L0-regularized compressed sensing with Mean-field Coherent Ising Machines

TL;DR

The results indicate that the proposed mean-field CIM (MF-CIM) model has similar performance to physically accurate SDEs in both artificial and magnetic resonance imaging data, paving the way for implementing CIM-based L0RBCS on digital hardware such as Field Programmable Gate Arrays.

Abstract

Coherent Ising Machine (CIM) is a network of optical parametric oscillators that solves combinatorial optimization problems by finding the ground state of an Ising Hamiltonian. As a practical application of CIM, Aonishi et al. proposed a quantum-classical hybrid system to solve optimization problems of L0-regularization-based compressed sensing (L0RBCS). Gunathilaka et al. has further enhanced the accuracy of the system. However, the computationally expensive CIM's stochastic differential equations (SDEs) limit the use of digital hardware implementations. As an alternative to Gunathilaka et al.'s CIM SDEs used previously, we propose using the mean-field CIM (MF-CIM) model, which is a physics-inspired heuristic solver without quantum noise. MF-CIM surmounts the high computational cost due to the simple nature of the differential equations (DEs). Furthermore, our results indicate that the proposed model has similar performance to physically accurate SDEs in both artificial and magnetic resonance imaging data, paving the way for implementing CIM-based L0RBCS on digital hardware such as Field Programmable Gate Arrays (FPGAs).

Figures (7)

• Figure 1: Amplitude evolution of CAC-CIM (Positive-$P$) and CAC-MFZs with continuous and binarized local fields.(a), (b) and (c) indicates the measured amplitude $\tilde{\mu}_r$ of CAC-CIM (Positive-$P$), amplitude $c_r$ evolution of CAC-MFZ with continuous and binarized local fields introduced in eq. (\ref{['localfieldMFZ']}) and eq. (\ref{['localfieldMFZBN']}) respectively. A slightly more chaotic behavior can be seen in the continuous case. The indicated amplitudes for each model come from the second alternative minimization process in Algorithm \ref{['algo']} under the same problem instance. The system size was set as $N = 2000$ while the compression and the sparseness were 0.6 and 0.2.
• Figure 2: CAC-MFZ-CDP's average RMSE compared to the theoretical limit of L0RBCS, when observation noise exists.(a) and (b) indicates the average performance for $N = 2000$ system where $\alpha = 0.6$ and $\alpha = 0.8$ respectively for $\nu = 0.05$. (c) and (d) states the average performance for $\nu = 0.1$. $\eta_{init} = 0.8$ was used for CAC-CIM-CDP and CAC-MFZ-CDP. (a) and (b)$\eta_{end}$ was set to 0.18 for all models. (c) and (d)$\eta_{end}$ was set to 0.35 for all models.
• Figure 3: Average performance of the models for different-size MRI data when L0-regularization parameter varies(a) Performance on $64\times64$. (b) Performance on $128\times128$. The black line indicates LASSO performance. On the blue box plot, CAC-CIM-CDP (Positive-$P$) results are shown, and on the red and green box plots, CAC-MFZ-CDP results with binarized and continuous local fields are shown. Each box plot illustrates the maximum, minimum, 25th percentile (bottom edge), 75th percentile (top edge), and median (central horizontal line) of RMSEs for each model at different threshold values. We display the maximum, minimum, and median of RMSE for CAC-MFZ-CDP. There are markers indicating outliers. The compression and sparseness for (a) were 0.4 and 0.212 respectively while for (b) were 0.3 and 0.178.
• Figure 4: Reconstructed Images for MRI data with $64\times64$.(a) Initial $64\times 64$ image. Sparseness and compression ratios were 0.212 and 0.4 respectively. (b) k-space data (grey points) and random undersampled points (red points). (c), (d), (e) and (f) correspond to the reconstructions obtained from LASSO, CAC-CIM-CDP (Positive-$P$) and CAC-MFZ-CDPs with continuous and binarized local fields (BN and CN) where RMSE values are 0.0292, 0.0182, 0.0176 and 0.0168 respectively. Pixel-wise differences between the reconstructions are depicted in the enlarged portions of the images. For (d), (e) and (f) a total of 11 alternating minimization processes were performed. And for (c), (d), (e) and (f)$\eta_{init} = \eta_{end}$ was 0.0003, 0.022, 0.025 and 0.022 respectively.
• Figure 5: Performance comparison between continuous and binarized local fields at each alternative minimization step.(a) Log average RMSE values for 10 random datasets for each alternative minimization step. CAC-CIM-CDP (Positive-$P$), CAC-MFZ-CDP (BN) and CAC-MFZ-CDP (CN) results are indicated in black, blue and red boxplots respectively. (b) Log average Hamming loss values for 10 random datasets for each alternative minimization step which calculates the average support estimation accuracy.