r}\times \mathbb {R}^{(m-r)\times r} \times \mathbb {R}^{r\times r}\) at (0, 0, G).
For that variant, it means that UG, \(VG^T\) and then G are updated.
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Acknowledgements
This research was funded by the RTI2018-093521-B-C32 grant from the Ministerio de 263 Ciencia, Innovación y Universidades and by the grant number INDI20/13 from Universidad CEU 264 Cardenal Herrera.
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A chart based splitting integrator
A chart based splitting integrator
Following the same lines as in [15], we justify how the chart based method introduced in Sect. 3.1.2 can be interpreted as a splitting scheme relying on the projection decomposition (3.4) as the sum of three contributions \(P_{T_Z(t)}= P_1+ P_2 + P_3 \). One integration step of the splitting method starting from \(t_0\) to \(t_1\) with initial guess \(Z(t_0) = U(t_0)G(t_0)V(t_0)^T\) proceeds as follows.
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(S1)
Find \(Z \in \mathcal{U}_{Z(t_0)}\) on \([t_0,t_1]\) such that \(\dot{Z} = P_{U} F(Z) P_{V}^T\) with initial condition \(Z(t_0)\).
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(S2)
Find \(Z \in \mathcal{U}_{Z(t_0)}\) on \([t_0,t_1]\) such that \(\dot{Z} = P_{U}^\perp F(Z) P_{V}^T\) with initial condition given by final condition of step (S1).
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(S3)
Find \(Z \in \mathcal{U}_{Z(t_0)}\) on \([t_0,t_1]\) such that \(\dot{Z} = P_{U} F(Z) (P^\perp _{V})^T \) with initial condition given by final condition of step (S2).
At each step (Si) of the splitting, Z belongs to the neighborhood of \(Z(t_0)\). Thus it is given by \(Z(t) = U(t)H(t)V(t)^T\) with \(U(t) = U(t_0)+U(t_0)_\perp X(t)\), \(Y(t)= V(t_0)+V(t_0)_\perp Y(t)\) provided by the ODE solved at Step i of the chart based splitting, as stated in the following proposition.
Proposition 5.1
The solution of (S1) is given by Z with
with \(H(t_0)=G(t_0)\), \(X(t_0)=0\) and \(Y(t_0)=0\). Set
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Letting \(H_1\) be the final condition of H from (S1), the solution of (S2) is given by Z with
$$\begin{aligned} \dot{X} H = {U}_\perp ^+ F (Z)(V^+)^T, \quad \dot{Y}= 0,\quad \dot{H}=0, \end{aligned}$$(A.2)with \(H(t_0)=H_1\), \(X(t_0)=0\) and \(Y(t_0)=0\). Set \(X_1 = X(t_1)\).
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Letting \(X_1\) be the final condition of X from (S2), the solution of (S3) is given by Z with
$$\begin{aligned} \dot{Y} H^T = {V}_\perp ^+ F (Z) (U^+)^T, \quad \dot{X}= 0,\quad \dot{H}=0, \end{aligned}$$(A.3)with \(H(t_0)=H_1\), \(X(t_0)=X_1\) and \(Y(t_0)=0\).
Proof
For each step (Si), Z admits the decomposition
with derivative
For (S1), the derivative satisfies \(\dot{Z} = P_{U} F(Z) P_{V}^T\). Then, multiplying on the left by \(U(t_0)^+\) and on the right by \((V(t_0)^+)^T\) the matrix \(\dot{Z}\) in both expressions leads to \(\dot{H}= U^+ F(Z) (V^+)^T\) and \(\dot{X} =0, \dot{Y} = 0.\) Now let us turn to (S2). The derivative satisfies \(\dot{Z} = P_{U}^\perp F(Z) P_{V}^T\). By multiplying on the right by \((V(t_0)^+)^T\), the equality is satisfied if \(\dot{X} H = U_\perp ^+ F(Z) (V^+)^T\) and \( \dot{Y}= 0, \dot{H} =0.\) The third point of the lemma is obtained from (S3) in the same manner, by multiplying the equation \(\dot{Z} = P_{U} F(Z) (P_{V}^\perp )^T\) on the left by \((U(t_0)+U(t_0)_\perp X_1)^+\) and setting \(\dot{X} =0\), \(\dot{H}=0\). \(\square \)
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Billaud-Friess, M., Falcó, A. & Nouy, A. A new splitting algorithm for dynamical low-rank approximation motivated by the fibre bundle structure of matrix manifolds. Bit Numer Math 62, 387–408 (2022). https://doi.org/10.1007/s10543-021-00884-x
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DOI: https://doi.org/10.1007/s10543-021-00884-x