Abstract
In this paper, we consider deep unfolding the standard iterative conjugate gradient (CG) algorithm to solve a linear system of equations. Instead of being adjusted with known rules, the parameters are learned via backpropagation to yield the optimal results. However, the proposed unfolded CG (UCG) is extended wherein a scalar parameter is substituted by a matrix-parameter to augment the degrees of freedom per layer. Once the training is completed, the UCG has revealed to require far a smaller number of layers than the number of iterations needed using the standard iterative CG. It is also shown to be very robust to noise and outperforms the standard CG in low signal to noise ratio (SNR) region. A key merit of the proposed approach is the fact that no explicit training data is dedicated to the learning phase as the optimization process relies on the residual error which is not explicitly expressed as a function of the desired data. As an example, the proposed UCG is applied to solve the reciprocity calibration problem encountered in massive MIMO (Multiple-Input Multiple-Output) systems.
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Acknowledgments
This work has been funded by the Natural Sciences and Engineering Research Council of Canada, Prompt and NUTAQ Innovation, and the CMC Microsystems for equipments. This work was supported by the Chaire de recherche sur les signaux et l’intelligence de systems haute performance for technical support.
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Appendix 1
Appendix 1
The reader must refer to the dependence tree shown in Fig. 1 and to Eqs. (19), (20) and (21) to understand the following derivations. The explicit forms of the gradient of the loss function e with respect to αk and λkare
where \( \overline{\frac{\partial \left(\cdot \right)}{\partial \left(\cdot \right)}} \) indicates the total derivative and \( \overline{\frac{\partial {r}_{n+1}^{(z)}}{\partial {\lambda}_k}} \) represents the total derivative of the zth element of rn + 1 with respect to λk where k ∈ [1 : n] for Eq. (A.1) and k ∈ [1 : n − 1] for Eq. (A.2). The formulas for the total derivative of rn + 1 with respect to αk and λk use
where the total derivative of rn + 1 with respect to αn corresponds to the only possible path on Fig. 1 that goes from rn + 1 to αn. Before going further, it is important to notice that the allowed paths are those following the same direction as the arrows in Fig. 1. Therefore, we have
where the chain rule is applied between the total derivative of rn + 1 with respect to rk + 1, which depends on every possible path on Fig. 1 between rn + 1 and rk + 1, and the straight unique path derivative between rk + 1 and αk with k ∈ [1 : n − 1]. In a similar way, we derive
where the two indices in parenthesis next to every term represent the row and the column of the resulting matrix with i, j, z ∈ [1 : 2M] and k ∈ [1 : n − 1]. The left-hand side of Eq. (A.5) shows that there is a distinct derivative for each element in the vector rn + 1 with respect to each (i, j) element in the λk matrix. The reason why the form of Eq. (A.5) is slightly different from the one of Eq. (A.4) is due to the fact that λk is a matrix instead of being a scalar. We then continue with the total derivative of rn + 1 with respect to pn
which is simply the direct path in Fig. 1 between rn + 1 and pn. Things get slightly more complicated for the general form of the derivative of rn + 1 with respect to pk with k ∈ [2 : n − 1]
where the chain rule is first applied between the total derivative of rn + 1 with respect to pk + 1, which depends on every possible path on Fig. 1 between these two nodes, and the straight path derivative between pk + 1 and pk. In a similar way, the chain rule is applied between the total derivative of rn + 1 with respect to pk + 1 and the straight paths derivatives going through pk + 1, sk, rk + 1 and pk respectively. Finally, the same process is done between the total derivative of rn + 1 with respect to rk + 1 and the unique straight path derivative between rk + 1 and pk. At that point, the only unknown we are left with is the total derivative of rn + 1 with respect to any other r. We begin with
where the first possible path is the direct one between rn + 1 and rn and the second possible path is the one that goes through rn + 1, pn, sn − 1 and rn respectively. We then continue with
where the three possible paths between rn + 1 and rn − 1 are considered. Finally, we have the general case for k ∈ [2 : n − 2]
where all possible paths going from rn + 1 to rkare selected. By differentiating Eqs. (19), (20) and (21), every symbolic derivative can be replaced by its true value
where I is a 2M × 2M identity matrix and \( \frac{\partial {\mathbf{p}}_{k+1}}{\partial {\boldsymbol{\uplambda}}_k} \) is a matrix with the ith column having a constant value corresponding to the ith element of pk.
To get better and faster results during the backpropagation process, the adaptation step for αk and λk can be set to different values and a momentum technique is used on λk:
where μ1 and μ2 are the fixed adaptation step, β is the momentum coefficient between 0 and 1 and t represents the current iteration step (epoch) in the backpropagation process.
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Sirois, S., Ahmed Ouameur, M. & Massicotte, D. Deep Unfolded Extended Conjugate Gradient Method for Massive MIMO Processing with Application to Reciprocity Calibration. J Sign Process Syst 93, 965–975 (2021). https://doi.org/10.1007/s11265-020-01631-1
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DOI: https://doi.org/10.1007/s11265-020-01631-1